Calculus: I can't seem to prove that a x b = -b x a?

Question

This is dealing in vectors and cross product. I have broken this down so far and I just seem to make it really complicated...

Answer 1

Do it in term of the components and the definition of the cross product.  here:

axb = <a2b3-a3b2,a3b1-a1b3,a1b2-a2b1>

-bxa=<-b2a3-(-b3)a2,-b3a1-(-b1)a3,-b1a...

Look at the fact that in each component you have -(-b)  so it comes to positive.  And if you look, it comes out to the positive part of axb.  Also note that each component has -b, as does each component of axb,  It's really just rearrangement from where you multiply out the components.