Vector products: Difference between revisions
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a x (b x c) = -267 204 -3 |
a x (b x c) = -267 204 -3 |
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</pre> |
</pre> |
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=={{header|PureBasic}}== |
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<lang PureBasic>Structure vector |
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x.f |
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y.f |
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z.f |
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EndStructure |
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;convert vector to a string for display |
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Procedure.s toString(*v.vector) |
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ProcedureReturn "[" + StrF(*v\x, 2) + ", " + StrF(*v\y, 2) + ", " + StrF(*v\z, 2) + "]" |
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EndProcedure |
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Procedure.f dotProduct(*a.vector, *b.vector) |
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ProcedureReturn *a\x * *b\x + *a\y * *b\y + *a\z * *b\z |
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EndProcedure |
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Procedure crossProduct(*a.vector, *b.vector, *r.vector) |
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*r\x = *a\y * *b\z - *a\z * *b\y |
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*r\y = *a\z * *b\x - *a\x * *b\z |
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*r\z = *a\x * *b\y - *a\y * *b\x |
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EndProcedure |
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Procedure.f scalarTriple(*a.vector, *b.vector, *c.vector) |
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Protected r.vector |
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crossProduct(*b, *c, r) |
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ProcedureReturn dotProduct(*a, r) |
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EndProcedure |
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Procedure vectorTriple(*a.vector, *b.vector, *c.vector, *r.vector) |
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Protected r.vector |
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crossProduct(*b, *c, r) |
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crossProduct(*a, r, *r) |
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EndProcedure |
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If OpenConsole() |
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Define.vector a, b, c, r |
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a\x = 3: a\y = 4: a\z = 5 |
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b\x = 4: b\y = 3: b\z = 5 |
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c\x = -5: c\y = -12: c\z = -13 |
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PrintN("a = " + toString(a) + ", b = " + toString(b) + ", c = " + toString(c)) |
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PrintN("a . b = " + StrF(dotProduct(a, b), 2)) |
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crossProduct(a, b, r) |
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PrintN("a x b = " + toString(r)) |
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PrintN("a . b x c = " + StrF(scalarTriple(a, b, c), 2)) |
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vectorTriple(a, b, c, r) |
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PrintN("a x b x c = " + toString(r)) |
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Print(#CRLF$ + #CRLF$ + "Press ENTER to exit"): Input() |
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CloseConsole() |
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EndIf</lang> |
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Sample output: |
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<pre>a = [3.00, 4.00, 5.00], b = [4.00, 3.00, 5.00], c = [-5.00, -12.00, -13.00] |
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a . b = 49.00 |
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a x b = [5.00, 5.00, -7.00] |
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a . b x c = 6.00 |
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a x b x c = [-267.00, 204.00, -3.00]</pre> |
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=={{header|Python}}== |
=={{header|Python}}== |
Revision as of 22:19, 21 June 2011
You are encouraged to solve this task according to the task description, using any language you may know.
Define a vector 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 vectors A = (a1, a2, a3); B = (b1, b2, b3);
and C = (c1, c2, c3);
then the following common vector products are defined:
- The dot product
- A • B =
a1b1 + a2b2 + a3b3;
a scalar quantity
- The cross product
- A x B =
(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1);
a vector quantity
- The scalar triple product
- A • (B x C); a scalar quantity
- The vector triple product
- A x (B x C); a vector quantity
- Task description
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.
- Create a function to compute the scalar triple product of three vectors.
- 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
using the scaler triple product function. - Compute and display:
a x b x c
using the vector triple product function.
- References
- Dot product here on RC.
- A starting page to the Wolfram Mathworld information on vector multiplication.
- Wikipedias dot product, cross product and triple product entries.
- C.f.
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: <lang Ada>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;</lang>
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.
<lang algol68>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)))
)</lang> 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)
C#
<lang csharp>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)); }
}</lang> Output: <lang>49 5;5;-7 6 -267;204;-3</lang>
C++
<lang cpp>#include <iostream>
template< class T > class D3Vector {
template< class U > friend std::ostream & operator<<( std::ostream & , const D3Vector & ) ;
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 ;
}</lang>
Output:
a . b : 49 a x b : ( 5 , 5 , -7 ) a . b x c : 6 a x b x c : ( -267 , 204 , -3 )
D
<lang d>import std.stdio, std.algorithm, std.conv;
struct V3 {
union { struct { double x, y, z; } double[3] v; }
pure nothrow double dot(in V3 rhs) const { return x*rhs.x + y*rhs.y + z*rhs.z; }
pure nothrow V3 cross(in V3 rhs) const { return V3(y*rhs.z - z*rhs.y, z*rhs.x - x*rhs.z, x*rhs.y - y*rhs.x); }
string toString() const { return text(v); }
}
pure nothrow double scalarTriple(in V3 a, in V3 b, in V3 c) {
return a.dot(b.cross(c));
}
pure nothrow V3 vectorTriple(in V3 a, in V3 b, in V3 c) {
return a.cross(b.cross(c));
}
void main() {
const V3 a = {3,4,5}, b = V3(4,3,5), c = V3(-5,-12,-13); writefln("a = %s, b = %s, c = %s", a, b, 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));
}</lang> 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]
Fortran
Specialized for 3-dimensional vectors.
<lang fortran>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</lang> Output
49.0000 5.00000 5.00000 -7.00000 6.00000 -267.000 204.000 -3.00000
GAP
<lang 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 ]</lang>
Go
<lang 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))
}</lang> Output:
49 {5 5 -7} 6 {-267 204 -3}
Icon and Unicon
<lang icon># record type to store a 3D vector record Vector3D(x, y, z)
- procedure to display vector as a string
procedure toString (vector)
return "(" || vector.x || ", " || vector.y || ", " || vector.z || ")"
end
procedure dotProduct (a, b)
return a.x * b.x + a.y * b.y + a.z * b.z
end
procedure crossProduct (a, b)
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 return Vector3D(x, y, z)
end
procedure scalarTriple (a, b, c)
return dotProduct (a, crossProduct (b, c))
end
procedure vectorTriple (a, b, c)
return crossProduct (a, crossProduct (b, c))
end
- main procedure, to run given test
procedure main ()
a := Vector3D(3, 4, 5) b := Vector3D(4, 3, 5) c := Vector3D(-5, -12, -13)
writes ("A.B : " || toString(a) || "." || toString(b) || " = ") write (dotProduct (a, b)) writes ("AxB : " || toString(a) || "x" || toString(b) || " = ") write (toString(crossProduct (a, b))) writes ("A.(BxC) : " || toString(a) || ".(" || toString(b) || "x" || toString(c) || ") = ") write (scalarTriple (a, b, c)) writes ("Ax(BxC) : " || toString(a) || "x(" || toString(b) || "x" || toString(c) || ") = ") write (toString(vectorTriple (a, b, c)))
end </lang>
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
Based on j:Essays/Complete Tensor: <lang j>CT=: C.!.2 @ (#:i.) @ $~ ip=: +/ .* NB. inner product cross=: ] ip CT@#@[ ip [</lang>
An alternative definition for cross (based on finding the determinant of a 3 by 3 matrix where one row is unit vectors) could be:
<lang j>cross=: [: > [: -&.>/ .(*&.>) (<"1=i.3) , ,:&:(<"0)</lang>
Implementation:<lang j>a=: 3 4 5 b=: 4 3 5 c=: -5 12 13
A=: 0 {:: ] NB. contents of the first box on the right B=: 1 {:: ] NB. contents of the second box on the right C=: 2 {:: ] NB. contents of the third box on the right
dotP=: A ip B crossP=: A cross B scTriP=: A ip B cross C veTriP=: A cross B cross C</lang>
Required example:
<lang j> dotP a;b 49
crossP a;b
5 5 _7
scTriP a;b;c
6
veTriP a;b;c
_267 204 _3</lang>
Java
All operations which return vectors give vectors containing Double
s.
<lang java5>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)); }
}</lang> Output:
49.0 <5.0, 5.0, -7.0> 6.0 <-267.0, 204.0, -3.0>
Liberty BASIC
<lang lb> 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 SUB </lang>
Lua
<lang 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 )</lang>
49 5 5 -7 6 -267 204 -3
Octave
Octave handles naturally vectors / matrices.
<lang octave>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)</lang>
PARI/GP
<lang parigp>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)</lang> Output:
49 [5, 5, -7] 6 [-267, 204, -3]
Perl
<lang 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 ^ $c), "\n";</lang>
Output:
a = (3 4 5) b = (4 3 5) c = (-5 -12 -13) (3 4 5) . (4 3 5) = 49 (3 4 5) x (4 3 5) = (5 5 -7) (3 4 5) . ((4 3 5) x (-5 -12 -13)) = 6 (3 4 5) x ((4 3 5) x (-5 -12 -13)) = (-267 204 -3)
Perl 6
<lang perl6>sub infix:<⋅> { [+] @^a »*« @^b }
sub infix:<⨯>([$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).perl; 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) }>";</lang> 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>
PicoLisp
<lang 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)) )</lang>
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
<lang 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; </lang> Results:
a . b = 49 a x b = 5 5 -7 a . (b x c) = 6 a x (b x c) = -267 204 -3
<lang> /* 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; </lang>
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
PureBasic
<lang 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</lang> 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. <lang python>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 =", dotp(a,b)) print("a x b =", crossp(a,b)) print("a . (b x c) =", scalartriplep(a, b, c)) print("a x (b x c) =", vectortriplep(a, b, c))</lang>
- Sample 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.
Ruby
Dot product is also known as inner product. The standard library already defines Vector#inner_product, so this program only defines the other three methods.
<lang ruby>require 'matrix'
class Vector
def cross_product(v) unless size == 3 && v.size == 3 raise ArgumentError, "Vectors must have size 3" end Vector[self[1] * v[2] - self[2] * v[1], self[2] * v[0] - self[0] * v[2], self[0] * v[1] - self[1] * v[0]] end
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}"</lang>
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]
Scheme
Using modified dot-product function from the Dot product task. <lang scheme>(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)</lang>
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)
Tcl
<lang 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]
}</lang> Demonstrating: <lang tcl>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]"</lang>
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 <lang vbnet>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</lang>
Module: Module1 <lang vbnet>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</lang>
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)