Vector products
You are encouraged to solve this task according to the task description, using any language you may know.
A vector is defined as having three dimensions as being represented by an ordered collection of three numbers: (X, Y, Z).
If you imagine a graph with the x and y axis being at right angles to each other and having a third, z axis coming out of the page, then a triplet of numbers, (X, Y, Z) would represent a point in the region, and a vector from the origin to the point.
Given the vectors:
A = (a1, a2, a3)
B = (b1, b2, b3)
C = (c1, c2, c3)
then the following common vector products are defined:
- The dot product (a scalar quantity)
- A • B = a1b1 + a2b2 + a3b3
- The cross product (a vector quantity)
- A x B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
- The scalar triple product (a scalar quantity)
- A • (B x C)
- The vector triple product (a vector quantity)
- A x (B x C)
- Task
Given the three vectors:
a = ( 3, 4, 5)
b = ( 4, 3, 5)
c = (-5, -12, -13)
- Create a named function/subroutine/method to compute the dot product of two vectors.
- Create a function to compute the cross product of two vectors.
- Optionally create a function to compute the scalar triple product of three vectors.
- Optionally create a function to compute the vector triple product of three vectors.
- Compute and display:
a • b - Compute and display:
a x b - Compute and display:
a • (b x c), the scalar triple product. - Compute and display:
a x (b x c), the vector triple product.
- References
- A starting page on Wolfram MathWorld is Vector Multiplication .
- Wikipedia dot product.
- Wikipedia cross product.
- Wikipedia triple product.
- Related tasks
11l
F scalartriplep(a, b, c)
return dot(a, cross(b, c))
F vectortriplep(a, b, c)
return cross(a, cross(b, c))
V a = (3, 4, 5)
V b = (4, 3, 5)
V c = (-5, -12, -13)
print(‘a = #.; b = #.; c = #.’.format(a, b, c))
print(‘a . b = #.’.format(dot(a, b)))
print(‘a x b = #.’.format(cross(a,b)))
print(‘a . (b x c) = #.’.format(scalartriplep(a, b, c)))
print(‘a x (b x c) = #.’.format(vectortriplep(a, b, c)))- Output:
a = (3, 4, 5); b = (4, 3, 5); c = (-5, -12, -13) a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
Action!
TYPE Vector=[INT x,y,z]
PROC CreateVector(INT vx,vy,vz Vector POINTER v)
v.x=vx v.y=vy v.z=vz
RETURN
PROC PrintVector(Vector POINTER v)
PrintF("(%I,%I,%I)",v.x,v.y,v.z)
RETURN
INT FUNC DotProduct(Vector POINTER v1,v2)
INT res
res=v1.x*v2.x ;calculation split into parts
res==+v1.y*v2.y ;otherwise incorrect result
res==+v1.z*v2.z ;is returned
RETURN (res)
PROC CrossProduct(Vector POINTER v1,v2,res)
res.x=v1.y*v2.z ;calculation split into parts
res.x==-v1.z*v2.y ;otherwise incorrect result
res.y=v1.z*v2.x ;is returned
res.y==-v1.x*v2.z
res.z=v1.x*v2.y
res.z==-v1.y*v2.x
RETURN
PROC Main()
Vector a,b,c,d,e
INT res
CreateVector(3,4,5,a)
CreateVector(4,3,5,b)
CreateVector(-5,-12,-13,c)
Print("a=") PrintVector(a) PutE()
Print("b=") PrintVector(b) PutE()
Print("c=") PrintVector(c) PutE()
PutE()
res=DotProduct(a,b)
PrintF("a.b=%I%E",res)
CrossProduct(a,b,d)
Print("axb=") PrintVector(d) PutE()
CrossProduct(b,c,d)
res=DotProduct(a,d)
PrintF("a.(bxc)=%I%E",res)
CrossProduct(b,c,d)
CrossProduct(a,d,e)
Print("ax(bxc)=") PrintVector(e) PutE()
RETURN- Output:
Screenshot from Atari 8-bit computer
a=(3,4,5) b=(4,3,5) c=(-5,-12,-13) a.b=49 axb=(5,5,-7) a.(bxc)=6 ax(bxc)=(-267,204,-3)
Ada
not using Ada.Numerics.Real_Arrays, to show some features of the language.
Ada determines which function to call not only on the types of the parameters, but also on the return type. That way we can use the same name for all multiplications (scalar and cross). But, if we add another one to stretch the vector, we get an ambiguity error, since the compiler can't know if A*(B*C) with result-type Vector is meant to be A stretched by the scalar product of B and C, or the cross product of A and the result of the cross product of B and C. Here, I used type qualification to tell the compiler that the result of (B*C) is of type Vector.
vector.adb:
with Ada.Text_IO;
procedure Vector is
type Float_Vector is array (Positive range <>) of Float;
package Float_IO is new Ada.Text_IO.Float_IO (Float);
procedure Vector_Put (X : Float_Vector) is
begin
Ada.Text_IO.Put ("(");
for I in X'Range loop
Float_IO.Put (X (I), Aft => 1, Exp => 0);
if I /= X'Last then
Ada.Text_IO.Put (", ");
end if;
end loop;
Ada.Text_IO.Put (")");
end Vector_Put;
-- cross product
function "*" (Left, Right : Float_Vector) return Float_Vector is
begin
if Left'Length /= Right'Length then
raise Constraint_Error with "vectors of different size in dot product";
end if;
if Left'Length /= 3 then
raise Constraint_Error with "dot product only implemented for R**3";
end if;
return Float_Vector'(Left (Left'First + 1) * Right (Right'First + 2) -
Left (Left'First + 2) * Right (Right'First + 1),
Left (Left'First + 2) * Right (Right'First) -
Left (Left'First) * Right (Right'First + 2),
Left (Left'First) * Right (Right'First + 1) -
Left (Left'First + 1) * Right (Right'First));
end "*";
-- scalar product
function "*" (Left, Right : Float_Vector) return Float is
Result : Float := 0.0;
I, J : Positive;
begin
if Left'Length /= Right'Length then
raise Constraint_Error with "vectors of different size in scalar product";
end if;
I := Left'First; J := Right'First;
while I <= Left'Last and then J <= Right'Last loop
Result := Result + Left (I) * Right (J);
I := I + 1; J := J + 1;
end loop;
return Result;
end "*";
-- stretching
function "*" (Left : Float_Vector; Right : Float) return Float_Vector is
Result : Float_Vector (Left'Range);
begin
for I in Left'Range loop
Result (I) := Left (I) * Right;
end loop;
return Result;
end "*";
A : constant Float_Vector := (3.0, 4.0, 5.0);
B : constant Float_Vector := (4.0, 3.0, 5.0);
C : constant Float_Vector := (-5.0, -12.0, -13.0);
begin
Ada.Text_IO.Put ("A: "); Vector_Put (A); Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("B: "); Vector_Put (B); Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("C: "); Vector_Put (C); Ada.Text_IO.New_Line;
Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("A dot B = "); Float_IO.Put (A * B, Aft => 1, Exp => 0);
Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("A x B = "); Vector_Put (A * B);
Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("A dot (B x C) = "); Float_IO.Put (A * (B * C), Aft => 1, Exp => 0);
Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("A x (B x C) = "); Vector_Put (A * Float_Vector'(B * C));
Ada.Text_IO.New_Line;
end Vector;
Output:
A: ( 3.0, 4.0, 5.0) B: ( 4.0, 3.0, 5.0) C: (-5.0, -12.0, -13.0) A dot B = 49.0 A x B = ( 5.0, 5.0, -7.0) A dot (B x C) = 6.0 A x (B x C) = (-267.0, 204.0, -3.0)
ALGOL 68
Note: This specimen retains the original Python coding style.
MODE FIELD = INT;
FORMAT field fmt = $g(-0)$;
MODE VEC = [3]FIELD;
FORMAT vec fmt = $"("f(field fmt)", "f(field fmt)", "f(field fmt)")"$;
PROC crossp = (VEC a, b)VEC:(
#Cross product of two 3D vectors#
CO ASSERT(LWB a = LWB b AND UPB a = UPB b AND UPB b = 3 # "For 3D vectors only" #); CO
(a[2]*b[3] - a[3]*b[2], a[3]*b[1] - a[1]*b[3], a[1]*b[2] - a[2]*b[1])
);
PRIO MAXLWB = 8, MINUPB=8;
OP MAXLWB = (VEC a, b)INT: (LWB a<LWB b|LWB a|LWB b);
OP MINUPB = (VEC a, b)INT: (UPB a>UPB b|UPB a|UPB b);
PROC dotp = (VEC a, b)FIELD:(
#Dot product of two vectors#
FIELD sum := 0;
FOR i FROM a MAXLWB b TO a MINUPB b DO sum +:= a[i]*b[i] OD;
sum
);
PROC scalartriplep = (VEC a, b, c)VEC:(
#Scalar triple product of three vectors: "a . (b x c)"#
dotp(a, crossp(b, c))
);
PROC vectortriplep = (VEC a, b, c)VEC:(
#Vector triple product of three vectors: "a x (b x c)"#
crossp(a, crossp(b, c))
);
# Declare some useful operators #
PRIO DOT = 5, X = 5;
OP (VEC, VEC)FIELD DOT = dotp;
OP (VEC, VEC)VEC X = crossp;
main:(
VEC a=(3, 4, 5), b=(4, 3, 5), c=(-5, -12, -13);
printf(($"a = "f(vec fmt)"; b = "f(vec fmt)"; c = "f(vec fmt)l$ , a, b, c));
printf($"Using PROCedures:"l$);
printf(($"a . b = "f(field fmt)l$, dotp(a,b)));
printf(($"a x b = "f(vec fmt)l$, crossp(a,b)));
printf(($"a . (b x c) = "f(field fmt)l$, scalartriplep(a, b, c)));
printf(($"a x (b x c) = "f(vec fmt)l$, vectortriplep(a, b, c)));
printf($"Using OPerators:"l$);
printf(($"a . b = "f(field fmt)l$, a DOT b));
printf(($"a x b = "f(vec fmt)l$, a X b));
printf(($"a . (b x c) = "f(field fmt)l$, a DOT (b X c)));
printf(($"a x (b x c) = "f(vec fmt)l$, a X (b X c)))
)Output:
a = (3, 4, 5); b = (4, 3, 5); c = (-5, -12, -13) Using PROCedures: a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3) Using OPerators: a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
ALGOL W
begin
% define the Vector record type %
record Vector( integer X, Y, Z );
% calculates the dot product of two Vectors %
integer procedure dotProduct( reference(Vector) value A, B ) ;
( X(A) * X(B) ) + ( Y(A) * Y(B) ) + ( Z(A) * Z(B) );
% calculates the cross product or two Vectors %
reference(Vector) procedure crossProduct( reference(Vector) value A, B ) ;
Vector( ( Y(A) * Z(B) ) - ( Z(A) * Y(B) )
, ( Z(A) * X(B) ) - ( X(A) * Z(B) )
, ( X(A) * Y(B) ) - ( Y(A) * X(B) )
);
% calculates the scaler triple product of two vectors %
integer procedure scalerTripleProduct( reference(Vector) value A, B, C ) ;
dotProduct( A, crossProduct( B, C ) );
% calculates the vector triple product of two vectors %
reference(Vector) procedure vectorTripleProduct( reference(Vector) value A, B, C ) ;
crossProduct( A, crossProduct( B, C ) );
% test the Vector routines %
begin
procedure writeonVector( reference(Vector) value v ) ;
writeon( "(", X(v), ", ", Y(v), ", ", Z(v), ")" );
Reference(Vector) a, b, c;
a := Vector( 3, 4, 5 );
b := Vector( 4, 3, 5 );
c := Vector( -5, -12, -13 );
i_w := 1; s_w := 0; % set output formatting %
write( " a: " ); writeonVector( a );
write( " b: " ); writeonVector( b );
write( " c: " ); writeonVector( c );
write( " a . b: ", dotProduct( a, b ) );
write( " a x b: " ); writeonVector( crossProduct( a, b ) );
write( "a . ( b x c ): ", scalerTripleProduct( a, b, c ) );
write( "a x ( b x c ): " ); writeonVector( vectorTripleProduct( a, b, c ) )
end
end.- Output:
a: (3, 4, 5)
b: (4, 3, 5)
c: (-5, -12, -13)
a . b: 49
a x b: (5, 5, -7)
a . ( b x c ): 6
a x ( b x c ): (-267, 204, -3)
APL
dot ← +.×
cross ← 1⌽(⊣×1⌽⊢)-⊢×1⌽⊣
- Output:
a←3 4 5
b←4 3 5
c←¯5 ¯12 ¯13
a dot b
49
a cross b
5 5 ¯7
a dot b cross c
6
a cross b cross c
¯267 204 ¯3
AppleScript
--------------------- VECTOR PRODUCTS ---------------------
-- dotProduct :: Num a => [a] -> [a] -> Either String a
on dotProduct(xs, ys)
-- Dot product of two vectors of equal dimension.
if length of xs = length of ys then
|Right|(sum(zipWith(my mul, xs, ys)))
else
|Left|("Dot product not defined for vectors of differing dimension.")
end if
end dotProduct
-- crossProduct :: Num a => (a, a, a) -> (a, a, a)
-- Either String -> (a, a, a)
on crossProduct(xs, ys)
-- The cross product of two 3D vectors.
if 3 ≠ length of xs or 3 ≠ length of ys then
|Left|("Cross product is defined only for 3d vectors.")
else
set {x1, x2, x3} to xs
set {y1, y2, y3} to ys
|Right|({¬
x2 * y3 - x3 * y2, ¬
x3 * y1 - x1 * y3, ¬
x1 * y2 - x2 * y1})
end if
end crossProduct
-- scalarTriple :: Num a => (a, a, a) -> (a, a, a) -> (a, a a) ->
-- Either String -> a
on scalarTriple(q, r, s)
-- The scalar triple product.
script go
on |λ|(ys)
dotProduct(q, ys)
end |λ|
end script
bindLR(crossProduct(r, s), go)
end scalarTriple
-- vectorTriple :: Num a => (a, a, a) -> (a, a, a) -> (a, a a) ->
-- Either String -> (a, a, a)
on vectorTriple(q, r, s)
-- The vector triple product.
script go
on |λ|(ys)
crossProduct(q, ys)
end |λ|
end script
bindLR(crossProduct(r, s), go)
end vectorTriple
-------------------------- TEST ---------------------------
on run
set a to {3, 4, 5}
set b to {4, 3, 5}
set c to {-5, -12, -13}
set d to {3, 4, 5, 6}
script test
on |λ|(f)
either(my identity, my show, ¬
mReturn(f)'s |λ|(a, b, c, d))
end |λ|
end script
tell test
unlines({¬
"a . b = " & |λ|(dotProduct), ¬
"a x b = " & |λ|(crossProduct), ¬
"a . (b x c) = " & |λ|(scalarTriple), ¬
"a x (b x c) = " & |λ|(vectorTriple), ¬
"a x d = " & either(my identity, my show, ¬
dotProduct(a, d)), ¬
"a . (b x d) = " & either(my identity, my show, ¬
scalarTriple(a, b, d)) ¬
})
end tell
end run
-------------------- GENERIC FUNCTIONS --------------------
-- Left :: a -> Either a b
on |Left|(x)
{type:"Either", |Left|:x, |Right|:missing value}
end |Left|
-- Right :: b -> Either a b
on |Right|(x)
{type:"Either", |Left|:missing value, |Right|:x}
end |Right|
-- bindLR (>>=) :: Either a -> (a -> Either b) -> Either b
on bindLR(m, mf)
if missing value is not |Left| of m then
m
else
mReturn(mf)'s |λ|(|Right| of m)
end if
end bindLR
-- either :: (a -> c) -> (b -> c) -> Either a b -> c
on either(lf, rf, e)
if missing value is |Left| of e then
tell mReturn(rf) to |λ|(|Right| of e)
else
tell mReturn(lf) to |λ|(|Left| of e)
end if
end either
-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- identity :: a -> a
on identity(x)
-- The argument unchanged.
x
end identity
-- intercalate :: String -> [String] -> String
on intercalate(delim, xs)
set {dlm, my text item delimiters} to ¬
{my text item delimiters, delim}
set str to xs as text
set my text item delimiters to dlm
str
end intercalate
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
-- The list obtained by applying f
-- to each element of xs.
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- min :: Ord a => a -> a -> a
on min(x, y)
if y < x then
y
else
x
end if
end min
-- mul :: Num a :: a -> a -> a
on mul(x, y)
x * y
end mul
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- show :: a -> String
on show(x)
if list is class of x then
showList(x)
else
str(x)
end if
end show
-- showList :: [a] -> String
on showList(xs)
"[" & intercalate(", ", map(my str, xs)) & "]"
end showList
-- str :: a -> String
on str(x)
x as string
end str
-- sum :: [Number] -> Number
on sum(xs)
script add
on |λ|(a, b)
a + b
end |λ|
end script
foldl(add, 0, xs)
end sum
-- unlines :: [String] -> String
on unlines(xs)
-- A single string formed by the intercalation
-- of a list of strings with the newline character.
set {dlm, my text item delimiters} to ¬
{my text item delimiters, linefeed}
set s to xs as text
set my text item delimiters to dlm
s
end unlines
-- zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
on zipWith(f, xs, ys)
set lng to min(length of xs, length of ys)
set lst to {}
tell mReturn(f)
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, item i of ys)
end repeat
return lst
end tell
end zipWith
- Output:
a . b = 49 a x b = [5, 5, -7] a . (b x c) = 6 a x (b x c) = [-267, 204, -3] a x d = Dot product not defined for vectors of differing dimension. a . (b x d) = Cross product is defined only for 3d vectors
Arturo
; dot product
dot: function [a b][
sum map couple a b => product
]
; cross product
cross: function [a b][
A: (a\1 * b\2) - a\2 * b\1
B: (a\2 * b\0) - a\0 * b\2
C: (a\0 * b\1) - a\1 * b\0
@[A B C]
]
; scalar triple product
stp: function [a b c][
dot a cross b c
]
; vector triple product
vtp: function [a b c][
cross a cross b c
]
; task
a: [3 4 5]
b: [4 3 5]
c: @[neg 5 neg 12 neg 13]
print ["a • b =", dot a b]
print ["a x b =", cross a b]
print ["a • (b x c) =", stp a b c]
print ["a x (b x c) =", vtp a b c]
- Output:
a • b = 49 a x b = [5 5 -7] a • (b x c) = 6 a x (b x c) = [-267 204 -3]
AutoHotkey
V := {a: [3, 4, 5], b: [4, 3, 5], c: [-5, -12, -13]}
for key, val in V
Out .= key " = (" val[1] ", " val[2] ", " val[3] ")`n"
CP := CrossProduct(V.a, V.b)
VTP := VectorTripleProduct(V.a, V.b, V.c)
MsgBox, % Out "`na • b = " DotProduct(V.a, V.b) "`n"
. "a x b = (" CP[1] ", " CP[2] ", " CP[3] ")`n"
. "a • b x c = " ScalerTripleProduct(V.a, V.b, V.c) "`n"
. "a x b x c = (" VTP[1] ", " VTP[2] ", " VTP[3] ")"
DotProduct(v1, v2) {
return, v1[1] * v2[1] + v1[2] * v2[2] + v1[3] * v2[3]
}
CrossProduct(v1, v2) {
return, [v1[2] * v2[3] - v1[3] * v2[2]
, v1[3] * v2[1] - v1[1] * v2[3]
, v1[1] * v2[2] - v1[2] * v2[1]]
}
ScalerTripleProduct(v1, v2, v3) {
return, DotProduct(v1, CrossProduct(v2, v3))
}
VectorTripleProduct(v1, v2, v3) {
return, CrossProduct(v1, CrossProduct(v2, v3))
}
Output:
a = (3, 4, 5) b = (4, 3, 5) c = (-5, -12, -13) a • b = 49 a x b = (5, 5, -7) a • b x c = 6 a x b x c = (-267, 204, -3)
AWK
#!/usr/bin/awk -f
BEGIN {
a[1] = 3; a[2]= 4; a[3] = 5;
b[1] = 4; b[2]= 3; b[3] = 5;
c[1] = -5; c[2]= -12; c[3] = -13;
print "a = ",printVec(a);
print "b = ",printVec(b);
print "c = ",printVec(c);
print "a.b = ",dot(a,b);
## upper case variables are used as temporary or intermediate results
cross(a,b,D);print "a.b = ",printVec(D);
cross(b,c,D);print "a.(b x c) = ",dot(a,D);
cross(b,c,D);cross(a,D,E); print "a x (b x c) = ",printVec(E);
}
function dot(A,B) {
return A[1]*B[1]+A[2]*B[2]+A[3]*B[3];
}
function cross(A,B,C) {
C[1] = A[2]*B[3]-A[3]*B[2];
C[2] = A[3]*B[1]-A[1]*B[3];
C[3] = A[1]*B[2]-A[2]*B[1];
}
function printVec(C) {
return "[ "C[1]" "C[2]" "C[3]" ]";
}
Output:
a = [ 3 4 5 ] b = [ 4 3 5 ] c = [ -5 -12 -13 ] A.b = 49 a.b = [ 5 5 -7 ] a.(b x c) = 6 a x (b x c) = [ -267 204 -3 ]
BASIC256
a={3,4,5}:b={4,3,5}:c={-5,-12,-13}
print "A.B = "+dot_product(ref(a),ref(b))
call cross_product(ref(a),ref(b),ref(y))
Print "AxB = ("+y[0]+","+y[1]+","+y[2]+")"
print "A.(BxC) = "+s_tri(ref(a),ref(b),ref(c))
call v_tri(ref(a),ref(b),ref(c),ref(x),ref(y))
Print "A x (BxC) = ("+y[0]+","+y[1]+","+y[2]+")"
function dot_product(ref(x1),ref(x2))
dot_product= 0
for t = 0 to 2
dot_product += x1[t]*x2[t]
next t
end function
subroutine cross_product(ref(x1),ref(x2),ref(y1))
y1={0,0,0}
y1[0]=x1[1]*x2[2]-x1[2]*x2[1]
y1[1]=x1[2]*x2[0]-x1[0]*x2[2]
y1[2]=x1[0]*x2[1]-x1[1]*x2[0]
end subroutine
function s_tri(ref(x1),ref(x2),ref(x3))
call cross_product(ref(x2),ref(x3),ref(y1))
s_tri=dot_product(ref(x1),ref(y1))
end function
subroutine v_tri(ref(x1),ref(x2),ref(x3),ref(y1),ref(y2))
call cross_product(ref(x2),ref(x3),ref(y1))
call cross_product(ref(x1),ref(y1),ref(y2))
end subroutineOutput:
A.B = 49 AxB = (5,5,-7) A.(BxC) = 6 A x (BxC) = (-267,204,-3)
BBC BASIC
DIM a(2), b(2), c(2), d(2)
a() = 3, 4, 5
b() = 4, 3, 5
c() = -5, -12, -13
PRINT "a . b = "; FNdot(a(),b())
PROCcross(a(),b(),d())
PRINT "a x b = (";d(0)", ";d(1)", ";d(2)")"
PRINT "a . (b x c) = "; FNscalartriple(a(),b(),c())
PROCvectortriple(a(),b(),c(),d())
PRINT "a x (b x c) = (";d(0)", ";d(1)", ";d(2)")"
END
DEF FNdot(A(),B())
LOCAL C() : DIM C(0,0)
C() = A().B()
= C(0,0)
DEF PROCcross(A(),B(),C())
C() = A(1)*B(2)-A(2)*B(1), A(2)*B(0)-A(0)*B(2), A(0)*B(1)-A(1)*B(0)
ENDPROC
DEF FNscalartriple(A(),B(),C())
LOCAL D() : DIM D(2)
PROCcross(B(),C(),D())
= FNdot(A(),D())
DEF PROCvectortriple(A(),B(),C(),D())
PROCcross(B(),C(),D())
PROCcross(A(),D(),D())
ENDPROC
Output:
a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
BQN
The cross product function here multiplies each vector pointwise by the other rotated by one. To align this result with the third index (the one not involved), it's rotated once more at the end. The APL solution 1⌽(⊣×1⌽⊢)-⊢×1⌽⊣ uses the same idea and works in BQN without modification.
Dot ← +´∘×
Cross ← 1⊸⌽⊸×{1⌽𝔽˜-𝔽}
Triple ← {𝕊a‿b‿c: a Dot b Cross c}
VTriple ← Cross´
a←3‿4‿5
b←4‿3‿5
c←¯5‿¯12‿¯13
Results:
a Dot b
49
a Cross b
⟨ 5 5 ¯7 ⟩
Triple a‿b‿c
6
VTriple a‿b‿c
⟨ ¯267 204 ¯3 ⟩
C
#include<stdio.h>
typedef struct{
float i,j,k;
}Vector;
Vector a = {3, 4, 5},b = {4, 3, 5},c = {-5, -12, -13};
float dotProduct(Vector a, Vector b)
{
return a.i*b.i+a.j*b.j+a.k*b.k;
}
Vector crossProduct(Vector a,Vector b)
{
Vector c = {a.j*b.k - a.k*b.j, a.k*b.i - a.i*b.k, a.i*b.j - a.j*b.i};
return c;
}
float scalarTripleProduct(Vector a,Vector b,Vector c)
{
return dotProduct(a,crossProduct(b,c));
}
Vector vectorTripleProduct(Vector a,Vector b,Vector c)
{
return crossProduct(a,crossProduct(b,c));
}
void printVector(Vector a)
{
printf("( %f, %f, %f)",a.i,a.j,a.k);
}
int main()
{
printf("\n a = "); printVector(a);
printf("\n b = "); printVector(b);
printf("\n c = "); printVector(c);
printf("\n a . b = %f",dotProduct(a,b));
printf("\n a x b = "); printVector(crossProduct(a,b));
printf("\n a . (b x c) = %f",scalarTripleProduct(a,b,c));
printf("\n a x (b x c) = "); printVector(vectorTripleProduct(a,b,c));
return 0;
}
Output:
a = ( 3.000000, 4.000000, 5.000000) b = ( 4.000000, 3.000000, 5.000000) c = ( -5.000000, -12.000000, -13.000000) a . b = 49.000000 a x b = ( 5.000000, 5.000000, -7.000000) a . (b x c) = 6.000000 a x (b x c) = ( -267.000000, 204.000000, -3.000000)
C#
using System;
using System.Windows.Media.Media3D;
class VectorProducts
{
static double ScalarTripleProduct(Vector3D a, Vector3D b, Vector3D c)
{
return Vector3D.DotProduct(a, Vector3D.CrossProduct(b, c));
}
static Vector3D VectorTripleProduct(Vector3D a, Vector3D b, Vector3D c)
{
return Vector3D.CrossProduct(a, Vector3D.CrossProduct(b, c));
}
static void Main()
{
var a = new Vector3D(3, 4, 5);
var b = new Vector3D(4, 3, 5);
var c = new Vector3D(-5, -12, -13);
Console.WriteLine(Vector3D.DotProduct(a, b));
Console.WriteLine(Vector3D.CrossProduct(a, b));
Console.WriteLine(ScalarTripleProduct(a, b, c));
Console.WriteLine(VectorTripleProduct(a, b, c));
}
}
Output:
49 5;5;-7 6 -267;204;-3
C++
#include <iostream>
template< class T >
class D3Vector {
template< class U >
friend std::ostream & operator<<( std::ostream & , const D3Vector<U> & ) ;
public :
D3Vector( T a , T b , T c ) {
x = a ;
y = b ;
z = c ;
}
T dotproduct ( const D3Vector & rhs ) {
T scalar = x * rhs.x + y * rhs.y + z * rhs.z ;
return scalar ;
}
D3Vector crossproduct ( const D3Vector & rhs ) {
T a = y * rhs.z - z * rhs.y ;
T b = z * rhs.x - x * rhs.z ;
T c = x * rhs.y - y * rhs.x ;
D3Vector product( a , b , c ) ;
return product ;
}
D3Vector triplevec( D3Vector & a , D3Vector & b ) {
return crossproduct ( a.crossproduct( b ) ) ;
}
T triplescal( D3Vector & a, D3Vector & b ) {
return dotproduct( a.crossproduct( b ) ) ;
}
private :
T x , y , z ;
} ;
template< class T >
std::ostream & operator<< ( std::ostream & os , const D3Vector<T> & vec ) {
os << "( " << vec.x << " , " << vec.y << " , " << vec.z << " )" ;
return os ;
}
int main( ) {
D3Vector<int> a( 3 , 4 , 5 ) , b ( 4 , 3 , 5 ) , c( -5 , -12 , -13 ) ;
std::cout << "a . b : " << a.dotproduct( b ) << "\n" ;
std::cout << "a x b : " << a.crossproduct( b ) << "\n" ;
std::cout << "a . b x c : " << a.triplescal( b , c ) << "\n" ;
std::cout << "a x b x c : " << a.triplevec( b , c ) << "\n" ;
return 0 ;
}
Output:
a . b : 49 a x b : ( 5 , 5 , -7 ) a . b x c : 6 a x b x c : ( -267 , 204 , -3 )
Ceylon
shared void run() {
alias Vector => Float[3];
function dot(Vector a, Vector b) =>
a[0] * b[0] + a[1] * b[1] + a[2] * b[2];
function cross(Vector a, Vector b) => [
a[1] * b[2] - a[2] * b[1],
a[2] * b[0] - a[0] * b[2],
a[0] * b[1] - a[1] * b[0]
];
function scalarTriple(Vector a, Vector b, Vector c) =>
dot(a, cross(b, c));
function vectorTriple(Vector a, Vector b, Vector c) =>
cross(a, cross(b, c));
value a = [ 3.0, 4.0, 5.0 ];
value b = [ 4.0, 3.0, 5.0 ];
value c = [-5.0, -12.0, -13.0 ];
print("``a`` . ``b`` = ``dot(a, b)``");
print("``a`` X ``b`` = ``cross(a, b)``");
print("``a`` . ``b`` X ``c`` = ``scalarTriple(a, b, c)``");
print("``a`` X ``b`` X ``c`` = ``vectorTriple(a, b, c)``");
}
- Output:
[3.0, 4.0, 5.0] . [4.0, 3.0, 5.0] = 49.0 [3.0, 4.0, 5.0] X [4.0, 3.0, 5.0] = [5.0, 5.0, -7.0] [3.0, 4.0, 5.0] . [4.0, 3.0, 5.0] X [-5.0, -12.0, -13.0] = 6.0 [3.0, 4.0, 5.0] X [4.0, 3.0, 5.0] X [-5.0, -12.0, -13.0] = [-267.0, 204.0, -3.0]
Clojure
(defrecord Vector [x y z])
(defn dot
[U V]
(+ (* (:x U) (:x V))
(* (:y U) (:y V))
(* (:z U) (:z V))))
(defn cross
[U V]
(new Vector
(- (* (:y U) (:z V)) (* (:z U) (:y V)))
(- (* (:z U) (:x V)) (* (:x U) (:z V)))
(- (* (:x U) (:y V)) (* (:y U) (:x V)))))
(let [a (new Vector 3 4 5)
b (new Vector 4 3 5)
c (new Vector -5 -12 -13)]
(doseq
[prod (list
(dot a b)
(cross a b)
(dot a (cross b c))
(cross a (cross b c)))]
(println prod)))
Output:
49
#:user.Vector{:x 5, :y 5, :z -7}
6
#:user.Vector{:x -267, :y 204, :z -3}
CLU
vector = cluster [T: type] is make, dot_product, cross_product,
equal, power, mul, unparse
where T has add: proctype (T,T) returns (T) signals (overflow),
sub: proctype (T,T) returns (T) signals (overflow),
mul: proctype (T,T) returns (T) signals (overflow),
equal: proctype (T,T) returns (bool),
unparse: proctype (T) returns (string)
rep = struct[x, y, z: T]
make = proc (x, y, z: T) returns (cvt)
return(rep${x:x, y:y, z:z})
end make
dot_product = proc (a, b: cvt) returns (T) signals (overflow)
return (a.x*b.x + a.y*b.y + a.z*b.z) resignal overflow
end dot_product
cross_product = proc (a, b: cvt) returns (cvt) signals (overflow)
begin
x: T := a.y * b.z - a.z * b.y
y: T := a.z * b.x - a.x * b.z
z: T := a.x * b.y - a.y * b.x
return(down(make(x,y,z)))
end resignal overflow
end cross_product
equal = proc (a, b: cvt) returns (bool)
return (a.x = b.x & a.y = b.y & a.z = b.z)
end equal
% Allow cross_product to be written as ** and dot_product to be written as *
power = proc (a, b: cvt) returns (cvt) signals (overflow)
return(down(cross_product(up(a),up(b)))) resignal overflow
end power
mul = proc (a, b: cvt) returns (T) signals (overflow)
return(dot_product(up(a),up(b))) resignal overflow
end mul
% Standard to_string routine. Properly, `parse' should also be defined,
% and x = parse(unparse(x)) forall x; but I'm not bothering here.
unparse = proc (v: cvt) returns (string)
return( "(" || T$unparse(v.x)
|| ", " || T$unparse(v.y)
|| ", " || T$unparse(v.z) || ")" )
end unparse
end vector
start_up = proc ()
vi = vector[int] % integer math is good enough for the examples
po: stream := stream$primary_output()
a, b, c: vi
a := vi$make(3, 4, 5)
b := vi$make(4, 3, 5)
c := vi$make(-5, -12, -13)
stream$putl(po, " a = " || vi$unparse(a))
stream$putl(po, " b = " || vi$unparse(b))
stream$putl(po, " c = " || vi$unparse(c))
stream$putl(po, " a . b = " || int$unparse(a * b))
stream$putl(po, " a x b = " || vi$unparse(a ** b))
stream$putl(po, "a . (b x c) = " || int$unparse(a * b ** c))
stream$putl(po, "a x (b x c) = " || vi$unparse(a ** b ** c))
end start_up- Output:
a = (3, 4, 5)
b = (4, 3, 5)
c = (-5, -12, -13)
a . b = 49
a x b = (5, 5, -7)
a . (b x c) = 6
a x (b x c) = (-267, 204, -3)
Common Lisp
Using the Common Lisp Object System.
(defclass 3d-vector ()
((x :type number :initarg :x)
(y :type number :initarg :y)
(z :type number :initarg :z)))
(defmethod print-object ((object 3d-vector) stream)
(print-unreadable-object (object stream :type t)
(with-slots (x y z) object
(format stream "~a ~a ~a" x y z))))
(defun make-3d-vector (x y z)
(make-instance '3d-vector :x x :y y :z z))
(defmethod dot-product ((a 3d-vector) (b 3d-vector))
(with-slots ((a1 x) (a2 y) (a3 z)) a
(with-slots ((b1 x) (b2 y) (b3 z)) b
(+ (* a1 b1) (* a2 b2) (* a3 b3)))))
(defmethod cross-product ((a 3d-vector)
(b 3d-vector))
(with-slots ((a1 x) (a2 y) (a3 z)) a
(with-slots ((b1 x) (b2 y) (b3 z)) b
(make-instance '3d-vector
:x (- (* a2 b3) (* a3 b2))
:y (- (* a3 b1) (* a1 b3))
:z (- (* a1 b2) (* a2 b1))))))
(defmethod scalar-triple-product ((a 3d-vector)
(b 3d-vector)
(c 3d-vector))
(dot-product a (cross-product b c)))
(defmethod vector-triple-product ((a 3d-vector)
(b 3d-vector)
(c 3d-vector))
(cross-product a (cross-product b c)))
(defun vector-products-example ()
(let ((a (make-3d-vector 3 4 5))
(b (make-3d-vector 4 3 5))
(c (make-3d-vector -5 -12 -13)))
(values (dot-product a b)
(cross-product a b)
(scalar-triple-product a b c)
(vector-triple-product a b c))))
Output:
CL-USER> (vector-products-example) 49 #<3D-VECTOR 5 5 -7> 6 #<3D-VECTOR -267 204 -3>
Using vector type
(defun cross (a b)
(when (and (equal (length a) 3) (equal (length b) 3))
(vector
(- (* (elt a 1) (elt b 2)) (* (elt a 2) (elt b 1)))
(- (* (elt a 2) (elt b 0)) (* (elt a 0) (elt b 2)))
(- (* (elt a 0) (elt b 1)) (* (elt a 1) (elt b 0))))))
(defun dot (a b)
(when (equal (length a) (length b))
(loop for ai across a for bi across b sum (* ai bi))))
(defun scalar-triple (a b c)
(dot a (cross b c)))
(defun vector-triple (a b c)
(cross a (cross b c)))
(defun task (a b c)
(values (dot a b)
(cross a b)
(scalar-triple a b c)
(vector-triple a b c)))
Output:
CL-USER> (task (vector 3 4 5) (vector 4 3 5) (vector -5 -12 -13)) 49 #(5 5 -7) 6 #(-267 204 -3)
Cowgol
include "cowgol.coh";
record Vector is
x: int32; # Cowgol does not have floating point types,
y: int32; # but for the examples it does not matter.
z: int32;
end record;
sub print_signed(n: int32) is
if n < 0 then
print_char('-');
n := -n;
end if;
print_i32(n as uint32);
end sub;
sub print_vector(v: [Vector]) is
print_char('(');
print_signed(v.x);
print(", ");
print_signed(v.y);
print(", ");
print_signed(v.z);
print_char(')');
print_nl();
end sub;
sub dot(a: [Vector], b: [Vector]): (r: int32) is
r := a.x * b.x + a.y * b.y + a.z * b.z;
end sub;
# Unfortunately it is impossible to return a complex type
# from a function. We have to have the caller pass in a pointer
# and have this function set its fields.
sub cross(a: [Vector], b: [Vector], r: [Vector]) is
r.x := a.y * b.z - a.z * b.y;
r.y := a.z * b.x - a.x * b.z;
r.z := a.x * b.y - a.y * b.x;
end sub;
sub scalarTriple(a: [Vector], b: [Vector], c: [Vector]): (r: int32) is
var v: Vector;
cross(b, c, &v);
r := dot(a, &v);
end sub;
sub vectorTriple(a: [Vector], b: [Vector], c: [Vector], r: [Vector]) is
var v: Vector;
cross(b, c, &v);
cross(a, &v, r);
end sub;
var a: Vector := {3, 4, 5};
var b: Vector := {4, 3, 5};
var c: Vector := {-5, -12, -13};
var scratch: Vector;
print(" a = "); print_vector(&a);
print(" b = "); print_vector(&b);
print(" c = "); print_vector(&c);
print(" a . b = "); print_signed(dot(&a, &b)); print_nl();
print(" a x b = "); cross(&a, &b, &scratch); print_vector(&scratch);
print("a . b x c = "); print_signed(scalarTriple(&a, &b, &c)); print_nl();
print("a x b x c = "); vectorTriple(&a, &b, &c, &scratch);
print_vector(&scratch);- Output:
a = (3, 4, 5)
b = (4, 3, 5)
c = (-5, -12, -13)
a . b = 49
a x b = (5, 5, -7)
a . b x c = 6
a x b x c = (-267, 204, -3)
Crystal
class Vector
property x, y, z
def initialize(@x : Int64, @y : Int64, @z : Int64) end
def dot_product(other : Vector)
(self.x * other.x) + (self.y * other.y) + (self.z * other.z)
end
def cross_product(other : Vector)
Vector.new(self.y * other.z - self.z * other.y,
self.z * other.x - self.x * other.z,
self.x * other.y - self.y * other.x)
end
def scalar_triple_product(b : Vector, c : Vector)
self.dot_product(b.cross_product(c))
end
def vector_triple_product(b : Vector, c : Vector)
self.cross_product(b.cross_product(c))
end
def to_s
"(#{self.x}, #{self.y}, #{self.z})\n"
end
end
a = Vector.new(3, 4, 5)
b = Vector.new(4, 3, 5)
c = Vector.new(-5, -12, -13)
puts "a = #{a.to_s}"
puts "b = #{b.to_s}"
puts "c = #{c.to_s}"
puts "a dot b = #{a.dot_product b}"
puts "a cross b = #{a.cross_product(b).to_s}"
puts "a dot (b cross c) = #{a.scalar_triple_product b, c}"
puts "a cross (b cross c) = #{a.vector_triple_product(b, c).to_s}"
- Output:
a = (3, 4, 5) b = (4, 3, 5) c = (-5, -12, -13) a dot b = 49 a cross b = (5, 5, -7) a dot (b cross c) = 6 a cross (b cross c) = (-267, 204, -3)
D
import std.stdio, std.conv, std.numeric;
struct V3 {
union {
immutable struct { double x, y, z; }
immutable double[3] v;
}
double dot(in V3 rhs) const pure nothrow /*@safe*/ @nogc {
return dotProduct(v, rhs.v);
}
V3 cross(in V3 rhs) const pure nothrow @safe @nogc {
return V3(y * rhs.z - z * rhs.y,
z * rhs.x - x * rhs.z,
x * rhs.y - y * rhs.x);
}
string toString() const { return v.text; }
}
double scalarTriple(in V3 a, in V3 b, in V3 c) /*@safe*/ pure nothrow {
return a.dot(b.cross(c));
// function vector_products.V3.cross (const(V3) rhs) immutable
// is not callable using argument types (const(V3)) const
}
V3 vectorTriple(in V3 a, in V3 b, in V3 c) @safe pure nothrow @nogc {
return a.cross(b.cross(c));
}
void main() {
immutable V3 a = {3, 4, 5},
b = {4, 3, 5},
c = {-5, -12, -13};
writeln("a = ", a);
writeln("b = ", b);
writeln("c = ", c);
writeln("a . b = ", a.dot(b));
writeln("a x b = ", a.cross(b));
writeln("a . (b x c) = ", scalarTriple(a, b, c));
writeln("a x (b x c) = ", vectorTriple(a, b, c));
}
- Output:
a = [3, 4, 5] b = [4, 3, 5] c = [-5, -12, -13] a . b = 49 a x b = [5, 5, -7] a . (b x c) = 6 a x (b x c) = [-267, 204, -3]
Delphi
See Pascal.
DuckDB
The list_dot_product() function provided by DuckDB will compute the dot product of any pair of numeric vectors provided they are both the same dimension.
The other functions of interest here expect 3D vectors and are defined as table-valued DuckDB functions because of certain current limitations of DuckDB.
create or replace function cross_product(a,b) as table (
select [ a[2]*b[3] - a[3]*b[2], a[3]*b[1] - a[1]*b[3], a[1]*b[2]-a[2]*b[1] ]);
create or replace function scalar_triple_product(a,b,c) as table (
select list_dot_product(a, (from cross_product(b, c))) );
create or replace function vector_triple_product(a,b,c) as table (
from cross_product(a, (from cross_product(b, c))) );
# Examples
select [3, 4, 5] as a,
[4, 3, 5] as b,
[-5, -12, -13] as c,
list_dot_product(a, b) as 'a . b',
((from cross_product(a, b) )) as 'a x b',
((from scalar_triple_product(a, b, c))) as 'a . (b x c)',
((from vector_triple_product(a, b, c))) as 'a x (b x c)' ;
- Output:
┌───────────┬───────────┬────────────────┬────────┬────────────┬─────────────┬─────────────────┐ │ a │ b │ c │ a . b │ a x b │ a . (b x c) │ a x (b x c) │ │ int32[] │ int32[] │ int32[] │ double │ int32[] │ double │ int32[] │ ├───────────┼───────────┼────────────────┼────────┼────────────┼─────────────┼─────────────────┤ │ [3, 4, 5] │ [4, 3, 5] │ [-5, -12, -13] │ 49.0 │ [5, 5, -7] │ 6.0 │ [-267, 204, -3] │ └───────────┴───────────┴────────────────┴────────┴────────────┴─────────────┴─────────────────┘
EasyLang
func vdot a[] b[] .
for i to len a[]
r += a[i] * b[i]
.
return r
.
func[] vcross a[] b[] .
r[] &= a[2] * b[3] - a[3] * b[2]
r[] &= a[3] * b[1] - a[1] * b[3]
r[] &= a[1] * b[2] - a[2] * b[1]
return r[]
.
a[] = [ 3 4 5 ]
b[] = [ 4 3 5 ]
c[] = [ -5 -12 -13 ]
#
print vdot a[] b[]
print vcross a[] b[]
print vdot a[] vcross b[] c[]
print vcross a[] vcross b[] c[]
- Output:
49 [ 5 5 -7 ] 6 [ -267 204 -3 ]
EchoLisp
The math library includes the dot-product and cross-product functions. They work on complex or real vectors.
(lib 'math)
(define (scalar-triple-product a b c)
(dot-product a (cross-product b c)))
(define (vector-triple-product a b c)
(cross-product a (cross-product b c)))
(define a #(3 4 5))
(define b #(4 3 5))
(define c #(-5 -12 -13))
(cross-product a b)
→ #( 5 5 -7)
(dot-product a b)
→ 49
(scalar-triple-product a b c)
→ 6
(vector-triple-product a b c)
→ #( -267 204 -3)
Elixir
defmodule Vector do
def dot_product({a1,a2,a3}, {b1,b2,b3}), do: a1*b1 + a2*b2 + a3*b3
def cross_product({a1,a2,a3}, {b1,b2,b3}), do: {a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1}
def scalar_triple_product(a, b, c), do: dot_product(a, cross_product(b, c))
def vector_triple_product(a, b, c), do: cross_product(a, cross_product(b, c))
end
a = {3, 4, 5}
b = {4, 3, 5}
c = {-5, -12, -13}
IO.puts "a = #{inspect a}"
IO.puts "b = #{inspect b}"
IO.puts "c = #{inspect c}"
IO.puts "a . b = #{inspect Vector.dot_product(a, b)}"
IO.puts "a x b = #{inspect Vector.cross_product(a, b)}"
IO.puts "a . (b x c) = #{inspect Vector.scalar_triple_product(a, b, c)}"
IO.puts "a x (b x c) = #{inspect Vector.vector_triple_product(a, b, c)}"
- Output:
a = {3, 4, 5}
b = {4, 3, 5}
c = {-5, -12, -13}
a . b = 49
a x b = {5, 5, -7}
a . (b x c) = 6
a x (b x c) = {-267, 204, -3}
Erlang
-module(vector).
-export([main/0]).
vector_product(X,Y)->
[X1,X2,X3]=X,
[Y1,Y2,Y3]=Y,
Ans=[X2*Y3-X3*Y2,X3*Y1-X1*Y3,X1*Y2-X2*Y1],
Ans.
dot_product(X,Y)->
[X1,X2,X3]=X,
[Y1,Y2,Y3]=Y,
Ans=X1*Y1+X2*Y2+X3*Y3,
io:fwrite("~p~n",[Ans]).
main()->
{ok, A} = io:fread("Enter vector A : ", "~d ~d ~d"),
{ok, B} = io:fread("Enter vector B : ", "~d ~d ~d"),
{ok, C} = io:fread("Enter vector C : ", "~d ~d ~d"),
dot_product(A,B),
Ans=vector_product(A,B),
io:fwrite("~p,~p,~p~n",Ans),
dot_product(C,vector_product(A,B)),
io:fwrite("~p,~p,~p~n",vector_product(C,vector_product(A,B))).
ERRE
PROGRAM VECTORPRODUCT
!$DOUBLE
TYPE TVECTOR=(X,Y,Z)
DIM A:TVECTOR,B:TVECTOR,C:TVECTOR
DIM AA:TVECTOR,BB:TVECTOR,CC:TVECTOR
DIM DD:TVECTOR,EE:TVECTOR,FF:TVECTOR
PROCEDURE DOTPRODUCT(DD.,EE.->DOTP)
DOTP=DD.X*EE.X+DD.Y*EE.Y+DD.Z*EE.Z
END PROCEDURE
PROCEDURE CROSSPRODUCT(DD.,EE.->FF.)
FF.X=DD.Y*EE.Z-DD.Z*EE.Y
FF.Y=DD.Z*EE.X-DD.X*EE.Z
FF.Z=DD.X*EE.Y-DD.Y*EE.X
END PROCEDURE
PROCEDURE SCALARTRIPLEPRODUCT(AA.,BB.,CC.->SCALARTP)
CROSSPRODUCT(BB.,CC.->FF.)
DOTPRODUCT(AA.,FF.->SCALARTP)
END PROCEDURE
PROCEDURE VECTORTRIPLEPRODUCT(AA.,BB.,CC.->FF.)
CROSSPRODUCT(BB.,CC.->FF.)
CROSSPRODUCT(AA.,FF.->FF.)
END PROCEDURE
PROCEDURE PRINTVECTOR(AA.)
PRINT("(";AA.X;",";AA.Y;",";AA.Z;")")
END PROCEDURE
BEGIN
A.X=3 A.Y=4 A.Z=5
B.X=4 B.Y=3 B.Z=5
C.X=-5 C.Y=-12 C.Z=-13
PRINT("A: ";) PRINTVECTOR(A.)
PRINT("B: ";) PRINTVECTOR(B.)
PRINT("C: ";) PRINTVECTOR(C.)
PRINT
DOTPRODUCT(A.,B.->DOTP)
PRINT("A.B =";DOTP)
CROSSPRODUCT(A.,B.->FF.)
PRINT("AxB =";) PRINTVECTOR(FF.)
SCALARTRIPLEPRODUCT(A.,B.,C.->SCALARTP)
PRINT("A.(BxC)=";SCALARTP)
VECTORTRIPLEPRODUCT(A.,B.,C.->FF.)
PRINT("Ax(BxC)=";) PRINTVECTOR(FF.)
END PROGRAMEuphoria
constant X = 1, Y = 2, Z = 3
function dot_product(sequence a, sequence b)
return a[X]*b[X] + a[Y]*b[Y] + a[Z]*b[Z]
end function
function cross_product(sequence a, sequence b)
return { a[Y]*b[Z] - a[Z]*b[Y],
a[Z]*b[X] - a[X]*b[Z],
a[X]*b[Y] - a[Y]*b[X] }
end function
function scalar_triple(sequence a, sequence b, sequence c)
return dot_product( a, cross_product( b, c ) )
end function
function vector_triple( sequence a, sequence b, sequence c)
return cross_product( a, cross_product( b, c ) )
end function
constant a = { 3, 4, 5 }, b = { 4, 3, 5 }, c = { -5, -12, -13 }
puts(1,"a = ")
? a
puts(1,"b = ")
? b
puts(1,"c = ")
? c
puts(1,"a dot b = ")
? dot_product( a, b )
puts(1,"a x b = ")
? cross_product( a, b )
puts(1,"a dot (b x c) = ")
? scalar_triple( a, b, c )
puts(1,"a x (b x c) = ")
? vector_triple( a, b, c )Output:
a = {3,4,5}
b = {4,3,5}
c = {-5,-12,-13}
a dot b = 49
a x b = {5,5,-7}
a dot (b x c) = 6
a x (b x c) = {-267,204,-3}
F#
let dot (ax, ay, az) (bx, by, bz) =
ax * bx + ay * by + az * bz
let cross (ax, ay, az) (bx, by, bz) =
(ay*bz - az*by, az*bx - ax*bz, ax*by - ay*bx)
let scalTrip a b c =
dot a (cross b c)
let vecTrip a b c =
cross a (cross b c)
[<EntryPoint>]
let main _ =
let a = (3.0, 4.0, 5.0)
let b = (4.0, 3.0, 5.0)
let c = (-5.0, -12.0, -13.0)
printfn "%A" (dot a b)
printfn "%A" (cross a b)
printfn "%A" (scalTrip a b c)
printfn "%A" (vecTrip a b c)
0 // return an integer exit code
- Output:
49.0 (5.0, 5.0, -7.0) 6.0 (-267.0, 204.0, -3.0)
Factor
Factor has a fantastic math.vectors vocabulary, but in the spirit of the task, it is not used.
USING: arrays io locals math prettyprint sequences ;
: dot-product ( a b -- dp ) [ * ] 2map sum ;
:: cross-product ( a b -- cp )
a first :> a1 a second :> a2 a third :> a3
b first :> b1 b second :> b2 b third :> b3
a2 b3 * a3 b2 * - ! X
a3 b1 * a1 b3 * - ! Y
a1 b2 * a2 b1 * - ! Z
3array ;
: scalar-triple-product ( a b c -- stp )
cross-product dot-product ;
: vector-triple-product ( a b c -- vtp )
cross-product cross-product ;
[let
{ 3 4 5 } :> a
{ 4 3 5 } :> b
{ -5 -12 -13 } :> c
"a: " write a .
"b: " write b .
"c: " write c . nl
"a . b: " write a b dot-product .
"a x b: " write a b cross-product .
"a . (b x c): " write a b c scalar-triple-product .
"a x (b x c): " write a b c vector-triple-product .
]
- Output:
a: { 3 4 5 }
b: { 4 3 5 }
c: { -5 -12 -13 }
a . b: 49
a x b: { 5 5 -7 }
a . (b x c): 6
a x (b x c): { -267 204 -3 }
Fantom
class Main
{
Int dot_product (Int[] a, Int[] b)
{
a[0]*b[0] + a[1]*b[1] + a[2]*b[2]
}
Int[] cross_product (Int[] a, Int[] b)
{
[a[1]*b[2] - a[2]*b[1], a[2]*b[0] - a[0]*b[2], a[0]*b[1]-a[1]*b[0]]
}
Int scalar_triple_product (Int[] a, Int[] b, Int[] c)
{
dot_product (a, cross_product (b, c))
}
Int[] vector_triple_product (Int[] a, Int[] b, Int[] c)
{
cross_product (a, cross_product (b, c))
}
Void main ()
{
a := [3, 4, 5]
b := [4, 3, 5]
c := [-5, -12, -13]
echo ("a . b = " + dot_product (a, b))
echo ("a x b = [" + cross_product(a, b).join (", ") + "]")
echo ("a . (b x c) = " + scalar_triple_product (a, b, c))
echo ("a x (b x c) = [" + vector_triple_product(a, b, c).join (", ") + "]")
}
}Output:
a . b = 49 a x b = [5, 5, -7] a . (b x c) = 6 a x (b x c) = [-267, 204, -3]
Forth
: 3f! ( &v - ) ( f: x y z - ) dup float+ dup float+ f! f! f! ;
: Vector \ Compiletime: ( f: x y z - ) ( <name> - )
create here [ 3 floats ] literal allot 3f! ; \ Runtime: ( - &v )
: >fx@ ( &v - ) ( f: - n ) postpone f@ ; immediate
: >fy@ ( &v - ) ( f: - n ) float+ f@ ;
: >fz@ ( &v - ) ( f: - n ) float+ float+ f@ ;
: .Vector ( &v - ) dup >fz@ dup >fy@ >fx@ f. f. f. ;
: Dot* ( &v1 &v2 - ) ( f - DotPrd )
2dup >fx@ >fx@ f*
2dup >fy@ >fy@ f* f+
>fz@ >fz@ f* f+ ;
: Cross* ( &v1 &v2 &vResult - )
>r 2dup >fz@ >fy@ f*
2dup >fy@ >fz@ f* f-
2dup >fx@ >fz@ f*
2dup >fz@ >fx@ f* f-
2dup >fy@ >fx@ f*
>fx@ >fy@ f* f-
r> 3f! ;
: ScalarTriple* ( &v1 &v2 &v3 - ) ( f: - ScalarTriple* )
>r pad Cross* pad r> Dot* ;
: VectorTriple* ( &v1 &v2 &v3 &vDest - )
>r swap r@ Cross* r> tuck Cross* ;
3e 4e 5e Vector A
4e 3e 5e Vector B
-5e -12e -13e Vector C
cr
cr .( a . b = ) A B Dot* f.
cr .( a x b = ) A B pad Cross* pad .Vector
cr .( a . [b x c] = ) A B C ScalarTriple* f.
cr .( a x [b x c] = ) A B C pad VectorTriple* pad .Vector
- Output:
a . b = 49.0000 a x b = 5.00000 5.00000 -7.00000 a . [b x c] = 6.00000 a x [b x c] = -267.000 204.000 -3.00000
S" fsl-util.fs" REQUIRED
: 3f! 3 SWAP }fput ;
: vector
CREATE
HERE 3 DUP FLOAT DUP , * ALLOT SWAP CELL+ }fput
DOES>
CELL+ ;
: >fx@ 0 } F@ ;
: >fy@ 1 } F@ ;
: >fz@ 2 } F@ ;
: .Vector 3 SWAP }fprint ;
0e 0e 0e vector pad \ NB: your system will be non-standard after this line
\ From here on is identical to the above example
Fortran
Specialized for 3-dimensional vectors.
program VectorProducts
real, dimension(3) :: a, b, c
a = (/ 3, 4, 5 /)
b = (/ 4, 3, 5 /)
c = (/ -5, -12, -13 /)
print *, dot_product(a, b)
print *, cross_product(a, b)
print *, s3_product(a, b, c)
print *, v3_product(a, b, c)
contains
function cross_product(a, b)
real, dimension(3) :: cross_product
real, dimension(3), intent(in) :: a, b
cross_product(1) = a(2)*b(3) - a(3)*b(2)
cross_product(2) = a(3)*b(1) - a(1)*b(3)
cross_product(3) = a(1)*b(2) - b(1)*a(2)
end function cross_product
function s3_product(a, b, c)
real :: s3_product
real, dimension(3), intent(in) :: a, b, c
s3_product = dot_product(a, cross_product(b, c))
end function s3_product
function v3_product(a, b, c)
real, dimension(3) :: v3_product
real, dimension(3), intent(in) :: a, b, c
v3_product = cross_product(a, cross_product(b, c))
end function v3_product
end program VectorProducts
Output
49.0000
5.00000 5.00000 -7.00000
6.00000
-267.000 204.000 -3.00000
FreeBASIC
'Construct only required operators for this.
Type V3
As double x,y,z
declare operator cast() as string
End Type
#define dot *
#define cross ^
#define Show(t1,t) ? #t1;tab(22);t
operator V3.cast() as string
return "("+str(x)+","+str(y)+","+str(z)+")"
end operator
Operator dot(v1 As v3,v2 As v3) As double
Return v1.x*v2.x+v1.y*v2.y+v1.z*v2.z
End Operator
Operator cross(v1 As v3,v2 As v3) As v3
Return type<v3>(v1.y*v2.z-v2.y*v1.z,-(v1.x*v2.z-v2.x*v1.z),v1.x*v2.y-v2.x*v1.y)
End Operator
dim as V3 a = (3, 4, 5), b = (4, 3, 5), c = (-5, -12, -13)
Show(a,a)
Show(b,b)
Show(c,c)
?
Show(a . b,a dot b)
Show(a X b,a cross b)
Show(a . b X c,a dot b cross c)
Show(a X (b X c),a cross (b cross c))
sleep
- Output:
a (3,4,5) b (4,3,5) c (-5,-12,-13) a . b 49 a X b (5,5,-7) a . b X c 6 a X (b X c) (-267,204,-3)
FunL
A = (3, 4, 5)
B = (4, 3, 5)
C = (-5, -12, -13)
def dot( u, v ) = sum( u(i)v(i) | i <- 0:u.>length() )
def cross( u, v ) = (u(1)v(2) - u(2)v(1), u(2)v(0) - u(0)v(2), u(0)v(1) - u(1)v(0) )
def scalarTriple( u, v, w ) = dot( u, cross(v, w) )
def vectorTriple( u, v, w ) = cross( u, cross(v, w) )
println( "A\u00b7B = ${dot(A, B)}" )
println( "A\u00d7B = ${cross(A, B)}" )
println( "A\u00b7(B\u00d7C) = ${scalarTriple(A, B, C)}" )
println( "A\u00d7(B\u00d7C) = ${vectorTriple(A, B, C)}" )- Output:
A·B = 49 A×B = (5, 5, -7) A·(B×C) = 6 A×(B×C) = (-267, 204, -3)
GAP
DotProduct := function(u, v)
return u*v;
end;
CrossProduct := function(u, v)
return [
u[2]*v[3] - u[3]*v[2],
u[3]*v[1] - u[1]*v[3],
u[1]*v[2] - u[2]*v[1] ];
end;
ScalarTripleProduct := function(u, v, w)
return DotProduct(u, CrossProduct(v, w));
end;
VectorTripleProduct := function(u, v, w)
return CrossProduct(u, CrossProduct(v, w));
end;
a := [3, 4, 5];
b := [4, 3, 5];
c := [-5, -12, -13];
DotProduct(a, b);
# 49
CrossProduct(a, b);
# [ 5, 5, -7 ]
ScalarTripleProduct(a, b, c);
# 6
# Another way to get it
Determinant([a, b, c]);
# 6
VectorTripleProduct(a, b, c);
# [ -267, 204, -3 ]
GLSL
vec3 a = vec3(3, 4, 5),b = vec3(4, 3, 5),c = vec3(-5, -12, -13);
float dotProduct(vec3 a, vec3 b)
{
return a.x*b.x+a.y*b.y+a.z*b.z;
}
vec3 crossProduct(vec3 a,vec3 b)
{
vec3 c = vec3(a.y*b.z - a.z*b.y, a.z*b.x - a.x*b.z, a.x*b.y- a.y*b.x);
return c;
}
float scalarTripleProduct(vec3 a,vec3 b,vec3 c)
{
return dotProduct(a,crossProduct(b,c));
}
vec3 vectorTripleProduct(vec3 a,vec3 b,vec3 c)
{
return crossProduct(a,crossProduct(b,c));
}
Go
package main
import "fmt"
type vector struct {
x, y, z float64
}
var (
a = vector{3, 4, 5}
b = vector{4, 3, 5}
c = vector{-5, -12, -13}
)
func dot(a, b vector) float64 {
return a.x*b.x + a.y*b.y + a.z*b.z
}
func cross(a, b vector) vector {
return vector{a.y*b.z - a.z*b.y, a.z*b.x - a.x*b.z, a.x*b.y - a.y*b.x}
}
func s3(a, b, c vector) float64 {
return dot(a, cross(b, c))
}
func v3(a, b, c vector) vector {
return cross(a, cross(b, c))
}
func main() {
fmt.Println(dot(a, b))
fmt.Println(cross(a, b))
fmt.Println(s3(a, b, c))
fmt.Println(v3(a, b, c))
}
Output:
49
{5 5 -7}
6
{-267 204 -3}
Groovy
Dot Product Solution:
def pairwiseOperation = { x, y, Closure binaryOp ->
assert x && y && x.size() == y.size()
[x, y].transpose().collect(binaryOp)
}
def pwMult = pairwiseOperation.rcurry { it[0] * it[1] }
def dotProduct = { x, y ->
assert x && y && x.size() == y.size()
pwMult(x, y).sum()
}
Cross Product Solution, using scalar operations:
def crossProductS = { x, y ->
assert x && y && x.size() == 3 && y.size() == 3
[x[1]*y[2] - x[2]*y[1], x[2]*y[0] - x[0]*y[2] , x[0]*y[1] - x[1]*y[0]]
}
Cross Product Solution, using "vector" operations:
def rotR = {
assert it && it.size() > 2
[it[-1]] + it[0..-2]
}
def rotL = {
assert it && it.size() > 2
it[1..-1] + [it[0]]
}
def pwSubtr = pairwiseOperation.rcurry { it[0] - it[1] }
def crossProductV = { x, y ->
assert x && y && x.size() == 3 && y.size() == 3
pwSubtr(pwMult(rotL(x), rotR(y)), pwMult(rotL(y), rotR(x)))
}
Test program (including triple products):
def test = { crossProduct ->
def scalarTripleProduct = { x, y, z ->
dotProduct(x, crossProduct(y, z))
}
def vectorTripleProduct = { x, y, z ->
crossProduct(x, crossProduct(y, z))
}
def a = [3, 4, 5]
def b = [4, 3, 5]
def c = [-5, -12, -13]
println(" a . b = " + dotProduct(a,b))
println(" a x b = " + crossProduct(a,b))
println("a . (b x c) = " + scalarTripleProduct(a,b,c))
println("a x (b x c) = " + vectorTripleProduct(a,b,c))
println()
}
test(crossProductS)
test(crossProductV)
Output:
a . b = 49
a x b = [5, 5, -7]
a . (b x c) = 6
a x (b x c) = [-267, 204, -3]
a . b = 49
a x b = [5, 5, -7]
a . (b x c) = 6
a x (b x c) = [-267, 204, -3]
Haskell
import Data.Monoid ((<>))
type Vector a = [a]
type Scalar a = a
a, b, c, d :: Vector Int
a = [3, 4, 5]
b = [4, 3, 5]
c = [-5, -12, -13]
d = [3, 4, 5, 6]
dot
:: (Num t)
=> Vector t -> Vector t -> Scalar t
dot u v
| length u == length v = sum $ zipWith (*) u v
| otherwise = error "Dotted Vectors must be of equal dimension."
cross
:: (Num t)
=> Vector t -> Vector t -> Vector t
cross u v
| length u == 3 && length v == 3 =
[ u !! 1 * v !! 2 - u !! 2 * v !! 1
, u !! 2 * head v - head u * v !! 2
, head u * v !! 1 - u !! 1 * head v
]
| otherwise = error "Crossed Vectors must both be three dimensional."
scalarTriple
:: (Num t)
=> Vector t -> Vector t -> Vector t -> Scalar t
scalarTriple q r s = dot q $ cross r s
vectorTriple
:: (Num t)
=> Vector t -> Vector t -> Vector t -> Vector t
vectorTriple q r s = cross q $ cross r s
main :: IO ()
main =
mapM_
putStrLn
[ "a . b = " <> show (dot a b)
, "a x b = " <> show (cross a b)
, "a . b x c = " <> show (scalarTriple a b c)
, "a x b x c = " <> show (vectorTriple a b c)
, "a . d = " <> show (dot a d)
]
Output:
a . b = 49 a x b = [5,5,-7] a . b x c = 6 a x b x c = [-267,204,-3] ** Exception: Dotted Vectors must be of equal dimension. a . d =
Or using Either and (>>=), rather than error, to pass on intelligible messages:
dotProduct
:: Num a
=> [a] -> [a] -> Either String a
dotProduct xs ys
| length xs /= length ys =
Left "Dot product not defined - vectors differ in dimension."
| otherwise = Right (sum $ zipWith (*) xs ys)
crossProduct
:: Num a
=> [a] -> [a] -> Either String [a]
crossProduct xs ys
| 3 /= length xs || 3 /= length ys =
Left "crossProduct is defined only for 3d vectors."
| otherwise = Right [x2 * y3 - x3 * y2, x3 * y1 - x1 * y3, x1 * y2 - x2 * y1]
where
[x1, x2, x3] = xs
[y1, y2, y3] = ys
scalarTriple
:: Num a
=> [a] -> [a] -> [a] -> Either String a
scalarTriple q r s = crossProduct r s >>= dotProduct q
vectorTriple
:: Num a
=> [a] -> [a] -> [a] -> Either String [a]
vectorTriple q r s = crossProduct r s >>= crossProduct q
-- TEST ---------------------------------------------------
a = [3, 4, 5]
b = [4, 3, 5]
c = [-5, -12, -13]
d = [3, 4, 5, 6]
main :: IO ()
main =
mapM_ putStrLn $
zipWith
(++)
["a . b", "a x b", "a . b x c", "a x b x c", "a . d", "a . (b x d)"]
[ sh $ dotProduct a b
, sh $ crossProduct a b
, sh $ scalarTriple a b c
, sh $ vectorTriple a b c
, sh $ dotProduct a d
, sh $ scalarTriple a b d
]
sh
:: Show a
=> Either String a -> String
sh = either (" => " ++) ((" = " ++) . show)
- Output:
a . b = 49 a x b = [5,5,-7] a . b x c = 6 a x b x c = [-267,204,-3] a . d => Dot product not defined - vectors differ in dimension. a . (b x d) => crossProduct is defined only for 3d vectors.
Icon and Unicon
Output:
A.B : (3, 4, 5).(4, 3, 5) = 49 AxB : (3, 4, 5)x(4, 3, 5) = (5, 5, -7) A.(BxC) : (3, 4, 5).((4, 3, 5)x(-5, -12, -13)) = 6 Ax(BxC) : (3, 4, 5)x((4, 3, 5)x(-5, -12, -13)) = (-267, 204, -3)
J
cross=: 1 _1 1*1-/ .*\. ,.
dot=: +/ .*
stp=: dot`cross/ NB. scalar triple product
vtp=: cross/ NB. vector triple product
- Output:
a=: 3 4 5
b=: 4 3 5
c=: -5 12 13
a dot b
49
a cross b
5 5 _7
stp a,b,:c
6
vtp a,b,:c
_267 204 _3
The cross product algorithm calculates the determinant (-/ .*) of each matrix of the 1-outfix (1 ]\. ]) of the element-wise pairing of the two vectors.
Naming the elements of vectors A and B:
A=:'a',"0'123' [B=:'b',"0'123'
we take the determinant of each matrix in the following 1-outfix:
1]\.A(,.' '&,"1)B
a2 b2
a3 b3
a1 b1
a3 b3
a1 b1
a2 b2
and negate the middle term.
Other cross product variations
Another common definition for cross product uses rotate, multiply and subtract:
cross=: (*1&|.){{1|.u-u~}}
or
cross=: 1|.(*1&|.)-(*1&|.)~
or
cross=: {{ ((1|.x)*2|.y) - (2|.x)*1|.y }}
However, there are other valid approaches. For example, a "generalized approach" based on j:Essays/Complete Tensor:
CT=: C.!.2 @ (#:i.) @ $~
ip=: +/ .* NB. inner product
cross=: ] ip CT@#@[ ip [
Note that there are a variety of other generalizations have cross products as a part of what they do. (For example, we could implement cross product using complex numbers in a Cayley Dickson implementation of quaternion product.)
An alternative definition for cross calculates the formal determinant
(where the first row of the given matrix consists of the unit vectors):
cross=: [: > [: -&.>/ .(*&.>) (<"1=i.3), ,:&:(<"0)
or
cross=: {{ >-L:0/ .(*L:0) (<"1=i.3), x,:&:(<"0) y}}
Java
All operations which return vectors give vectors containing Doubles.
public class VectorProds{
public static class Vector3D<T extends Number>{
private T a, b, c;
public Vector3D(T a, T b, T c){
this.a = a;
this.b = b;
this.c = c;
}
public double dot(Vector3D<?> vec){
return (a.doubleValue() * vec.a.doubleValue() +
b.doubleValue() * vec.b.doubleValue() +
c.doubleValue() * vec.c.doubleValue());
}
public Vector3D<Double> cross(Vector3D<?> vec){
Double newA = b.doubleValue()*vec.c.doubleValue() - c.doubleValue()*vec.b.doubleValue();
Double newB = c.doubleValue()*vec.a.doubleValue() - a.doubleValue()*vec.c.doubleValue();
Double newC = a.doubleValue()*vec.b.doubleValue() - b.doubleValue()*vec.a.doubleValue();
return new Vector3D<Double>(newA, newB, newC);
}
public double scalTrip(Vector3D<?> vecB, Vector3D<?> vecC){
return this.dot(vecB.cross(vecC));
}
public Vector3D<Double> vecTrip(Vector3D<?> vecB, Vector3D<?> vecC){
return this.cross(vecB.cross(vecC));
}
@Override
public String toString(){
return "<" + a.toString() + ", " + b.toString() + ", " + c.toString() + ">";
}
}
public static void main(String[] args){
Vector3D<Integer> a = new Vector3D<Integer>(3, 4, 5);
Vector3D<Integer> b = new Vector3D<Integer>(4, 3, 5);
Vector3D<Integer> c = new Vector3D<Integer>(-5, -12, -13);
System.out.println(a.dot(b));
System.out.println(a.cross(b));
System.out.println(a.scalTrip(b, c));
System.out.println(a.vecTrip(b, c));
}
}
Output:
49.0 <5.0, 5.0, -7.0> 6.0 <-267.0, 204.0, -3.0>
This solution uses Java SE new Stream API
import java.util.Arrays;
import java.util.stream.IntStream;
public class VectorsOp {
// Vector dot product using Java SE 8 stream abilities
// the method first create an array of size values,
// and map the product of each vectors components in a new array (method map())
// and transform the array to a scalr by summing all elements (method reduce)
// the method parallel is there for optimization
private static int dotProduct(int[] v1, int[] v2,int length) {
int result = IntStream.range(0, length)
.parallel()
.map( id -> v1[id] * v2[id])
.reduce(0, Integer::sum);
return result;
}
// Vector Cross product using Java SE 8 stream abilities
// here we map in a new array where each element is equal to the cross product
// With Stream is is easier to handle N dimensions vectors
private static int[] crossProduct(int[] v1, int[] v2,int length) {
int result[] = new int[length] ;
//result[0] = v1[1] * v2[2] - v1[2]*v2[1] ;
//result[1] = v1[2] * v2[0] - v1[0]*v2[2] ;
// result[2] = v1[0] * v2[1] - v1[1]*v2[0] ;
result = IntStream.range(0, length)
.parallel()
.map( i -> v1[(i+1)%length] * v2[(i+2)%length] - v1[(i+2)%length]*v2[(i+1)%length])
.toArray();
return result;
}
public static void main (String[] args)
{
int[] vect1 = {3, 4, 5};
int[] vect2 = {4, 3, 5};
int[] vect3 = {-5, -12, -13};
System.out.println("dot product =:" + dotProduct(vect1,vect2,3));
int[] prodvect = new int[3];
prodvect = crossProduct(vect1,vect2,3);
System.out.println("cross product =:[" + prodvect[0] + ","
+ prodvect[1] + ","
+ prodvect[2] + "]");
prodvect = crossProduct(vect2,vect3,3);
System.out.println("scalar product =:" + dotProduct(vect1,prodvect,3));
prodvect = crossProduct(vect1,prodvect,3);
System.out.println("triple product =:[" + prodvect[0] + ","
+ prodvect[1] + ","
+ prodvect[2] + "]");
}
}result is the same as above , fortunately
dot product =:49 cross product =:[5,5,-7] scalar product =:6 triple product =:[-267,204,-3]
JavaScript
ES5
The dotProduct() function is generic and will create a dot product of any set of vectors provided they are all the same dimension.
The crossProduct() function expects two 3D vectors.
function dotProduct() {
var len = arguments[0] && arguments[0].length;
var argsLen = arguments.length;
var i, j = len;
var prod, sum = 0;
// If no arguments supplied, return undefined
if (!len) {
return;
}
// If all vectors not same length, return undefined
i = argsLen;
while (i--) {
if (arguments[i].length != len) {
return; // return undefined
}
}
// Sum terms
while (j--) {
i = argsLen;
prod = 1;
while (i--) {
prod *= arguments[i][j];
}
sum += prod;
}
return sum;
}
function crossProduct(a, b) {
// Check lengths
if (a.length != 3 || b.length != 3) {
return;
}
return [a[1]*b[2] - a[2]*b[1],
a[2]*b[0] - a[0]*b[2],
a[0]*b[1] - a[1]*b[0]];
}
function scalarTripleProduct(a, b, c) {
return dotProduct(a, crossProduct(b, c));
}
function vectorTripleProduct(a, b, c) {
return crossProduct(a, crossProduct(b, c));
}
// Run tests
(function () {
var a = [3, 4, 5];
var b = [4, 3, 5];
var c = [-5, -12, -13];
alert(
'A . B: ' + dotProduct(a, b) +
'\n' +
'A x B: ' + crossProduct(a, b) +
'\n' +
'A . (B x C): ' + scalarTripleProduct(a, b, c) +
'\n' +
'A x (B x C): ' + vectorTripleProduct(a, b, c)
);
}());
- Output:
A . B: 49 A x B: 5,5,-7 A . (B x C): 6 A x (B x C): -267,204,-3
ES6
(() => {
'use strict';
// dotProduct :: [a] -> [a] -> Either String a
const dotProduct = xs =>
// Dot product of two vectors of equal dimension.
ys => xs.length !== ys.length ? (
Left('Dot product not defined - vectors differ in dimension.')
) : Right(sum(
zipWith(mul)(Array.from(xs))(Array.from(ys))
));
// crossProduct :: Num a => (a, a, a) -> (a, a, a)
// Either String -> (a, a, a)
const crossProduct = xs =>
// Cross product of two 3D vectors.
ys => 3 !== xs.length || 3 !== ys.length ? (
Left('crossProduct is defined only for 3d vectors.')
) : Right((() => {
const [x1, x2, x3] = Array.from(xs);
const [y1, y2, y3] = Array.from(ys);
return [
x2 * y3 - x3 * y2,
x3 * y1 - x1 * y3,
x1 * y2 - x2 * y1
];
})());
// scalarTriple :: Num a => (a, a, a) -> (a, a, a) -> (a, a a) ->
// Either String -> a
const scalarTriple = q =>
// The scalar triple product.
r => s => bindLR(crossProduct(r)(s))(
dotProduct(q)
);
// vectorTriple :: Num a => (a, a, a) -> (a, a, a) -> (a, a a) ->
// Either String -> (a, a, a)
const vectorTriple = q =>
// The vector triple product.
r => s => bindLR(crossProduct(r)(s))(
crossProduct(q)
);
// main :: IO ()
const main = () => {
// TEST -------------------------------------------
const
a = [3, 4, 5],
b = [4, 3, 5],
c = [-5, -12, -13],
d = [3, 4, 5, 6];
console.log(unlines(
zipWith(k => f => k + show(
saturated(f)([a, b, c])
))(['a . b', 'a x b', 'a . (b x c)', 'a x (b x c)'])(
[dotProduct, crossProduct, scalarTriple, vectorTriple]
)
.concat([
'a . d' + show(
dotProduct(a)(d)
),
'a . (b x d)' + show(
scalarTriple(a)(b)(d)
)
])
));
};
// GENERIC FUNCTIONS ----------------------------------
// Left :: a -> Either a b
const Left = x => ({
type: 'Either',
Left: x
});
// Right :: b -> Either a b
const Right = x => ({
type: 'Either',
Right: x
});
// bindLR (>>=) :: Either a -> (a -> Either b) -> Either b
const bindLR = m => mf =>
undefined !== m.Left ? (
m
) : mf(m.Right);
// either :: (a -> c) -> (b -> c) -> Either a b -> c
const either = fl => fr => e =>
'Either' === e.type ? (
undefined !== e.Left ? (
fl(e.Left)
) : fr(e.Right)
) : undefined;
// identity :: a -> a
const identity = x => x;
// mul (*) :: Num a => a -> a -> a
const mul = a => b => a * b;
// Curried function -> [Argument] -> a more saturated value
const saturated = f =>
// A curried function applied successively to
// a list of arguments up to, but not beyond,
// the point of saturation.
args => 0 < args.length ? (
args.slice(1).reduce(
(a, x) => 'function' !== typeof a ? (
a
) : a(x),
f(args[0])
)
) : f;
// show :: Either String a -> String
const show = x =>
either(x => ' => ' + x)(
x => ' = ' + JSON.stringify(x)
)(x);
// sum :: [Num] -> Num
const sum = xs => xs.reduce((a, x) => a + x, 0);
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// zipWith:: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = f => xs => ys =>
xs.slice(
0, Math.min(xs.length, ys.length)
).map((x, i) => f(x)(ys[i]));
// MAIN ---
return main();
})();
- Output:
a . b = 49 a x b = [5,5,-7] a . (b x c) = 6 a x (b x c) = [-267,204,-3] a . d => Dot product not defined - vectors differ in dimension. a . (b x d) => crossProduct is defined only for 3d vectors.
jq
The dot_product() function is generic and will create a dot product of any pair of vectors provided they are both the same dimension. The other functions expect 3D vectors.
def dot_product(a; b):
reduce range(0;a|length) as $i (0; . + (a[$i] * b[$i]) );
# for 3d vectors
def cross_product(a;b):
[ a[1]*b[2] - a[2]*b[1], a[2]*b[0] - a[0]*b[2], a[0]*b[1]-a[1]*b[0] ];
def scalar_triple_product(a;b;c):
dot_product(a; cross_product(b; c));
def vector_triple_product(a;b;c):
cross_product(a; cross_product(b; c));
def main:
[3, 4, 5] as $a
| [4, 3, 5] as $b
| [-5, -12, -13] as $c
| "a . b = \(dot_product($a; $b))",
"a x b = [\( cross_product($a; $b) | map(tostring) | join (", ") )]" ,
"a . (b x c) = \( scalar_triple_product ($a; $b; $c)) )",
"a x (b x c) = [\( vector_triple_product($a; $b; $c)|map(tostring)|join (", ") )]" ;Output:
"a . b = 49"
"a x b = [5, 5, -7]"
"a . (b x c) = 6 )"
"a x (b x c) = [-267, 204, -3]"Julia
Julia provides dot and cross products with LinearAlgebra. It's easy enough to use these to construct the triple products.
using LinearAlgebra
const a = [3, 4, 5]
const b = [4, 3, 5]
const c = [-5, -12, -13]
println("Test Vectors:")
@show a b c
println("\nVector Products:")
@show a ⋅ b
@show a × b
@show a ⋅ (b × c)
@show a × (b × c)
- Output:
Test Vectors: a = [3, 4, 5] b = [4, 3, 5] c = [-5, -12, -13] Vector Products: a ⋅ b = 49 a × b = [5, 5, -7] a ⋅ (b × c) = 6 a × (b × c) = [-267, 204, -3]
K
rot:{,/|![#y;0,x]_y} / rotate
cross:{rot[1;]@(x*rot[1;y])-y*rot[1;x]}
dot:{+/x*y}
- Output:
a:3 4 5 b:4 3 5 c:-(5 12 13) dot[a;b] 49 cross[a;b] 5 5 -7 dot[a;]@cross[b;c] /scalar triple product 6 cross[a;]@cross[b;c] /vector triple product -267 204 -3
Kotlin
// version 1.1.2
class Vector3D(val x: Double, val y: Double, val z: Double) {
infix fun dot(v: Vector3D) = x * v.x + y * v.y + z * v.z
infix fun cross(v: Vector3D) =
Vector3D(y * v.z - z * v.y, z * v.x - x * v.z, x * v.y - y * v.x)
fun scalarTriple(v: Vector3D, w: Vector3D) = this dot (v cross w)
fun vectorTriple(v: Vector3D, w: Vector3D) = this cross (v cross w)
override fun toString() = "($x, $y, $z)"
}
fun main(args: Array<String>) {
val a = Vector3D(3.0, 4.0, 5.0)
val b = Vector3D(4.0, 3.0, 5.0)
val c = Vector3D(-5.0, -12.0, -13.0)
println("a = $a")
println("b = $b")
println("c = $c")
println()
println("a . b = ${a dot b}")
println("a x b = ${a cross b}")
println("a . b x c = ${a.scalarTriple(b, c)}")
println("a x b x c = ${a.vectorTriple(b, c)}")
}
- Output:
a = (3.0, 4.0, 5.0) b = (4.0, 3.0, 5.0) c = (-5.0, -12.0, -13.0) a . b = 49.0 a x b = (5.0, 5.0, -7.0) a . b x c = 6.0 a x b x c = (-267.0, 204.0, -3.0)
Ksh
#!/bin/ksh
# Vector products
# # dot product (a scalar quantity) A • B = a1b1 + a2b2 + a3b3 + ...
# # cross product (a vector quantity) A x B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
# # scalar triple product (a scalar quantity) A • (B x C)
# # vector triple product (a vector quantity) A x (B x C)
# # Variables:
#
typeset -a A=( 3 4 5 )
typeset -a B=( 4 3 5 )
typeset -a C=( -5 -12 -13 )
# # Functions:
#
# # Function _dotprod(vec1, vec2) - Return the (scalar) dot product of 2 vectors
#
function _dotprod {
typeset _vec1 ; nameref _vec1="$1" # Input vector 1
typeset _vec2 ; nameref _vec2="$2" # Input vector 2
typeset _i ; typeset -si _i
typeset _dotp ; integer _dotp=0
for ((_i=0; _i<${#_vec1[*]}; _i++)); do
(( _dotp+=(_vec1[_i] * _vec2[_i]) ))
done
echo ${_dotp}
}
# # Function _crossprod(vec1, vec2, vec) - Return the (vector) cross product of 2 vectors
#
function _crossprod {
typeset _vec1 ; nameref _vec1="$1" # Input vector 1
typeset _vec2 ; nameref _vec2="$2" # Input vector 2
typeset _vec3 ; nameref _vec3="$3" # Output vector
_vec3+=( $(( _vec1[1]*_vec2[2] - _vec1[2]*_vec2[1] )) )
_vec3+=( $(( _vec1[2]*_vec2[0] - _vec1[0]*_vec2[2] )) )
_vec3+=( $(( _vec1[0]*_vec2[1] - _vec1[1]*_vec2[0] )) )
}
# # Function _scal3prod(vec1, vec2, vec3) - Return the (scalar) scalar triple product of 3 vectors
#
function _scal3prod {
typeset _vec1 ; nameref _vec1="$1" # Input vector 1
typeset _vec2 ; nameref _vec2="$2" # Input vector 2
typeset _vec3 ; nameref _vec3="$3" # Input vector 3
typeset _vect ; typeset -a _vect # temp vector
_crossprod _vec2 _vec3 _vect # (B x C)
echo $(_dotprod _vec1 _vect) # A • (B x C)
}
# # Function _vect3prod(vec1, vec2, vec3, vec) - Return the (vector) vector triple product of 3 vectors
#
function _vect3prod {
typeset _vec1 ; nameref _vec1="$1" # Input vector 1
typeset _vec2 ; nameref _vec2="$2" # Input vector 2
typeset _vec3 ; nameref _vec3="$3" # Input vector 3
typeset _vec4 ; nameref _vec4="$4" # Output vector
typeset _vect ; typeset -a _vect # temp vector
_crossprod _vec2 _vec3 _vect # (B x C)
_crossprod _vec1 _vect _vec4 # A x (B x C)
}
######
# main #
######
print "The dot product A • B = $(_dotprod A B)"
typeset -a arr
_crossprod A B arr
print "The cross product A x B = ( ${arr[@]} )"
print "The scalar triple product A • (B x C) = $(_scal3prod A B C)"
typeset -m crossprod=arr ; typeset -a arr
_vect3prod A B C arr
print "The vector triple product A x (B x C) = ( ${arr[@]} )"
- Output:
The dot product A • B = 49 The cross product A x B = ( 5 5 -7 ) The scalar triple product A • (B x C) = 6
The vector triple product A x (B x C) = ( -267 204 -3 )
Lambdatalk
{def dotProduct
{lambda {:a :b}
{+ {* {A.get 0 :a} {A.get 0 :b}}
{* {A.get 1 :a} {A.get 1 :b}}
{* {A.get 2 :a} {A.get 2 :b}}}}}
-> dotProduct
{def crossProduct
{lambda {:a :b}
{A.new {- {* {A.get 1 :a} {A.get 2 :b}}
{* {A.get 2 :a} {A.get 1 :b}}}
{- {* {A.get 2 :a} {A.get 0 :b}}
{* {A.get 0 :a} {A.get 2 :b}}}
{- {* {A.get 0 :a} {A.get 1 :b}}
{* {A.get 1 :a} {A.get 0 :b}}} }}}
-> crossProduct
{def A {A.new 3 4 5}} -> A = [3,4,5]
{def B {A.new 4 3 5}} -> B = [4,3,5]
{def C {A.new -5 -12 -13}} -> C = [4,3,5]
A.B : {dotProduct {A} {B}} -> 49
AxB : {crossProduct {A} {B}} -> [5,5,-7]
A.(BxC) : {dotProduct {A} {crossProduct {B} {C}}} -> 6
Ax(BxC) : {crossProduct {A} {crossProduct {B} {C}}} -> [-267,204,-3]
Liberty BASIC
print "Vector products of 3-D vectors"
print "Dot product of 3,4,5 and 4,3,5 is "
print DotProduct( "3,4,5", "4,3,5")
print "Cross product of 3,4,5 and 4,3,5 is "
print CrossProduct$( "3,4,5", "4,3,5")
print "Scalar triple product of 3,4,5, 4,3,5 -5, -12, -13 is "
print ScalarTripleProduct( "3,4,5", "4,3,5", "-5, -12, -13")
print "Vector triple product of 3,4,5, 4,3,5 -5, -12, -13 is "
print VectorTripleProduct$( "3,4,5", "4,3,5", "-5, -12, -13")
end
function DotProduct( i$, j$)
ix =val( word$( i$, 1, ","))
iy =val( word$( i$, 2, ","))
iz =val( word$( i$, 3, ","))
jx =val( word$( j$, 1, ","))
jy =val( word$( j$, 2, ","))
jz =val( word$( j$, 3, ","))
DotProduct = ix *jx +iy *jy + iz *jz
end function
function CrossProduct$( i$, j$)
ix =val( word$( i$, 1, ","))
iy =val( word$( i$, 2, ","))
iz =val( word$( i$, 3, ","))
jx =val( word$( j$, 1, ","))
jy =val( word$( j$, 2, ","))
jz =val( word$( j$, 3, ","))
cpx =iy *jz -iz *jy
cpy =iz *jx -ix *jz
cpz =ix *jy -iy *jx
CrossProduct$ =str$( cpx); ","; str$( cpy); ","; str$( cpz)
end function
function ScalarTripleProduct( i$, j$, k$))
ScalarTripleProduct =DotProduct( i$, CrossProduct$( j$, k$))
end function
function VectorTripleProduct$( i$, j$, k$))
VectorTripleProduct$ =CrossProduct$( i$, CrossProduct$( j$, k$))
end function
END SUBLingo
Lingo has a built-in vector data type that supports calculation of both dot and cross products:
a = vector(1,2,3)
b = vector(4,5,6)
put a * b
-- 32.0000
put a.dot(b)
-- 32.0000
put a.cross(b)
-- vector( -3.0000, 6.0000, -3.0000 )Lua
Vector = {}
function Vector.new( _x, _y, _z )
return { x=_x, y=_y, z=_z }
end
function Vector.dot( A, B )
return A.x*B.x + A.y*B.y + A.z*B.z
end
function Vector.cross( A, B )
return { x = A.y*B.z - A.z*B.y,
y = A.z*B.x - A.x*B.z,
z = A.x*B.y - A.y*B.x }
end
function Vector.scalar_triple( A, B, C )
return Vector.dot( A, Vector.cross( B, C ) )
end
function Vector.vector_triple( A, B, C )
return Vector.cross( A, Vector.cross( B, C ) )
end
A = Vector.new( 3, 4, 5 )
B = Vector.new( 4, 3, 5 )
C = Vector.new( -5, -12, -13 )
print( Vector.dot( A, B ) )
r = Vector.cross(A, B )
print( r.x, r.y, r.z )
print( Vector.scalar_triple( A, B, C ) )
r = Vector.vector_triple( A, B, C )
print( r.x, r.y, r.z )
49 5 5 -7 6 -267 204 -3
M2000 Interpreter
Private members can be handled from another's object member if that object has the same type
Module checkit {
Class Vector {
private:
single a,b,c
public:
Property ToString$ {
Value {
link parent a,b,c to a,b,c
value$=format$("({0}, {1}, {2})",a,b,c)
}
}
Operator "==" {
read n as vector
push .a==n.a and .b==n.b and .c==n.c
}
Operator Unary {
.a-! : .b-! : .c-!
}
Operator "+" {
Read v2 as Vector
For this, v2 {
.a+=..a :.b+=..b:.c+=..c:
}
}
Function Mul(r as single) {
vv=this
for vv {
.a*=r:.b*=r:.c*=r
}
=vv
}
Function Dot(v2 as Vector) {
def double sum
for this, v2 {
sum=.a*..a+.b*..b+.c*..c
}
=sum
}
Operator "*" (v2 as Vector) {
For This, v2 {
Push .b*..c-.c*..b
Push .c*..a-.a*..c
.c<=.a*..b-.b*..a
Read .b, .a
}
}
Class:
module Vector {
if match("NNN") then {
Read .a,.b,.c
}
}
}
A=Vector(3,4,5)
B=Vector(4,3,5)
C=Vector(-5,-12,-13)
Print "A=";A.toString$
Print "B=";B.toString$
Print "C=";C.toString$
Print "A dot B="; A.dot(B)
AxB=A*B
Print "A x B="; AxB.toString$
Print "A dot (B x C)=";A.dot(B*C)
AxBxC=A*(B*C)
Print "A x (B x C)=";AxBxC.toString$
Def ToString$(a)=a.toString$
Print "A x (B x C)=";ToString$(A*(B*C))
}
Checkit
- Output:
A=(3, 4, 5) B=(4, 3, 5) C=(-5, -12, -13) A dot B=49 A x B=(5, 5, -7) A dot (B x C)=6 A x (B x C)=(-267, 204, -3) A x (B x C)=(-267, 204, -3)
Maple
with(LinearAlgebra):
A := Vector([3,4,5]):
B := Vector([4,3,5]):
C := Vector([-5,-12,-13]):
>>>A.B;
49
>>>CrossProduct(A,B);
Vector([5, 5, -7])
>>>A.(CrossProduct(B,C));
6
>>>CrossProduct(A,CrossProduct(B,C));
Vector([-267, 204, -3])Mathematica /Wolfram Language
a={3,4,5};
b={4,3,5};
c={-5,-12,-13};
a.b
Cross[a,b]
a.Cross[b,c]
Cross[a,Cross[b,c]]
- Output:
49
{5,5,-7}
6
{-267,204,-3}
MATLAB / Octave
Matlab / Octave use double precesion numbers per default, and pi is a builtin constant value. Arbitrary precision is only implemented in some additional toolboxes (e.g. symbolic toolbox).
% Create a named function/subroutine/method to compute the dot product of two vectors.
dot(a,b)
% Create a function to compute the cross product of two vectors.
cross(a,b)
% Optionally create a function to compute the scalar triple product of three vectors.
dot(a,cross(b,c))
% Optionally create a function to compute the vector triple product of three vectors.
cross(a,cross(b,c))
% Compute and display: a • b
cross(a,b)
% Compute and display: a x b
cross(a,b)
% Compute and display: a • b x c, the scaler triple product.
dot(a,cross(b,c))
% Compute and display: a x b x c, the vector triple product.
cross(a,cross(b,c))
Code for testing:
A = [ 3.0, 4.0, 5.0] B = [ 4.0, 3.0, 5.0] C = [-5.0, -12.0, -13.0] dot(A,B) cross(A,B) dot(A,cross(B,C)) cross(A,cross(B,C))
Output:
>> A = [ 3.0, 4.0, 5.0] >> B = [ 4.0, 3.0, 5.0] >> C = [-5.0, -12.0, -13.0] >> dot(A,B) ans = 49 >> cross(A,B) ans = 5 5 -7 >> dot(A,cross(B,C)) ans = 6 >> cross(A,cross(B,C)) ans = -267 204 -3
Mercury
:- module vector_product.
:- interface.
:- import_module io.
:- pred main(io::di, io::uo) is det.
:- implementation.
:- import_module int, list, string.
main(!IO) :-
A = vector3d(3, 4, 5),
B = vector3d(4, 3, 5),
C = vector3d(-5, -12, -13),
io.format("A . B = %d\n", [i(A `dot_product` B)], !IO),
io.format("A x B = %s\n", [s(to_string(A `cross_product` B))], !IO),
io.format("A . (B x C) = %d\n", [i(scalar_triple_product(A, B, C))], !IO),
io.format("A x (B x C) = %s\n", [s(to_string(vector_triple_product(A, B, C)))], !IO).
:- type vector3d ---> vector3d(int, int, int).
:- func dot_product(vector3d, vector3d) = int.
dot_product(vector3d(A1, A2, A3), vector3d(B1, B2, B3)) =
A1 * B1 + A2 * B2 + A3 * B3.
:- func cross_product(vector3d, vector3d) = vector3d.
cross_product(vector3d(A1, A2, A3), vector3d(B1, B2, B3)) =
vector3d(A2 * B3 - A3 * B2, A3 * B1 - A1 * B3, A1 * B2 - A2 * B1).
:- func scalar_triple_product(vector3d, vector3d, vector3d) = int.
scalar_triple_product(A, B, C) = A `dot_product` (B `cross_product` C).
:- func vector_triple_product(vector3d, vector3d, vector3d) = vector3d.
vector_triple_product(A, B, C) = A `cross_product` (B `cross_product` C).
:- func to_string(vector3d) = string.
to_string(vector3d(X, Y, Z)) =
string.format("(%d, %d, %d)", [i(X), i(Y), i(Z)]).
MiniScript
vectorA = [3, 4, 5]
vectorB = [4, 3, 5]
vectorC = [-5, -12, -13]
dotProduct = function(x, y)
return x[0]*y[0] + x[1]*y[1] + x[2]*y[2]
end function
crossProduct = function(x, y)
return [x[1]*y[2] - x[2]*y[1], x[2]*y[0] - x[0]*y[2], x[0]*y[1] - x[1]*y[0]]
end function
print "Dot Product = " + dotProduct(vectorA, vectorB)
print "Cross Product = " + crossProduct(vectorA, vectorB)
print "Scalar Triple Product = " + dotProduct(vectorA, crossProduct(vectorB,vectorC))
print "Vector Triple Product = " + crossProduct(vectorA, crossProduct(vectorB,vectorC))
- Output:
Dot Product = 49 Cross Product = [5, 5, -7] Scalar Triple Product = 6 Vector Triple Product = [-267, 204, -3]
МК-61/52
ПП 54 С/П ПП 66 С/П
ИП0 ИП3 ИП6 П3 -> П0 -> П6
ИП1 ИП4 ИП7 П4 -> П1 -> П7
ИП2 ИП5 ИП8 П5 -> П2 -> П8
ПП 66
ИП6 ИП7 ИП8 П2 -> П1 -> П0
ИП9 ИПA ИПB П5 -> П4 -> П3
ПП 54 С/П ПП 66 С/П
ИП0 ИП3 * ИП1 ИП4 * + ИП2 ИП5 * + В/О
ИП1 ИП5 * ИП2 ИП4 * - П9
ИП2 ИП3 * ИП0 ИП5 * - ПA
ИП0 ИП4 * ИП1 ИП3 * - ПB В/О
Instruction: Р0 - a1, Р1 - a2, Р2 - a3, Р3 - b1, Р4 - b2, Р5 - b3, Р6 - c1, Р7 - c2, Р8 - c3; В/О С/П.
Modula-2
MODULE VectorProducts;
FROM RealStr IMPORT RealToStr;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE WriteReal(r : REAL);
VAR buf : ARRAY[0..31] OF CHAR;
BEGIN
RealToStr(r, buf);
WriteString(buf)
END WriteReal;
TYPE Vector = RECORD
a,b,c : REAL;
END;
PROCEDURE Dot(u,v : Vector) : REAL;
BEGIN
RETURN u.a * v.a
+ u.b * v.b
+ u.c * v.c
END Dot;
PROCEDURE Cross(u,v : Vector) : Vector;
BEGIN
RETURN Vector{
u.b*v.c - u.c*v.b,
u.c*v.a - u.a*v.c,
u.a*v.b - u.b*v.a
}
END Cross;
PROCEDURE ScalarTriple(u,v,w : Vector) : REAL;
BEGIN
RETURN Dot(u, Cross(v, w))
END ScalarTriple;
PROCEDURE VectorTriple(u,v,w : Vector) : Vector;
BEGIN
RETURN Cross(u, Cross(v, w))
END VectorTriple;
PROCEDURE WriteVector(v : Vector);
BEGIN
WriteString("<");
WriteReal(v.a);
WriteString(", ");
WriteReal(v.b);
WriteString(", ");
WriteReal(v.c);
WriteString(">")
END WriteVector;
VAR a,b,c : Vector;
BEGIN
a := Vector{3.0, 4.0, 5.0};
b := Vector{4.0, 3.0, 5.0};
c := Vector{-5.0, -12.0, -13.0};
WriteVector(a);
WriteString(" dot ");
WriteVector(b);
WriteString(" = ");
WriteReal(Dot(a,b));
WriteLn;
WriteVector(a);
WriteString(" cross ");
WriteVector(b);
WriteString(" = ");
WriteVector(Cross(a,b));
WriteLn;
WriteVector(a);
WriteString(" cross (");
WriteVector(b);
WriteString(" cross ");
WriteVector(c);
WriteString(") = ");
WriteVector(VectorTriple(a,b,c));
WriteLn;
ReadChar
END VectorProducts.
Nemerle
using System.Console;
module VectorProducts3d
{
Dot(x : int * int * int, y : int * int * int) : int
{
def (x1, x2, x3) = x;
def (y1, y2, y3) = y;
(x1 * y1) + (x2 * y2) + (x3 * y3)
}
Cross(x : int * int * int, y : int * int * int) : int * int * int
{
def (x1, x2, x3) = x;
def (y1, y2, y3) = y;
((x2 * y3 - x3 * y2), (x3 * y1 - x1 * y3), (x1 * y2 - x2 * y1))
}
ScalarTriple(a : int * int * int, b : int * int * int, c : int * int * int) : int
{
Dot(a, Cross(b, c))
}
VectorTriple(a : int * int * int, b : int * int * int, c : int * int * int) : int * int * int
{
Cross(a, Cross(b, c))
}
Main() : void
{
def a = (3, 4, 5); def b = (4, 3, 5); def c = (-5, -12, -13);
WriteLine(Dot(a, b)); WriteLine(Cross(a, b));
WriteLine(ScalarTriple(a, b, c));
WriteLine(VectorTriple(a, b, c));
}
}
Outputs
49 (5, 5, -7) 6 (-267, 204, -3)
Never
func printv(a[d] : float) -> int {
prints("[" + a[0] + ", " + a[1] + ", " + a[2] + "]\n");
0
}
func dot(a[d1] : float, b[d2] : float) -> float {
a[0] * b[0] + a[1] * b[1] + a[2] * b[2]
}
func cross(a[d1] : float, b[d2] : float) -> [_] : float {
[ a[1] * b[2] - a[2] * b[1], a[2] * b[0] - a[0] * b[2], a[0] * b[1] - a[1] * b[0] ] : float
}
func scalar_triple(a[d1] : float, b[d2] : float, c[d3] : float) -> float {
dot(a, cross(b, c))
}
func vector_triple(a[d1] : float, b[d2] : float, c[d3] : float) -> [_] : float {
cross(a, cross(b, c))
}
func main() -> int {
var a = [ 3.0, 4.0, 5.0 ] : float;
var b = [ 4.0, 3.0, 5.0 ] : float;
var c = [ -5.0, -12.0, -13.0 ] : float;
printv(a);
printv(b);
printv(c);
printf(dot(a, b));
printv(cross(a, b));
printf(scalar_triple(a, b, c));
printv(vector_triple(a, b, c));
0
}
Output:
[3.00, 4.00, 5.00] [4.00, 3.00, 5.00] [-5.00, -12.00, -13.00] 49.00 [5.00, 5.00, -7.00] 6.00 [-267.00, 204.00, -3.00]
Nim
import strformat, strutils
type Vector3 = array[1..3, float]
proc `$`(a: Vector3): string =
result = "("
for x in a:
result.addSep(", ", 1)
result.add &"{x}"
result.add ')'
proc cross(a, b: Vector3): Vector3 =
result = [a[2]*b[3] - a[3]*b[2], a[3]*b[1] - a[1]*b[3], a[1]*b[2] - a[2]*b[1]]
proc dot(a, b: Vector3): float =
for i in a.low..a.high:
result += a[i] * b[i]
proc scalarTriple(a, b, c: Vector3): float = a.dot(b.cross(c))
proc vectorTriple(a, b, c: Vector3): Vector3 = a.cross(b.cross(c))
let
a = [3.0, 4.0, 5.0]
b = [4.0, 3.0, 5.0]
c = [-5.0, -12.0, -13.0]
echo &"a ⨯ b = {a.cross(b)}"
echo &"a . b = {a.dot(b)}"
echo &"a . (b ⨯ c) = {scalarTriple(a, b, c)}"
echo &"a ⨯ (b ⨯ c) = {vectorTriple(a, b, c)}"
- Output:
a ⨯ b = (5.0, 5.0, -7.0) a . b = 49.0 a . (b ⨯ c) = 6.0 a ⨯ (b ⨯ c) = (-267.0, 204.0, -3.0)
Objeck
bundle Default {
class VectorProduct {
function : Main(args : String[]) ~ Nil {
a := Vector3D->New(3.0, 4.0, 5.0);
b := Vector3D->New(4.0, 3.0, 5.0);
c := Vector3D->New(-5.0, -12.0, -13.0);
a->Dot(b)->Print();
a->Cross(b)->Print();
a->ScaleTrip(b, c)->Print();
a->VectorTrip(b, c)->Print();
}
}
class Vector3D {
@a : Float;
@b : Float;
@c : Float;
New(a : Float, b : Float, c : Float) {
@a := a;
@b := b;
@c := c;
}
method : GetA() ~ Float {
return @a;
}
method : GetB() ~ Float {
return @b;
}
method : GetC() ~ Float {
return @c;
}
method : public : Dot(vec : Vector3D) ~ Float {
return @a * vec->GetA() + @b * vec->GetB() + @c * vec->GetC();
}
method : public : Cross(vec : Vector3D) ~ Vector3D {
newA := @b * vec->GetC() - @c * vec->GetB();
newB := @c * vec->GetA() - @a * vec->GetC();
newC := @a * vec->GetB() - @b * vec->GetA();
return Vector3D->New(newA, newB, newC);
}
method : public : ScaleTrip(vec_b: Vector3D, vec_c : Vector3D) ~ Float {
return Dot(vec_b->Cross(vec_c));
}
method : public : Print() ~ Nil {
IO.Console->Print('<')->Print(@a)->Print(" ,")
->Print(@b)->Print(", ")->Print(@c)->PrintLine('>');
}
method : public : VectorTrip(vec_b: Vector3D, vec_c : Vector3D) ~ Vector3D {
return Cross(vec_b->Cross(vec_c));
}
}
}Output:
49<5 ,5, -7> 6<-267 ,204, -3>
OCaml
let a = (3.0, 4.0, 5.0)
let b = (4.0, 3.0, 5.0)
let c = (-5.0, -12.0, -13.0)
let string_of_vector (x,y,z) =
Printf.sprintf "(%g, %g, %g)" x y z
let dot (a1, a2, a3) (b1, b2, b3) =
(a1 *. b1) +. (a2 *. b2) +. (a3 *. b3)
let cross (a1, a2, a3) (b1, b2, b3) =
(a2 *. b3 -. a3 *. b2,
a3 *. b1 -. a1 *. b3,
a1 *. b2 -. a2 *. b1)
let scalar_triple a b c =
dot a (cross b c)
let vector_triple a b c =
cross a (cross b c)
let () =
Printf.printf "a: %s\n" (string_of_vector a);
Printf.printf "b: %s\n" (string_of_vector b);
Printf.printf "c: %s\n" (string_of_vector c);
Printf.printf "a . b = %g\n" (dot a b);
Printf.printf "a x b = %s\n" (string_of_vector (cross a b));
Printf.printf "a . (b x c) = %g\n" (scalar_triple a b c);
Printf.printf "a x (b x c) = %s\n" (string_of_vector (vector_triple a b c));
;;
outputs:
a: (3, 4, 5) b: (4, 3, 5) c: (-5, -12, -13) a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
Octave
Octave handles naturally vectors / matrices.
a = [3, 4, 5];
b = [4, 3, 5];
c = [-5, -12, -13];
function r = s3prod(a, b, c)
r = dot(a, cross(b, c));
endfunction
function r = v3prod(a, b, c)
r = cross(a, cross(b, c));
endfunction
% 49
dot(a, b)
% or matrix-multiplication between row and column vectors
a * b'
% 5 5 -7
cross(a, b) % only for 3d-vectors
% 6
s3prod(a, b, c)
% -267 204 -3
v3prod(a, b, c)
ooRexx
a = .vector~new(3, 4, 5);
b = .vector~new(4, 3, 5);
c = .vector~new(-5, -12, -13);
say a~dot(b)
say a~cross(b)
say a~scalarTriple(b, c)
say a~vectorTriple(b, c)
::class vector
::method init
expose x y z
use arg x, y, z
::attribute x get
::attribute y get
::attribute z get
-- dot product operation
::method dot
expose x y z
use strict arg other
return x * other~x + y * other~y + z * other~z
-- cross product operation
::method cross
expose x y z
use strict arg other
newX = y * other~z - z * other~y
newY = z * other~x - x * other~z
newZ = x * other~y - y * other~x
return self~class~new(newX, newY, newZ)
-- scalar triple product
::method scalarTriple
use strict arg vectorB, vectorC
return self~dot(vectorB~cross(vectorC))
-- vector triple product
::method vectorTriple
use strict arg vectorB, vectorC
return self~cross(vectorB~cross(vectorC))
::method string
expose x y z
return "<"||x", "y", "z">"
Output:
49 <5, 5, -7> 6 <-267, 204, -3>
PARI/GP
dot(u,v)={
sum(i=1,#u,u[i]*v[i])
};
cross(u,v)={
[u[2]*v[3] - u[3]*v[2], u[3]*v[1] - u[1]*v[3], u[1]*v[2] - u[2]*v[1]]
};
striple(a,b,c)={
dot(a,cross(b,c))
};
vtriple(a,b,c)={
cross(a,cross(b,c))
};
a = [3,4,5]; b = [4,3,5]; c = [-5,-12,-13];
dot(a,b)
cross(a,b)
striple(a,b,c)
vtriple(a,b,c)Output:
49 [5, 5, -7] 6 [-267, 204, -3]
Pascal
Program VectorProduct (output);
type
Tvector = record
x, y, z: double
end;
function dotProduct(a, b: Tvector): double;
begin
dotProduct := a.x*b.x + a.y*b.y + a.z*b.z;
end;
function crossProduct(a, b: Tvector): Tvector;
begin
crossProduct.x := a.y*b.z - a.z*b.y;
crossProduct.y := a.z*b.x - a.x*b.z;
crossProduct.z := a.x*b.y - a.y*b.x;
end;
function scalarTripleProduct(a, b, c: Tvector): double;
begin
scalarTripleProduct := dotProduct(a, crossProduct(b, c));
end;
function vectorTripleProduct(a, b, c: Tvector): Tvector;
begin
vectorTripleProduct := crossProduct(a, crossProduct(b, c));
end;
procedure printVector(a: Tvector);
begin
writeln(a.x:15:8, a.y:15:8, a.z:15:8);
end;
var
a: Tvector = (x: 3; y: 4; z: 5);
b: Tvector = (x: 4; y: 3; z: 5);
c: Tvector = (x:-5; y:-12; z:-13);
begin
write('a: '); printVector(a);
write('b: '); printVector(b);
write('c: '); printVector(c);
writeln('a . b: ', dotProduct(a,b):15:8);
write('a x b: '); printVector(crossProduct(a,b));
writeln('a . (b x c): ', scalarTripleProduct(a,b,c):15:8);
write('a x (b x c): '); printVector(vectorTripleProduct(a,b,c));
end.
Output:
a: 3.00000000 4.00000000 5.00000000 b: 4.00000000 3.00000000 5.00000000 c: -5.00000000 -12.00000000 -13.00000000 a . b: 49.00000000 a x b: 5.00000000 5.00000000 -7.00000000 a . (b x c): 6.00000000 a x (b x c): -267.00000000 204.00000000 -3.00000000
PascalABC.NET
uses System.Numerics;
function DotProduct(v1,v2: Vector3): real
:= v1.x * v2.x + v1.y * v2.y + v1.z * v2.z;
function CrossProduct(v1,v2: Vector3): Vector3
:= new Vector3(v1.y * v2.z - v1.z * v2.y,
v1.z * v2.x - v1.x * v2.z,
v1.x * v2.y - v1.y * v2.x);
function ScalarTripleProduct(a,b,c: Vector3): real
:= DotProduct(a, CrossProduct(b, c));
function VectorTripleProduct(a,b,c: Vector3): Vector3
:= CrossProduct(a, CrossProduct(b, c));
begin
var a := new Vector3(3,4,5);
var b := new Vector3(4,3,5);
var c := new Vector3(-5,-12,-13);
Writeln(DotProduct(a, b));
Writeln(CrossProduct(a, b));
Writeln(ScalarTripleProduct(a, b, c));
Writeln(VectorTripleProduct(a, b, c));
end.
- Output:
49 <5. 5. -7> 6 <-267. 204. -3>
Perl
package Vector;
use List::Util 'sum';
use List::MoreUtils 'pairwise';
sub new { shift; bless [@_] }
use overload (
'""' => sub { "(@{+shift})" },
'&' => sub { sum pairwise { $a * $b } @{+shift}, @{+shift} },
'^' => sub {
my @a = @{+shift};
my @b = @{+shift};
bless [ $a[1]*$b[2] - $a[2]*$b[1],
$a[2]*$b[0] - $a[0]*$b[2],
$a[0]*$b[1] - $a[1]*$b[0] ]
},
);
package main;
my $a = Vector->new(3, 4, 5);
my $b = Vector->new(4, 3, 5);
my $c = Vector->new(-5, -12, -13);
print "a = $a b = $b c = $c\n";
print "$a . $b = ", $a & $b, "\n";
print "$a x $b = ", $a ^ $b, "\n";
print "$a . ($b x $c) = ", $a & ($b ^ $c), "\n";
print "$a x ($b x $c) = ", $a ^ ($b ^ $c), "\n";
Output:
a = (3 4 5) b = (4 3 5) c = (-5 -12 -13) (3 4 5) . (4 3 5) = 49 (3 4 5) x (4 3 5) = (5 5 -7) (3 4 5) . ((4 3 5) x (-5 -12 -13)) = 6 (3 4 5) x ((4 3 5) x (-5 -12 -13)) = (-267 204 -3)
Phix
function dot_product(sequence a, b) return sum(sq_mul(a,b)) end function function cross_product(sequence a, b) integer {a1,a2,a3} = a, {b1,b2,b3} = b return {a2*b3-a3*b2, a3*b1-a1*b3, a1*b2-a2*b1} end function function scalar_triple_product(sequence a, b, c) return dot_product(a,cross_product(b,c)) end function function vector_triple_product(sequence a, b, c) return cross_product(a,cross_product(b,c)) end function constant a = {3, 4, 5}, b = {4, 3, 5}, c = {-5, -12, -13} printf(1," a . b = %v\n",{dot_product(a,b)}) printf(1," a x b = %v\n",{cross_product(a,b)}) printf(1,"a . (b x c) = %v\n",{scalar_triple_product(a,b,c)}) printf(1,"a x (b x c) = %v\n",{vector_triple_product(a,b,c)})
- Output:
a . b = 49
a x b = {5,5,-7}
a . (b x c) = 6
a x (b x c) = {-267,204,-3}
Phixmonti
include ..\Utilitys.pmt
( 3 4 5 ) var vectorA
( 4 3 5 ) var vectorB
( -5 -12 -13 ) var vectorC
def dotProduct /# x y -- n #/
0 >ps
len for var i
i get rot i get rot * ps> + >ps
endfor
drop drop
ps>
enddef
def crossProduct /# x y -- z #/
1 get rot 2 get rot * >ps
1 get rot 2 get rot * >ps
3 get rot 1 get rot * >ps
3 get rot 1 get rot * >ps
2 get rot 3 get rot * >ps
2 get rot 3 get rot * ps> - ps> ps> - ps> ps> - 3 tolist
nip nip
enddef
"Dot Product = " print vectorA vectorB dotProduct ?
"Cross Product = " print vectorA vectorB crossProduct ?
"Scalar Triple Product = " print vectorB vectorC crossProduct vectorA swap dotProduct ?
"Vector Triple Product = " print vectorB vectorC crossProduct vectorA swap crossProduct ?- Output:
Dot Product = 49 Cross Product = [5, 5, -7] Scalar Triple Product = 6 Vector Triple Product = [-267, 204, -3] === Press any key to exit ===
PHP
<?php
class Vector
{
private $values;
public function setValues(array $values)
{
if (count($values) != 3)
throw new Exception('Values must contain exactly 3 values');
foreach ($values as $value)
if (!is_int($value) && !is_float($value))
throw new Exception('Value "' . $value . '" has an invalid type');
$this->values = $values;
}
public function getValues()
{
if ($this->values == null)
$this->setValues(array (
0,
0,
0
));
return $this->values;
}
public function Vector(array $values)
{
$this->setValues($values);
}
public static function dotProduct(Vector $va, Vector $vb)
{
$a = $va->getValues();
$b = $vb->getValues();
return ($a[0] * $b[0]) + ($a[1] * $b[1]) + ($a[2] * $b[2]);
}
public static function crossProduct(Vector $va, Vector $vb)
{
$a = $va->getValues();
$b = $vb->getValues();
return new Vector(array (
($a[1] * $b[2]) - ($a[2] * $b[1]),
($a[2] * $b[0]) - ($a[0] * $b[2]),
($a[0] * $b[1]) - ($a[1] * $b[0])
));
}
public static function scalarTripleProduct(Vector $va, Vector $vb, Vector $vc)
{
return self::dotProduct($va, self::crossProduct($vb, $vc));
}
public static function vectorTrippleProduct(Vector $va, Vector $vb, Vector $vc)
{
return self::crossProduct($va, self::crossProduct($vb, $vc));
}
}
class Program
{
public function Program()
{
$a = array (
3,
4,
5
);
$b = array (
4,
3,
5
);
$c = array (
-5,
-12,
-13
);
$va = new Vector($a);
$vb = new Vector($b);
$vc = new Vector($c);
$result1 = Vector::dotProduct($va, $vb);
$result2 = Vector::crossProduct($va, $vb)->getValues();
$result3 = Vector::scalarTripleProduct($va, $vb, $vc);
$result4 = Vector::vectorTrippleProduct($va, $vb, $vc)->getValues();
printf("\n");
printf("A = (%0.2f, %0.2f, %0.2f)\n", $a[0], $a[1], $a[2]);
printf("B = (%0.2f, %0.2f, %0.2f)\n", $b[0], $b[1], $b[2]);
printf("C = (%0.2f, %0.2f, %0.2f)\n", $c[0], $c[1], $c[2]);
printf("\n");
printf("A · B = %0.2f\n", $result1);
printf("A × B = (%0.2f, %0.2f, %0.2f)\n", $result2[0], $result2[1], $result2[2]);
printf("A · (B × C) = %0.2f\n", $result3);
printf("A × (B × C) =(%0.2f, %0.2f, %0.2f)\n", $result4[0], $result4[1], $result4[2]);
}
}
new Program();
?>
Output:
A = (3.00, 4.00, 5.00) B = (4.00, 3.00, 5.00) C = (-5.00, -12.00, -13.00) A · B = 49.00 A × B = (5.00, 5.00, -7.00) A · (B × C) = 6.00 A × (B × C) =(-267.00, 204.00, -3.00)
Picat
go =>
A = [3, 4, 5],
B = [4, 3, 5],
C = [-5, -12, -13],
println(a=A),
println(b=B),
println(c=C),
println("A . B"=dot(A,B)),
println("A x B"=cross(A,B)),
println("A . (B x C)"=scalar_triple(A,B,C)),
println("A X (B X C)"=vector_triple(A,B,C)),
nl.
dot(A,B) = sum([ AA*BB : {AA,BB} in zip(A,B)]).
cross(A,B) = [A[2]*B[3]-A[3]*B[2], A[3]*B[1]-A[1]*B[3], A[1]*B[2]-A[2]*B[1]].
scalar_triple(A,B,C) = dot(A,cross(B,C)).
vector_triple(A,B,C) = cross(A,cross(B,C)).- Output:
a = [3,4,5] b = [4,3,5] c = [-5,-12,-13] A . B = 49 A x B = [5,5,-7] A . (B x C) = 6 A X (B X C) = [-267,204,-3]
PicoLisp
(de dotProduct (A B)
(sum * A B) )
(de crossProduct (A B)
(list
(- (* (cadr A) (caddr B)) (* (caddr A) (cadr B)))
(- (* (caddr A) (car B)) (* (car A) (caddr B)))
(- (* (car A) (cadr B)) (* (cadr A) (car B))) ) )
(de scalarTriple (A B C)
(dotProduct A (crossProduct B C)) )
(de vectorTriple (A B C)
(crossProduct A (crossProduct B C)) )Test:
(setq A ( 3 4 5) B ( 4 3 5) C (-5 -12 -13) ) : (dotProduct A B) -> 49 : (crossProduct A B) -> (5 5 -7) : (scalarTriple A B C) -> 6 : (vectorTriple A B C) -> (-267 204 -3)
PL/I
/* dot product, cross product, etc. 4 June 2011 */
test_products: procedure options (main);
declare a(3) fixed initial (3, 4, 5);
declare b(3) fixed initial (4, 3, 5);
declare c(3) fixed initial (-5, -12, -13);
declare e(3) fixed;
put skip list ('a . b =', dot_product(a, b));
call cross_product(a, b, e); put skip list ('a x b =', e);
put skip list ('a . (b x c) =', scalar_triple_product(a, b, c));
call vector_triple_product(a, b, c, e); put skip list ('a x (b x c) =', e);
dot_product: procedure (a, b) returns (fixed);
declare (a, b) (*) fixed;
return (sum(a*b));
end dot_product;
cross_product: procedure (a, b, c);
declare (a, b, c) (*) fixed;
c(1) = a(2)*b(3) - a(3)*b(2);
c(2) = a(3)*b(1) - a(1)*b(3);
c(3) = a(1)*b(2) - a(2)*b(1);
end cross_product;
scalar_triple_product: procedure (a, b, c) returns (fixed);
declare (a, b, c)(*) fixed;
declare t(hbound(a, 1)) fixed;
call cross_product(b, c, t);
return (dot_product(a, t));
end scalar_triple_product;
vector_triple_product: procedure (a, b, c, e);
declare (a, b, c, e)(*) fixed;
declare t(hbound(a,1)) fixed;
call cross_product(b, c, t);
call cross_product(a, t, e);
end vector_triple_product;
end test_products;Results:
a . b = 49 a x b = 5 5 -7 a . (b x c) = 6 a x (b x c) = -267 204 -3
/* This version uses the ability of PL/I to return arrays. */
/* dot product, cross product, etc. 6 June 2011 */
test_products: procedure options (main);
define structure 1 vector, 2 vec(3) fixed;
declare (a, b, c) type(vector);
a.vec(1) = 3; a.vec(2) = 4; a.vec(3) = 5;
b.vec(1) = 4; b.vec(2) = 3; b.vec(3) = 5;
c.vec(1) = -5; c.vec(2) = -12; c.vec(3) = -13;
put skip list ('a . b =', dot_product (a, b) );
put skip list ('a x b =', cross_product(a, b).vec);
put skip list ('a . (b x c) =', scalar_triple_product(a, b, c) );
put skip list ('a x (b x c) =', vector_triple_product(a, b, c).vec);
dot_product: procedure (a, b) returns (fixed);
declare (a, b) type(vector);
return (sum(a.vec*b.vec));
end dot_product;
cross_product: procedure (a, b) returns (type(vector));
declare (a, b) type(vector);
declare c type vector;
c.vec(1) = a.vec(2)*b.vec(3) - a.vec(3)*b.vec(2);
c.vec(2) = a.vec(3)*b.vec(1) - a.vec(1)*b.vec(3);
c.vec(3) = a.vec(1)*b.vec(2) - a.vec(2)*b.vec(1);
return (c);
end cross_product;
scalar_triple_product: procedure (a, b, c) returns (fixed);
declare (a, b, c) type(vector);
declare t type (vector);
t = cross_product(b, c);
return (dot_product(a, t));
end scalar_triple_product;
vector_triple_product: procedure (a, b, c) returns (type(vector));
declare (a, b, c) type(vector);
declare (t, e) type (vector);
t = cross_product(b, c);
e = cross_product(a, t);
return (e);
end vector_triple_product;
end test_products;The output is:
a . b = 49 a x b = 5 5 -7 a . (b x c) = 6 a x (b x c) = -267 204 -3
Plain English
To run:
Start up.
Make a vector from 3 and 4 and 5.
Make another vector from 4 and 3 and 5.
Make a third vector from -5 and -12 and -13.
Write "A vector: " then the vector on the console.
Write "Another vector: " then the other vector on the console.
Write "A third vector: " then the third vector on the console.
Write "" on the console.
Compute a dot product of the vector and the other vector.
Write "Dot product between the vector and the other vector: " then the dot product on the console.
Compute a cross product of the vector and the other vector.
Write "Cross product between the vector and the other vector: " then the cross product on the console.
Compute a scalar triple product of the vector and the other vector and the third vector.
Write "Scalar triple product between the vector and the other vector and the third vector: " then the scalar triple product on the console.
Compute a vector triple product of the vector and the other vector and the third vector.
Write "Vector triple product between the vector and the other vector and the third vector: " then the vector triple product on the console.
Wait for the escape key.
Shut down.
A vector has a first number, a second number, and a third number.
To make a vector from a first number and a second number and a third number:
Put the first into the vector's first.
Put the second into the vector's second.
Put the third into the vector's third.
To put a vector into another vector:
Put the vector's first into the other vector's first.
Put the vector's second into the other vector's second.
Put the vector's third into the other vector's third.
To convert a vector into a string:
Append "(" then the vector's first then ", " then the vector's second then ", " then the vector's third then ")" to the string.
A dot product is a number.
To compute a dot product of a vector and another vector:
Put the vector's first times the other vector's first into a first number.
Put the vector's second times the other vector's second into a second number.
Put the vector's third times the other vector's third into a third number.
Put the first plus the second plus the third into the dot product.
A cross product is a vector.
To compute a cross product of a vector and another vector:
Put the vector's second times the other vector's third into a first number.
Put the vector's third times the other vector's second into a second number.
Put the vector's third times the other vector's first into a third number.
Put the vector's first times the other vector's third into a fourth number.
Put the vector's first times the other vector's second into a fifth number.
Put the vector's second times the other vector's first into a sixth number.
Make a result vector from the first minus the second and the third minus the fourth and the fifth minus the sixth.
Put the result into the cross product.
A scalar triple product is a number.
To compute a scalar triple product of a vector and another vector and a third vector:
Compute a cross product of the other vector and the third vector.
Compute a dot product of the vector and the cross product.
Put the dot product into the scalar triple product.
A vector triple product is a vector.
To compute a vector triple product of a vector and another vector and a third vector:
Compute a cross product of the other vector and the third vector.
Compute another cross product of the vector and the cross product.
Put the other cross product into the vector triple product.- Output:
A vector: (3, 4, 5) Another vector: (4, 3, 5) A third vector: (-5, -12, -13) Dot product between the vector and the other vector: 49 Cross product between the vector and the other vector: (5, 5, -7) Scalar triple product between the vector and the other vector and the third vector: 6 Vector triple product between the vector and the other vector and the third vector: (-267, 204, -3)
PowerShell
function dot-product($a,$b) {
$a[0]*$b[0] + $a[1]*$b[1] + $a[2]*$b[2]
}
function cross-product($a,$b) {
$v1 = $a[1]*$b[2] - $a[2]*$b[1]
$v2 = $a[2]*$b[0] - $a[0]*$b[2]
$v3 = $a[0]*$b[1] - $a[1]*$b[0]
@($v1,$v2,$v3)
}
function scalar-triple-product($a,$b,$c) {
dot-product $a (cross-product $b $c)
}
function vector-triple-product($a,$b) {
cross-product $a (cross-product $b $c)
}
$a = @(3, 4, 5)
$b = @(4, 3, 5)
$c = @(-5, -12, -13)
"a.b = $(dot-product $a $b)"
"axb = $(cross-product $a $b)"
"a.(bxc) = $(scalar-triple-product $a $b $c)"
"ax(bxc) = $(vector-triple-product $a $b $c)"
Output:
a.b = 49 axb = 5 5 -7 a.(bxc) = 6 ax(bxc) = -267 204 -3
Prolog
Works with SWI-Prolog.
dot_product([A1, A2, A3], [B1, B2, B3], Ans) :-
Ans is A1 * B1 + A2 * B2 + A3 * B3.
cross_product([A1, A2, A3], [B1, B2, B3], Ans) :-
T1 is A2 * B3 - A3 * B2,
T2 is A3 * B1 - A1 * B3,
T3 is A1 * B2 - A2 * B1,
Ans = [T1, T2, T3].
scala_triple(A, B, C, Ans) :-
cross_product(B, C, Temp),
dot_product(A, Temp, Ans).
vector_triple(A, B, C, Ans) :-
cross_product(B, C, Temp),
cross_product(A, Temp, Ans).
Output:
?- dot_product([3.0, 4.0, 5.0], [4.0, 3.0, 5.0], Ans). Ans = 49.0. ?- cross_product([3.0, 4.0, 5.0], [4.0, 3.0, 5.0], Ans). Ans = [5.0, 5.0, -7.0]. ?- scala_triple([3.0, 4.0, 5.0], [4.0, 3.0, 5.0], [-5.0, -12.0, -13.0], Ans). Ans = 6.0. ?- vector_triple([3.0, 4.0, 5.0], [4.0, 3.0, 5.0], [-5.0, -12.0, -13.0], Ans). Ans = [-267.0, 204.0, -3.0].
PureBasic
Structure vector
x.f
y.f
z.f
EndStructure
;convert vector to a string for display
Procedure.s toString(*v.vector)
ProcedureReturn "[" + StrF(*v\x, 2) + ", " + StrF(*v\y, 2) + ", " + StrF(*v\z, 2) + "]"
EndProcedure
Procedure.f dotProduct(*a.vector, *b.vector)
ProcedureReturn *a\x * *b\x + *a\y * *b\y + *a\z * *b\z
EndProcedure
Procedure crossProduct(*a.vector, *b.vector, *r.vector)
*r\x = *a\y * *b\z - *a\z * *b\y
*r\y = *a\z * *b\x - *a\x * *b\z
*r\z = *a\x * *b\y - *a\y * *b\x
EndProcedure
Procedure.f scalarTriple(*a.vector, *b.vector, *c.vector)
Protected r.vector
crossProduct(*b, *c, r)
ProcedureReturn dotProduct(*a, r)
EndProcedure
Procedure vectorTriple(*a.vector, *b.vector, *c.vector, *r.vector)
Protected r.vector
crossProduct(*b, *c, r)
crossProduct(*a, r, *r)
EndProcedure
If OpenConsole()
Define.vector a, b, c, r
a\x = 3: a\y = 4: a\z = 5
b\x = 4: b\y = 3: b\z = 5
c\x = -5: c\y = -12: c\z = -13
PrintN("a = " + toString(a) + ", b = " + toString(b) + ", c = " + toString(c))
PrintN("a . b = " + StrF(dotProduct(a, b), 2))
crossProduct(a, b, r)
PrintN("a x b = " + toString(r))
PrintN("a . b x c = " + StrF(scalarTriple(a, b, c), 2))
vectorTriple(a, b, c, r)
PrintN("a x b x c = " + toString(r))
Print(#CRLF$ + #CRLF$ + "Press ENTER to exit"): Input()
CloseConsole()
EndIf
Sample output:
a = [3.00, 4.00, 5.00], b = [4.00, 3.00, 5.00], c = [-5.00, -12.00, -13.00] a . b = 49.00 a x b = [5.00, 5.00, -7.00] a . b x c = 6.00 a x b x c = [-267.00, 204.00, -3.00]
Python
The solution is in the form of an Executable library.
def crossp(a, b):
'''Cross product of two 3D vectors'''
assert len(a) == len(b) == 3, 'For 3D vectors only'
a1, a2, a3 = a
b1, b2, b3 = b
return (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1)
def dotp(a,b):
'''Dot product of two eqi-dimensioned vectors'''
assert len(a) == len(b), 'Vector sizes must match'
return sum(aterm * bterm for aterm,bterm in zip(a, b))
def scalartriplep(a, b, c):
'''Scalar triple product of three vectors: "a . (b x c)"'''
return dotp(a, crossp(b, c))
def vectortriplep(a, b, c):
'''Vector triple product of three vectors: "a x (b x c)"'''
return crossp(a, crossp(b, c))
if __name__ == '__main__':
a, b, c = (3, 4, 5), (4, 3, 5), (-5, -12, -13)
print("a = %r; b = %r; c = %r" % (a, b, c))
print("a . b = %r" % dotp(a,b))
print("a x b = %r" % (crossp(a,b),))
print("a . (b x c) = %r" % scalartriplep(a, b, c))
print("a x (b x c) = %r" % (vectortriplep(a, b, c),))
- Output:
a = (3, 4, 5); b = (4, 3, 5); c = (-5, -12, -13) a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
- Note
The popular numpy package has functions for dot and cross products.
Quackery
[ 0 unrot witheach
[ over i^ peek *
rot + swap ]
drop ] is dotproduct ( [ [ --> n )
[ join
dup 1 peek over 5 peek *
swap
dup 2 peek over 4 peek *
swap dip -
dup 2 peek over 3 peek *
swap
dup 0 peek over 5 peek *
swap dip -
dup 0 peek over 4 peek *
swap
dup 1 peek swap 3 peek *
- join join ] is crossproduct ( [ [ --> [ )
[ crossproduct dotproduct ] is scalartriple ( [ [ [ --> n )
[ crossproduct crossproduct ] is vectortriple ( [ [ [ --> [ )
[ ' [ 3 4 5 ] ] is a ( --> [ )
[ ' [ 4 3 5 ] ] is b ( --> [ )
[ ' [ -5 -12 -13 ] ] is c ( --> [ )
a b dotproduct echo cr
a b crossproduct echo cr
a b c scalartriple echo cr
a b c vectortriple echo cr- Output:
49 [ 5 5 -7 ] 6 [ -267 204 -3 ]
R
#===============================================================
# Vector products
# R implementation
#===============================================================
a <- c(3, 4, 5)
b <- c(4, 3, 5)
c <- c(-5, -12, -13)
#---------------------------------------------------------------
# Dot product
#---------------------------------------------------------------
dotp <- function(x, y) {
if (length(x) == length(y)) {
sum(x*y)
}
}
#---------------------------------------------------------------
# Cross product
#---------------------------------------------------------------
crossp <- function(x, y) {
if (length(x) == 3 && length(y) == 3) {
c(x[2]*y[3] - x[3]*y[2], x[3]*y[1] - x[1]*y[3], x[1]*y[2] - x[2]*y[1])
}
}
#---------------------------------------------------------------
# Scalar triple product
#---------------------------------------------------------------
scalartriplep <- function(x, y, z) {
if (length(x) == 3 && length(y) == 3 && length(z) == 3) {
dotp(x, crossp(y, z))
}
}
#---------------------------------------------------------------
# Vector triple product
#---------------------------------------------------------------
vectortriplep <- function(x, y, z) {
if (length(x) == 3 && length(y) == 3 && length(z) == 3) {
crosssp(x, crossp(y, z))
}
}
#---------------------------------------------------------------
# Compute and print
#---------------------------------------------------------------
cat("a . b =", dotp(a, b))
cat("a x b =", crossp(a, b))
cat("a . (b x c) =", scalartriplep(a, b, c))
cat("a x (b x c) =", vectortriplep(a, b, c))
- Output:
a . b = 49 a x b = 5 5 -7 a . (b x c) = 6 a x (b x c) = -267 204 -3
Note: R has built-in functions for vector and matrix multiplications. Examples: "crossprod", %*% for inner and %o% for outer product.
Racket
#lang racket
(define (dot-product X Y)
(for/sum ([x (in-vector X)] [y (in-vector Y)]) (* x y)))
(define (cross-product X Y)
(define len (vector-length X))
(for/vector ([n len])
(define (ref V i) (vector-ref V (modulo (+ n i) len)))
(- (* (ref X 1) (ref Y 2)) (* (ref X 2) (ref Y 1)))))
(define (scalar-triple-product X Y Z)
(dot-product X (cross-product Y Z)))
(define (vector-triple-product X Y Z)
(cross-product X (cross-product Y Z)))
(define A '#(3 4 5))
(define B '#(4 3 5))
(define C '#(-5 -12 -13))
(printf "A = ~s\n" A)
(printf "B = ~s\n" B)
(printf "C = ~s\n" C)
(newline)
(printf "A . B = ~s\n" (dot-product A B))
(printf "A x B = ~s\n" (cross-product A B))
(printf "A . B x C = ~s\n" (scalar-triple-product A B C))
(printf "A x B x C = ~s\n" (vector-triple-product A B C))
Raku
(formerly Perl 6)
sub infix:<⋅> { [+] @^a »*« @^b }
sub infix:<⨯>([$a1, $a2, $a3], [$b1, $b2, $b3]) {
[ $a2*$b3 - $a3*$b2,
$a3*$b1 - $a1*$b3,
$a1*$b2 - $a2*$b1 ];
}
sub scalar-triple-product { @^a ⋅ (@^b ⨯ @^c) }
sub vector-triple-product { @^a ⨯ (@^b ⨯ @^c) }
my @a = <3 4 5>;
my @b = <4 3 5>;
my @c = <-5 -12 -13>;
say (:@a, :@b, :@c);
say "a ⋅ b = { @a ⋅ @b }";
say "a ⨯ b = <{ @a ⨯ @b }>";
say "a ⋅ (b ⨯ c) = { scalar-triple-product(@a, @b, @c) }";
say "a ⨯ (b ⨯ c) = <{ vector-triple-product(@a, @b, @c) }>";
- Output:
("a" => ["3", "4", "5"], "b" => ["4", "3", "5"], "c" => ["-5", "-12", "-13"])
a ⋅ b = 49
a ⨯ b = <5 5 -7>
a ⋅ (b ⨯ c) = 6
a ⨯ (b ⨯ c) = <-267 204 -3>
REXX
/*REXX program computes the products: dot, cross, scalar triple, and vector triple.*/
a= 3 4 5
b= 4 3 5 /*(positive numbers don't need quotes.)*/
c= '-5 -12 -13'
Call tellV 'vector A =', a /*show the A vector, aligned numbers.*/
Call tellV "vector B =", b /* " " B " " " */
Call tellV "vector C =", c /* " " C " " " */
Say ''
Call tellV ' dot product [A·B] =', dot(a,b)
Call tellV 'cross product [AxB] =', cross(a,b)
Call tellV 'scalar triple product [A·(BxC)] =', dot(a,cross(b,c))
Call tellV 'vector triple product [Ax(BxC)] =', cross(a,cross(b,c))
Exit /*stick a fork in it, we're all done. */
/*---------------------------------------------------------------------------*/
cross: Procedure
Arg a b c, u v w
Return b*w-c*v c*u-a*w a*v-b*u
dot: Procedure
Arg a b c, u v w
Return a*u + b*v + c*w
/*---------------------------------------------------------------------------*/
tellV: Procedure
Parse Arg name,x y z
w=max(4,length(x),length(y),length(z)) /*max width */
Say right(name,33) right(x,w) right(y,w) right(z,w) /*show vector. */
Return
- output when using the default internal inputs:
vector A = 3 4 5
vector B = 4 3 5
vector C = -5 -12 -13
dot product [AÀB] = 49
cross product [AxB] = 5 5 -7
scalar triple product [AÀ(BxC)] = 6
vector triple product [Ax(BxC)] = -267 204 -3
Ring
# Project : Vector products
d = list(3)
e = list(3)
a = [3, 4, 5]
b = [4, 3, 5]
c = [-5, -12, -13]
see "a . b = " + dot(a,b) + nl
cross(a,b,d)
see "a x b = (" + d[1] + ", " + d[2] + ", " + d[3] + ")" + nl
see "a . (b x c) = " + scalartriple(a,b,c) + nl
vectortriple(a,b,c,d)
def dot(a,b)
sum = 0
for n=1 to len(a)
sum = sum + a[n]*b[n]
next
return sum
func cross(a,b,d)
d = [a[2]*b[3]-a[3]*b[2], a[3]*b[1]-a[1]*b[3], a[1]*b[2]-a[2]*b[1]]
func scalartriple(a,b,c)
cross(b,c,d)
return dot(a,d)
func vectortriple(a,b,c,d)
cross(b,c,d)
cross(a,d,e)
see "a x (b x c) = (" + e[1] + ", " +e[2] + ", " + e[3] + ")"Output:
a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
RPL
Dot and cross products are built-in functions in RPL.
≪ → a b c
≪ a b DOT
a b CROSS
a b c CROSS DOT
a b c CROSS CROSS
≫ ≫
‘VPROD’ STO
[3 4 5] [4 3 5] [-5 -12 -13] VPROD
- Output:
4: 49 3: [ 5 5 -7 ] 2: 6 1: [ -267 204 -3 ]
Ruby
Dot product is also known as inner product. The standard library already defines Vector#inner_product and Vector# cross_product, so this program only defines the other two methods.
require 'matrix'
class Vector
def scalar_triple_product(b, c)
self.inner_product(b.cross_product c)
end
def vector_triple_product(b, c)
self.cross_product(b.cross_product c)
end
end
a = Vector[3, 4, 5]
b = Vector[4, 3, 5]
c = Vector[-5, -12, -13]
puts "a dot b = #{a.inner_product b}"
puts "a cross b = #{a.cross_product b}"
puts "a dot (b cross c) = #{a.scalar_triple_product b, c}"
puts "a cross (b cross c) = #{a.vector_triple_product b, c}"
Output:
a dot b = 49 a cross b = Vector[5, 5, -7] a dot (b cross c) = 6 a cross (b cross c) = Vector[-267, 204, -3]
Rust
#[derive(Debug)]
struct Vector {
x: f64,
y: f64,
z: f64,
}
impl Vector {
fn new(x: f64, y: f64, z: f64) -> Self {
Vector {
x: x,
y: y,
z: z,
}
}
fn dot_product(&self, other: &Vector) -> f64 {
(self.x * other.x) + (self.y * other.y) + (self.z * other.z)
}
fn cross_product(&self, other: &Vector) -> Vector {
Vector::new(self.y * other.z - self.z * other.y,
self.z * other.x - self.x * other.z,
self.x * other.y - self.y * other.x)
}
fn scalar_triple_product(&self, b: &Vector, c: &Vector) -> f64 {
self.dot_product(&b.cross_product(&c))
}
fn vector_triple_product(&self, b: &Vector, c: &Vector) -> Vector {
self.cross_product(&b.cross_product(&c))
}
}
fn main(){
let a = Vector::new(3.0, 4.0, 5.0);
let b = Vector::new(4.0, 3.0, 5.0);
let c = Vector::new(-5.0, -12.0, -13.0);
println!("a . b = {}", a.dot_product(&b));
println!("a x b = {:?}", a.cross_product(&b));
println!("a . (b x c) = {}", a.scalar_triple_product(&b, &c));
println!("a x (b x c) = {:?}", a.vector_triple_product(&b, &c));
}
Output:
a . b = 49
a x b = Vector { x: 5, y: 5, z: -7 }
a . (b x c) = 6
a x (b x c) = Vector { x: -267, y: 204, z: -3 }
Scala
case class Vector3D(x:Double, y:Double, z:Double) {
def dot(v:Vector3D):Double=x*v.x + y*v.y + z*v.z;
def cross(v:Vector3D)=Vector3D(y*v.z - z*v.y, z*v.x - x*v.z, x*v.y - y*v.x)
def scalarTriple(v1:Vector3D, v2:Vector3D)=this dot (v1 cross v2)
def vectorTriple(v1:Vector3D, v2:Vector3D)=this cross (v1 cross v2)
}
object VectorTest {
def main(args:Array[String])={
val a=Vector3D(3,4,5)
val b=Vector3D(4,3,5)
val c=Vector3D(-5,-12,-13)
println(" a . b : " + (a dot b))
println(" a x b : " + (a cross b))
println("a . (b x c) : " + (a scalarTriple(b, c)))
println("a x (b x c) : " + (a vectorTriple(b, c)))
}
}
- Output:
a . b : 49.0
a x b : Vector3D(5.0,5.0,-7.0)
a . (b x c) : 6.0
a x (b x c) : Vector3D(-267.0,204.0,-3.0)
Scheme
Using modified dot-product function from the Dot product task.
(define (dot-product A B)
(apply + (map * (vector->list A) (vector->list B))))
(define (cross-product A B)
(define len (vector-length A))
(define xp (make-vector (vector-length A) #f))
(let loop ((n 0))
(vector-set! xp n (-
(* (vector-ref A (modulo (+ n 1) len))
(vector-ref B (modulo (+ n 2) len)))
(* (vector-ref A (modulo (+ n 2) len))
(vector-ref B (modulo (+ n 1) len)))))
(if (eqv? len (+ n 1))
xp
(loop (+ n 1)))))
(define (scalar-triple-product A B C)
(dot-product A (cross-product B C)))
(define (vector-triple-product A B C)
(cross-product A (cross-product B C)))
(define A #( 3 4 5))
(define B #(4 3 5))
(define C #(-5 -12 -13))
(display "A = ")(display A)(newline)
(display "B = ")(display B)(newline)
(display "C = ")(display C)(newline)
(newline)
(display "A . B = ")(display (dot-product A B))(newline)
(display "A x B = ")(display (cross-product A B))(newline)
(display "A . B x C = ")(display (scalar-triple-product A B C))(newline)
(display "A x B x C = ") (display (vector-triple-product A B C))(newline)
Output:
A = #(3 4 5) B = #(4 3 5) C = #(-5 -12 -13) A . B = 49 A x B = #(5 5 -7) A . B x C = 6 A x B x C = #(-267 204 -3)
Seed7
The program below uses Seed7s capaibility to define operator symbols. The operators dot and X are defined with with priority 6 and assiciativity left-to-right.
$ include "seed7_05.s7i";
include "float.s7i";
const type: vec3 is new struct
var float: x is 0.0;
var float: y is 0.0;
var float: z is 0.0;
end struct;
const func vec3: vec3 (in float: x, in float: y, in float: z) is func
result
var vec3: aVector is vec3.value;
begin
aVector.x := x;
aVector.y := y;
aVector.z := z;
end func;
$ syntax expr: .(). dot .() is -> 6;
const func float: (in vec3: a) dot (in vec3: b) is
return a.x*b.x + a.y*b.y + a.z*b.z;
$ syntax expr: .(). X .() is -> 6;
const func vec3: (in vec3: a) X (in vec3: b) is
return vec3(a.y*b.z - a.z*b.y,
a.z*b.x - a.x*b.z,
a.x*b.y - a.y*b.x);
const func string: str (in vec3: v) is
return "(" <& v.x <& ", " <& v.y <& ", " <& v.z <& ")";
enable_output(vec3);
const func float: scalarTriple (in vec3: a, in vec3: b, in vec3: c) is
return a dot (b X c);
const func vec3: vectorTriple (in vec3: a, in vec3: b, in vec3: c) is
return a X (b X c);
const proc: main is func
local
const vec3: a is vec3(3.0, 4.0, 5.0);
const vec3: b is vec3(4.0, 3.0, 5.0);
const vec3: c is vec3(-5.0, -12.0, -13.0);
begin
writeln("a = " <& a <& ", b = " <& b <& ", c = " <& c);
writeln("a . b = " <& a dot b);
writeln("a x b = " <& a X b);
writeln("a .(b x c) = " <& scalarTriple(a, b, c));
writeln("a x(b x c) = " <& vectorTriple(a, b, c));
end func;- Output:
a = (3.0, 4.0, 5.0), b = (4.0, 3.0, 5.0), c = (-5.0, -12.0, -13.0) a . b = 49.0 a x b = (5.0, 5.0, -7.0) a .(b x c) = 6.0 a x(b x c) = (-267.0, 204.0, -3.0)
SETL
program vector_products;
a := [3, 4, 5];
b := [4, 3, 5];
c := [-5, -12, -13];
print(" a:", a);
print(" b:", b);
print(" c:", c);
print(" a . b:", a dot b);
print(" a x b:", a cross b);
print("a . (b x c):", a dot (b cross c));
print("a x (b x c):", a cross (b cross c));
op dot(a, b);
return a(1)*b(1) + a(2)*b(2) + a(3)*b(3);
end op;
op cross(a, b);
return [a(2)*b(3) - a(3)*b(2),
a(3)*b(1) - a(1)*b(3),
a(1)*b(2) - a(2)*b(1)];
end op;
end program;- Output:
a: [3 4 5]
b: [4 3 5]
c: [-5 -12 -13]
a . b: 49
a x b: [5 5 -7]
a . (b x c): 6
a x (b x c): [-267 204 -3]
Sidef
class MyVector(x, y, z) {
method ∙(vec) {
[self{:x,:y,:z}] »*« [vec{:x,:y,:z}] «+»
}
method ⨉(vec) {
MyVector(self.y*vec.z - self.z*vec.y,
self.z*vec.x - self.x*vec.z,
self.x*vec.y - self.y*vec.x)
}
method to_s {
"(#{x}, #{y}, #{z})"
}
}
var a = MyVector(3, 4, 5)
var b = MyVector(4, 3, 5)
var c = MyVector(-5, -12, -13)
say "a=#{a}; b=#{b}; c=#{c};"
say "a ∙ b = #{a ∙ b}"
say "a ⨉ b = #{a ⨉ b}"
say "a ∙ (b ⨉ c) = #{a ∙ (b ⨉ c)}"
say "a ⨉ (b ⨉ c) = #{a ⨉ (b ⨉ c)}"
- Output:
a=(3, 4, 5); b=(4, 3, 5); c=(-5, -12, -13); a ∙ b = 49 a ⨉ b = (5, 5, -7) a ∙ (b ⨉ c) = 6 a ⨉ (b ⨉ c) = (-267, 204, -3)
Simula
BEGIN
CLASS VECTOR(I,J,K); REAL I,J,K;;
REAL PROCEDURE DOTPRODUCT(A,B); REF(VECTOR) A,B;
DOTPRODUCT := A.I*B.I+A.J*B.J+A.K*B.K;
REF(VECTOR) PROCEDURE CROSSPRODUCT(A,B); REF(VECTOR) A,B;
CROSSPRODUCT :- NEW VECTOR(A.J*B.K - A.K*B.J,
A.K*B.I - A.I*B.K,
A.I*B.J - A.J*B.I);
REAL PROCEDURE SCALARTRIPLEPRODUCT(A,B,C); REF(VECTOR) A,B,C;
SCALARTRIPLEPRODUCT := DOTPRODUCT(A,CROSSPRODUCT(B,C));
REF(VECTOR) PROCEDURE VECTORTRIPLEPRODUCT(A,B,C); REF(VECTOR) A,B,C;
VECTORTRIPLEPRODUCT :- CROSSPRODUCT(A,CROSSPRODUCT(B,C));
PROCEDURE OUTR(X); REAL X;
OUTFIX(X,6,0);
PROCEDURE OUTVECTOR(A); REF(VECTOR) A;
BEGIN
OUTTEXT("("); OUTR(A.I);
OUTTEXT(", "); OUTR(A.J);
OUTTEXT(", "); OUTR(A.K); OUTTEXT(")");
END;
BEGIN
REF(VECTOR) A,B,C;
A :- NEW VECTOR(3, 4, 5);
B :- NEW VECTOR(4, 3, 5);
C :- NEW VECTOR(-5, -12, -13);
OUTTEXT("A = "); OUTVECTOR(A);
OUTIMAGE;
OUTTEXT("B = "); OUTVECTOR(B);
OUTIMAGE;
OUTTEXT("C = "); OUTVECTOR(C);
OUTIMAGE;
OUTTEXT("A . B = "); OUTR(DOTPRODUCT(A,B));
OUTIMAGE;
OUTTEXT("A X B = "); OUTVECTOR(CROSSPRODUCT(A,B));
OUTIMAGE;
OUTTEXT("A . (B X C) = "); OUTR(SCALARTRIPLEPRODUCT(A,B,C));
OUTIMAGE;
OUTTEXT("A X (B X C) = "); OUTVECTOR(VECTORTRIPLEPRODUCT(A,B,C));
OUTIMAGE;
END;
END;- Output:
A = (3.000000, 4.000000, 5.000000) B = (4.000000, 3.000000, 5.000000) C = (-5.000000, -12.000000, -13.000000) A . B = 49.000000 A X B = (5.000000, 5.000000, -7.000000) A . (B X C) = 6.000000 A X (B X C) = (-267.000000, 204.000000, -3.000000)
Stata
mata
real scalar sprod(real colvector u, real colvector v) {
return(u[1]*v[1] + u[2]*v[2] + u[3]*v[3])
}
real colvector vprod(real colvector u, real colvector v) {
return(u[2]*v[3]-u[3]*v[2]\u[3]*v[1]-u[1]*v[3]\u[1]*v[2]-u[2]*v[1])
}
real scalar striple(real colvector u, real colvector v, real colvector w) {
return(sprod(u, vprod(v, w)))
}
real colvector vtriple(real colvector u, real colvector v, real colvector w) {
return(vprod(u, vprod(v, w)))
}
a = 3\4\5
b = 4\3\5
c = -5\-12\-13
sprod(a, b)
49
vprod(a, b)
1
+------+
1 | 5 |
2 | 5 |
3 | -7 |
+------+
striple(a, b, c)
6
vtriple(a, b, c)
1
+--------+
1 | -267 |
2 | 204 |
3 | -3 |
+--------+
end
Swift
import Foundation
infix operator • : MultiplicationPrecedence
infix operator × : MultiplicationPrecedence
public struct Vector {
public var x = 0.0
public var y = 0.0
public var z = 0.0
public init(x: Double, y: Double, z: Double) {
(self.x, self.y, self.z) = (x, y, z)
}
public static func • (lhs: Vector, rhs: Vector) -> Double {
return lhs.x * rhs.x + lhs.y * rhs.y + lhs.z * rhs.z
}
public static func × (lhs: Vector, rhs: Vector) -> Vector {
return Vector(
x: lhs.y * rhs.z - lhs.z * rhs.y,
y: lhs.z * rhs.x - lhs.x * rhs.z,
z: lhs.x * rhs.y - lhs.y * rhs.x
)
}
}
let a = Vector(x: 3, y: 4, z: 5)
let b = Vector(x: 4, y: 3, z: 5)
let c = Vector(x: -5, y: -12, z: -13)
print("a: \(a)")
print("b: \(b)")
print("c: \(c)")
print()
print("a • b = \(a • b)")
print("a × b = \(a × b)")
print("a • (b × c) = \(a • (b × c))")
print("a × (b × c) = \(a × (b × c))")
- Output:
a: Vector(x: 3.0, y: 4.0, z: 5.0) b: Vector(x: 4.0, y: 3.0, z: 5.0) c: Vector(x: -5.0, y: -12.0, z: -13.0) a • b = 49.0 a × b = Vector(x: 5.0, y: 5.0, z: -7.0) a • (b × c) = 6.0 a × (b × c) = Vector(x: -267.0, y: 204.0, z: -3.0)
Tcl
proc dot {A B} {
lassign $A a1 a2 a3
lassign $B b1 b2 b3
expr {$a1*$b1 + $a2*$b2 + $a3*$b3}
}
proc cross {A B} {
lassign $A a1 a2 a3
lassign $B b1 b2 b3
list [expr {$a2*$b3 - $a3*$b2}] \
[expr {$a3*$b1 - $a1*$b3}] \
[expr {$a1*$b2 - $a2*$b1}]
}
proc scalarTriple {A B C} {
dot $A [cross $B $C]
}
proc vectorTriple {A B C} {
cross $A [cross $B $C]
}
Demonstrating:
set a {3 4 5}
set b {4 3 5}
set c {-5 -12 -13}
puts "a • b = [dot $a $b]"
puts "a x b = [cross $a $b]"
puts "a • b x c = [scalarTriple $a $b $c]"
puts "a x b x c = [vectorTriple $a $b $c]"
Output:
a • b = 49 a x b = 5 5 -7 a • b x c = 6 a x b x c = -267 204 -3
uBasic/4tH
Since uBasic/4tH has only one single array, we use its variables to hold the offsets of the vectors. A similar problem arises when local vectors are required.
a = 0 ' use variables for vector addresses
b = a + 3
c = b + 3
d = c + 3
Proc _Vector (a, 3, 4, 5) ' initialize the vectors
Proc _Vector (b, 4, 3, 5)
Proc _Vector (c, -5, -12, -13)
Print "a . b = "; FUNC(_FNdot(a, b))
Proc _Cross (a, b, d)
Print "a x b = (";@(d+0);", ";@(d+1);", ";@(d+2);")"
Print "a . (b x c) = "; FUNC(_FNscalarTriple(a, b, c))
Proc _VectorTriple (a, b, c, d)
Print "a x (b x c) = (";@(d+0);", ";@(d+1);", ";@(d+2);")"
End
_FNdot Param (2)
Return ((@(a@+0)*@(b@+0))+(@(a@+1)*@(b@+1))+(@(a@+2)*@(b@+2)))
_Vector Param (4) ' initialize a vector
@(a@ + 0) = b@
@(a@ + 1) = c@
@(a@ + 2) = d@
Return
_Cross Param (3)
@(c@+0) = @(a@ + 1) * @(b@ + 2) - @(a@ + 2) * @(b@ + 1)
@(c@+1) = @(a@ + 2) * @(b@ + 0) - @(a@ + 0) * @(b@ + 2)
@(c@+2) = @(a@ + 0) * @(b@ + 1) - @(a@ + 1) * @(b@ + 0)
Return
_FNscalarTriple Param (3)
Local (1) ' a "local" vector
d@ = d + 3 ' (best effort) ;-)
Proc _Cross(b@, c@, d@)
Return (FUNC(_FNdot(a@, d@)))
_VectorTriple Param(4)
Local (1) ' a "local" vector
e@ = d + 3 ' (best effort) ;-)
Proc _Cross (b@, c@, e@)
Proc _Cross (a@, e@, d@)
Return
- Output:
a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3) 0 OK, 0:1370
VBA
Option Base 1
Function dot_product(a As Variant, b As Variant) As Variant
dot_product = WorksheetFunction.SumProduct(a, b)
End Function
Function cross_product(a As Variant, b As Variant) As Variant
cross_product = Array(a(2) * b(3) - a(3) * b(2), a(3) * b(1) - a(1) * b(3), a(1) * b(2) - a(2) * b(1))
End Function
Function scalar_triple_product(a As Variant, b As Variant, c As Variant) As Variant
scalar_triple_product = dot_product(a, cross_product(b, c))
End Function
Function vector_triple_product(a As Variant, b As Variant, c As Variant) As Variant
vector_triple_product = cross_product(a, cross_product(b, c))
End Function
Public Sub main()
a = [{3, 4, 5}]
b = [{4, 3, 5}]
c = [{-5, -12, -13}]
Debug.Print " a . b = "; dot_product(a, b)
Debug.Print " a x b = "; "("; Join(cross_product(a, b), ", "); ")"
Debug.Print "a . (b x c) = "; scalar_triple_product(a, b, c)
Debug.Print "a x (b x c) = "; "("; Join(vector_triple_product(a, b, c), ", "); ")"
End Sub
- Output:
a . b = 49
a x b = (5, 5, -7)
a . (b x c) = 6
a x (b x c) = (-267, 204, -3)
Visual Basic .NET
Class: Vector3D
Public Class Vector3D
Private _x, _y, _z As Double
Public Sub New(ByVal X As Double, ByVal Y As Double, ByVal Z As Double)
_x = X
_y = Y
_z = Z
End Sub
Public Property X() As Double
Get
Return _x
End Get
Set(ByVal value As Double)
_x = value
End Set
End Property
Public Property Y() As Double
Get
Return _y
End Get
Set(ByVal value As Double)
_y = value
End Set
End Property
Public Property Z() As Double
Get
Return _z
End Get
Set(ByVal value As Double)
_z = value
End Set
End Property
Public Function Dot(ByVal v2 As Vector3D) As Double
Return (X * v2.X) + (Y * v2.Y) + (Z * v2.Z)
End Function
Public Function Cross(ByVal v2 As Vector3D) As Vector3D
Return New Vector3D((Y * v2.Z) - (Z * v2.Y), _
(Z * v2.X) - (X * v2.Z), _
(X * v2.Y) - (Y * v2.X))
End Function
Public Function ScalarTriple(ByVal v2 As Vector3D, ByVal v3 As Vector3D) As Double
Return Me.Dot(v2.Cross(v3))
End Function
Public Function VectorTriple(ByRef v2 As Vector3D, ByVal v3 As Vector3D) As Vector3D
Return Me.Cross(v2.Cross(v3))
End Function
Public Overrides Function ToString() As String
Return String.Format("({0}, {1}, {2})", _x, _y, _z)
End Function
End Class
Module: Module1
Module Module1
Sub Main()
Dim v1 As New Vector3D(3, 4, 5)
Dim v2 As New Vector3D(4, 3, 5)
Dim v3 As New Vector3D(-5, -12, -13)
Console.WriteLine("v1: {0}", v1.ToString())
Console.WriteLine("v2: {0}", v2.ToString())
Console.WriteLine("v3: {0}", v3.ToString())
Console.WriteLine()
Console.WriteLine("v1 . v2 = {0}", v1.Dot(v2))
Console.WriteLine("v1 x v2 = {0}", v1.Cross(v2).ToString())
Console.WriteLine("v1 . (v2 x v3) = {0}", v1.ScalarTriple(v2, v3))
Console.WriteLine("v1 x (v2 x v3) = {0}", v1.VectorTriple(v2, v3))
End Sub
End Module
Output:
v1: (3, 4, 5) v2: (4, 3, 5) v3: (-5, -12, -13) v1 . v2 = 49 v1 x v2 = (5, 5, -7) v1 . (v2 x v3) = 6 v1 x (v2 x v3) = (-267, 204, -3)
V (Vlang)
struct Vector {
x f64
y f64
z f64
}
const (
a = Vector{3, 4, 5}
b = Vector{4, 3, 5}
c = Vector{-5, -12, -13}
)
fn dot(a Vector, b Vector) f64 {
return a.x*b.x + a.y*b.y + a.z*b.z
}
fn cross(a Vector, b Vector) Vector {
return Vector{a.y*b.z - a.z*b.y, a.z*b.x - a.x*b.z, a.x*b.y - a.y*b.x}
}
fn s3(a Vector, b Vector, c Vector) f64 {
return dot(a, cross(b, c))
}
fn v3(a Vector, b Vector, c Vector) Vector {
return cross(a, cross(b, c))
}
fn main() {
println(dot(a, b))
println(cross(a, b))
println(s3(a, b, c))
println(v3(a, b, c))
}- Output:
49.
Vector{
x: 5
y: 5
z: -7
}
6.
Vector{
x: -267
y: 204
z: -3
}
Wortel
@let {
dot &[a b] @sum @mapm ^* [a b]
cross &[a b] [[
-*a.1 b.2 *a.2 b.1
-*a.2 b.0 *a.0 b.2
-*a.0 b.1 *a.1 b.0
]]
scalarTripleProduct &[a b c] !!dot a !!cross b c
vectorTripleProduct &[a b c] !!cross a !!cross b c
a [3 4 5]
b [4 3 5]
c [5N 12N 13N]
[[
!!dot a b
!!cross a b
@!scalarTripleProduct [a b c]
@!vectorTripleProduct [a b c]
]]
}Returns:
[49 [5 5 -7] 6 [-267 204 -3]]
Wren
class Vector3D {
construct new(x, y, z) {
if (x.type != Num || y.type != Num || z.type != Num) Fiber.abort("Arguments must be numbers.")
_x = x
_y = y
_z = z
}
x { _x }
y { _y }
z { _z }
dot(v) {
if (v.type != Vector3D) Fiber.abort("Argument must be a Vector3D.")
return _x * v.x + _y * v.y + _z * v.z
}
cross(v) {
if (v.type != Vector3D) Fiber.abort("Argument must be a Vector3D.")
return Vector3D.new(_y*v.z - _z*v.y, _z*v.x - _x*v.z, _x*v.y - _y*v.x)
}
scalarTriple(v, w) {
if ((v.type != Vector3D) || (w.type != Vector3D)) Fiber.abort("Arguments must be Vector3Ds.")
return this.dot(v.cross(w))
}
vectorTriple(v, w) {
if ((v.type != Vector3D) || (w.type != Vector3D)) Fiber.abort("Arguments must be Vector3Ds.")
return this.cross(v.cross(w))
}
toString { [_x, _y, _z].toString }
}
var a = Vector3D.new(3, 4, 5)
var b = Vector3D.new(4, 3, 5)
var c = Vector3D.new(-5, -12, -13)
System.print("a = %(a)")
System.print("b = %(b)")
System.print("c = %(c)")
System.print()
System.print("a . b = %(a.dot(b))")
System.print("a x b = %(a.cross(b))")
System.print("a . b x c = %(a.scalarTriple(b, c))")
System.print("a x b x c = %(a.vectorTriple(b, c))")
- Output:
a = [3, 4, 5] b = [4, 3, 5] c = [-5, -12, -13] a . b = 49 a x b = [5, 5, -7] a . b x c = 6 a x b x c = [-267, 204, -3]
Alternatively, using the above module to produce exactly the same output as before:
import "./vector" for Vector3
var a = Vector3.new(3, 4, 5)
var b = Vector3.new(4, 3, 5)
var c = Vector3.new(-5, -12, -13)
System.print("a = %(a)")
System.print("b = %(b)")
System.print("c = %(c)")
System.print()
System.print("a . b = %(a.dot(b))")
System.print("a x b = %(a.cross(b))")
System.print("a . b x c = %(a.scalarTripleProd(b, c))")
System.print("a x b x c = %(a.vectorTripleProd(b, c))")
XBS
#> Vector3 Class <#
class Vector3 {
construct=func(self,x,y,z){
self:x=x;
self:y=y;
self:z=z;
};
ToString=func(self){
send self.x+", "+self.y+", "+self.z;
};
Magnitude=func(self){
send math.sqrt((self.x^2)+(self.y^2)+(self.z^2));
};
Normalize=func(self){
set Mag = self::Magnitude();
send new Vector3 with [self.x/Mag,self.y/Mag,self.z/Mag];
};
Dot=func(self,v2){
send (self.x*v2.x)+(self.y*v2.y)+(self.z*v2.z);
};
__add=func(self,x){
if (type(x)=="object"){
send new Vector3 with [self.x+x.x,self.y+x.y,self.z+x.z];
} else {
send new Vector3 with [self.x+x,self.y+x,self.z+x];
}
};
__sub=func(self,x){
if (type(x)=="object"){
send new Vector3 with [self.x-x.x,self.y-x.y,self.z-x.z];
} else {
send new Vector3 with [self.x-x,self.y-x,self.z-x];
}
};
__mul=func(self,x){
if (type(x)=="object"){
send new Vector3 with [self.x*x.x,self.y*x.y,self.z*x.z];
} else {
send new Vector3 with [self.x*x,self.y*x,self.z*x];
}
};
__div=func(self,x){
if (type(x)=="object"){
send new Vector3 with [self.x/x.x,self.y/x.y,self.z/x.z];
} else {
send new Vector3 with [self.x/x,self.y/x,self.z/x];
}
};
__pow=func(self,x){
if (type(x)=="object"){
send new Vector3 with [self.x^x.x,self.y^x.y,self.z^x.z];
} else {
send new Vector3 with [self.x^x,self.y^x,self.z^x];
}
};
__mod=func(self,x){
if (type(x)=="object"){
send new Vector3 with [self.x%x.x,self.y%x.y,self.z%x.z];
} else {
send new Vector3 with [self.x%x,self.y%x,self.z%x];
}
};
Reflect=func(self,v2){
set Normal = self::Normalize();
set Direction = v2::Normalize();
send (Normal*(2*Normal::Dot(Direction)))-Direction;
};
Cross=func(self,v2){
send new Vector3 with [(self.y*v2.z)-(self.z*v2.y),(self.z*v2.x)-(self.x*v2.z),(self.x*v2.y)-(self.y*v2.x)];
};
}
math:deg=func(x){
send x*(180/math.PI);
}
math:rad=func(x){
send x*(math.PI/180);
}
set a = new Vector3 with [3,4,5];
set b = new Vector3 with [4,3,5];
set c = new Vector3 with [-5,-12,-13];
log("Dot: ",a::Dot(b));
log("Cross: ",a::Cross(b)::ToString());
log("Scalar Triple: ",a::Dot(b::Cross(c)));
log("Vector Triple: ",a::Cross(b::Cross(c))::ToString());- Output:
Dot: 49 Cross: 5, 5, -7 Scalar Triple: 6 Vector Triple: -267, 204, -3
XPL0
include c:\cxpl\codes; \intrinsic 'code' declarations
func DotProd(A, B); \Return the dot product of two 3D vectors
int A, B; \A ù B
return A(0)*B(0) + A(1)*B(1) + A(2)*B(2);
proc CrossProd(A, B, C); \Calculate the cross product of two 3D vectors
int A, B, C; \C:= A x B
[C(0):= A(1)*B(2) - A(2)*B(1);
C(1):= A(2)*B(0) - A(0)*B(2);
C(2):= A(0)*B(1) - A(1)*B(0);
]; \CrossProd
func ScalarTriProd(A, B, C); \Return the scalar triple product
int A, B, C; \A ù (B x C)
int D(3);
[CrossProd(B, C, D);
return DotProd(A, D);
]; \ScalarTriProd
proc VectTriProd(A, B, C, D); \Calculate the vector triple product
int A, B, C, D; \D:= A x (B x C)
int E(3);
[CrossProd(B, C, E);
CrossProd(A, E, D);
]; \CrossProd
int A, B, C, D(3);
[A:= [3, 4, 5]; B:= [4, 3, 5]; C:= [-5, -12, -13];
IntOut(0, DotProd(A,B)); CrLf(0);
CrossProd(A, B, D);
IntOut(0, D(0)); ChOut(0, 9\tab\);
IntOut(0, D(1)); ChOut(0, 9\tab\);
IntOut(0, D(2)); CrLf(0);
IntOut(0, ScalarTriProd(A,B,C)); CrLf(0);
VectTriProd(A, B, C, D);
IntOut(0, D(0)); ChOut(0, 9\tab\);
IntOut(0, D(1)); ChOut(0, 9\tab\);
IntOut(0, D(2)); CrLf(0);
]Output:
49 5 5 -7 6 -267 204 -3
zkl
Since the input vectors are all int, the output is int. For a float output, use float data (or convert) in the input vectors and change sum() to sum(0.0) (in dotp).
The [(a1,a2,a3)] parameter notation just means add a preamble to the function body to do list assignment: a1,a2,a3:=arglist[0]. Since we don't need the vector as such, don't bother to name it (in the parameter list)
fcn dotp(a,b){ a.zipWith('*,b).sum() } //1 slow but concise
fcn crossp([(a1,a2,a3)],[(b1,b2,b3)]) //2
{ return(a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1) }a,b,c := T(3,4,5), T(4,3,5), T(-5,-12,-13);
dotp(a,b).println(); //5 --> 49
crossp(a,b).println(); //6 --> (5,5,-7)
dotp(a, crossp(b,c)).println(); //7 --> 6
crossp(a, crossp(b,c)).println(); //8 --> (-267,204,-3)- Output:
49 L(5,5,-7) 6 L(-267,204,-3)
Or, using the GNU Scientific Library:
var [const] GSL=Import("zklGSL"); // libGSL (GNU Scientific Library)
a:=GSL.VectorFromData( 3, 4, 5);
b:=GSL.VectorFromData( 4, 3, 5);
c:=GSL.VectorFromData(-5,-12,-13);
(a*b).println(); // 49, dot product
a.copy().crossProduct(b) // (5,5,-7) cross product, in place
.format().println();
(a*(b.copy().crossProduct(c))).println(); // 6 scalar triple product
(a.crossProduct(b.crossProduct(c))) // (-267,204,-3) vector triple product, in place
.format().println();- Output:
49 5.00,5.00,-7.00 6 -267.00,204.00,-3.00
Categories:
- Programming Tasks
- Solutions by Programming Task
- Wolfram Linked
- 11l
- Action!
- Ada
- ALGOL 68
- ALGOL W
- APL
- AppleScript
- Arturo
- AutoHotkey
- AWK
- BASIC256
- BBC BASIC
- BQN
- C
- C sharp
- C++
- Ceylon
- Clojure
- CLU
- Common Lisp
- Cowgol
- Crystal
- D
- Delphi
- DuckDB
- EasyLang
- EchoLisp
- Elixir
- Erlang
- ERRE
- Euphoria
- F Sharp
- Factor
- Fantom
- Forth
- Forth Scientific Library
- Fortran
- FreeBASIC
- FunL
- GAP
- GLSL
- Go
- Groovy
- Haskell
- Icon
- Unicon
- J
- Java
- JavaScript
- Jq
- Julia
- K
- Kotlin
- Ksh
- Lambdatalk
- Liberty BASIC
- Lingo
- Lua
- M2000 Interpreter
- Maple
- Mathematica
- Wolfram Language
- MATLAB
- Octave
- Mercury
- MiniScript
- МК-61/52
- Modula-2
- Nemerle
- Never
- Nim
- Objeck
- OCaml
- OoRexx
- PARI/GP
- Pascal
- PascalABC.NET
- Perl
- Phix
- Phix/basics
- Phixmonti
- PHP
- Picat
- PicoLisp
- PL/I
- Plain English
- PowerShell
- Prolog
- PureBasic
- Python
- Quackery
- R
- Racket
- Raku
- REXX
- Ring
- RPL
- Ruby
- Rust
- Scala
- Scheme
- Seed7
- SETL
- Sidef
- Simula
- Stata
- Swift
- Tcl
- UBasic/4tH
- VBA
- Visual Basic .NET
- V (Vlang)
- Wortel
- Wren
- Wren-vector
- XBS
- XPL0
- Zkl
- Geometry