Talk:Geometric algebra: Difference between revisions
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::::::::::::But "geometric product" is not a part of the definition of "vector". You do not need a "geometric product" for a vector to be a vector, and in fact in the general case vectors exist without any geometric product being defined. Meanwhile, the axioms of the of the algebra require that a scalar product can be applied to i, j and k. So the reader should indeed be able to assume that the scalar product formula can be applied to them. |
::::::::::::But "geometric product" is not a part of the definition of "vector". You do not need a "geometric product" for a vector to be a vector, and in fact in the general case vectors exist without any geometric product being defined. Meanwhile, the axioms of the of the algebra require that a scalar product can be applied to i, j and k. So the reader should indeed be able to assume that the scalar product formula can be applied to them. |
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::::::::::::Still, yes, you are using a particular definition of "orthogonality" - it's not what I would have expected, and I can easily imagine other definitions which might serve in other contexts. So that means that that definition of orthogonality needs to be in the task description. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 20:48, 19 October 2015 (UTC) |
::::::::::::Still, yes, you are using a particular definition of "orthogonality" - it's not what I would have expected, and I can easily imagine other definitions which might serve in other contexts. So that means that that definition of orthogonality needs to be in the task description. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 20:48, 19 October 2015 (UTC) |
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:::::::::::::No, the axioms do not require that a scalar product can be applied to i, j and k. I think you're confusing things. The axioms imply that the ''geometric product'' can be applied to them, not the scalar product. In fact, the axioms do not mention a scalar product at all. Also I don't understand where you got the idea that I was suggesting that the geometric product is required for the definition of a vector.--[[User:Grondilu|Grondilu]] ([[User talk:Grondilu|talk]]) 20:57, 19 October 2015 (UTC) |
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== "Orthonormal basis" == |
== "Orthonormal basis" == |
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