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Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.
Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.
I tend to see it the otherway around. It is not an invention, it simply is there. You first deal with vectors and the diagram comes next, when you try to explain it to someone else. It's not really needed. And the same holds for the cross-product. It is simply some determinants which happen to represent a size (length, area, volume). E.g. the orthogonality is forced by definition: ##\begin{bmatrix} * \\ * \\ 0 \end{bmatrix} \times \begin{bmatrix} * \\ * \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ * \end{bmatrix}.##I just want to know how it came to their mind, there is a idea for every thing , just like we know that A + B = C (A,B,C are vectors) and this can be represented by the diagram given below
That's the point! If someone asks you ... then you draw a diagram ... In case of the cross-product it turns out to be the right hand rule that often appears in nature.If someone ask, why you represented addition of vector like given in above diagram.
No. It exists without a diagram, a right hand rule, an area or whatever. It is just as useful as complex numbers are useful without having a complex plane, or integers are without being an accountant or bookmaker.Then there is a answer ... therefore I think that there may be a good reason for the cross product question
For me its a cool invention , but I got your point.I tend to see it the otherway around. It is not an invention, it simply is there. You first deal with vectors and the diagram comes next, when you try to explain it to someone else. It's not really needed. And the same holds for the cross-product. It is simply some determinants which happen to represent a size (length, area, volume). E.g. the orthogonality is forced by definition: ##\begin{bmatrix} * \\ * \\ 0 \end{bmatrix} \times \begin{bmatrix} * \\ * \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ * \end{bmatrix}.##
That's the point! If someone asks you ... then you draw a diagram ... In case of the cross-product it turns out to be the right hand rule that often appears in nature.
No. It exists without a diagram, a right hand rule, an area or whatever. It is just as useful as complex numbers are useful without having a complex plane, or integers are without being an accountant or bookmaker.
A difference on the philosophical point of view. Sometimes as well the difference between more "applied" and more "pure" scientists.For me its a cool invention , but I got your point.
I think of it this way. We have a different product, the dot product, which takes a pair of vectors and gives a scalar. So we want the cross product to give a vector.Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.