My professor in E&M told us that the cross product is the only possible product of vectors that produces another vector. I didn't ask him what he meant by that because he is Russian and can't always translate what he means into English very well. How is the cross product the unique product which produces another vector? Couldn't we define a bunch of other products like $(a,b,c)\times_1(d,e,f) = (ad,be,cf)$ or $(a,b,c)\times_2(d,e,f) = (ae,2bd,cf)$ or even just $(a,b,c)\times_3(d,e,f) = \frac{1}{2}(a,b,c)\times (d,e,f)$ (where the RHS is half the regular cross product)? Are these things somehow not vectors?
EDIT: I started wondering if their was some invariant of vectors that the cross product preserves that other products don't. Searching this site I found that cross products are invariant under rotations. I wonder if that is it. Is it true that the cross product is the only product of vectors which produces a vector and is invariant under rotations? If so, how could that be proven? If not, is there some other invariant that is preserved under the cross product, but not any other product of vectors on $\Bbb R^3$?