High School What is the cross product of two vectors?

[Mark44]

@Mark44 I think what @wrobel is saying, is that the cross product of two vectors is not itself a vector, because it does not transform like a vector. To get the right transformation properties (i.e. to turn it into a vector) you also need to multiply by the determinant of the transformation matrix. But before you do that it's still just a tensor density.

Here's what wikipedia has to say (https://en.wikipedia.org/wiki/Cross_product#:~:text=In mathematics, the cross product or vector product,is denoted by the symbol {displaystyle%20times%20}:

Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them.

Cross products are typically introduced in intro physics courses and in precalculus mathematics courses -- and there is not usually any mention of tensors.

 

[suremarc]

I mean, I prefer to think of the cross product as a pseudovector, but I guess it’s probably not helpful for OP.

I think the most succinct explanation of the cross product at this level probably comes from the volume formula, as others have said. Namely, that the “signed volume” $V(\mathbf{a,b,c})$ of a parallelepiped with sides $\mathbf{a,b,c}$ is linear in each of its arguments, and that $\mathbf{a\times b}$ is the unique (pseudo)vector such that $V(\mathbf{a,b,c})=(\mathbf{a\times b})\cdot\mathbf{c}.$ While most of us know from linear algebra that in any given basis such a vector exists, it can be verified in Cartesian coordinates.

 

[Mark44]

I mean, I prefer to think of the cross product as a pseudovector, but I guess it’s probably not helpful for OP.

I agree. Several posts that are well beyond B-level have been removed. Since the OP has not returned since his initial post, I'm closing this thread. If he/she has additional questions that haven't been addressed in this thread, I will reopen it.