I Cross product

May I ask you, what kind of answer do you expect? The way you phrased the question is a tempting pass-play to create jokes.

 

It is a useful operation that way.

 

Several off topic posts have been removed. Please try to keep responses on topic

 

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.
Your question brought me to think about the following: Who first used it and in which context? (That's why I was asking in #2.... and to keep me from joking like saying: It has been to annoy physicists )

Has it been an ancient geometer like Euklid or Aristoteles? Was it Graßmann? Descartes? To be honest, I don't know and have not really an idea where to search for the answer.

Edit: The book about history I have (J. Dieudonné) says: Graßmann and he was indeed driven by geometric considerations like generalizing the one-dimensional length of a vector.

 

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.

Two vectors define a plane of vectors which are linear combinations of the two vectors, so there is no point in making our cross product give any of those vectors. So if you want to get out of that plane, then the only sensible vector is a vector normal to the plane (any other vector is a linear combination of that and a vector in the plane). That vector is necessarily perpendicular to any vector in the plane, including the original two.