Cross Product: is there an inverse?

Question

I learning vector calculus and electromagnetism, from which I noticed cross product is used extensively. I.e. $\phi = \Delta \times \mathbf E$. I'm curious, is there an inverse to the cross product? It seems that for two vectors it would be a plane and for the case of $\phi$ above maybe some integral? My goal is solving for different variables in these equations and am curious if such an operation is used.

Answer 1

There is no inverse for the cross product. To see why, consider two vectors $A$ and $B$. The cross product is $|A||B|\sin{\theta}$ where $\theta$ is the angle between them. You can rotate each of those vectors to a different position maintaining the angle between them. Since their magnitudes don't change in a rotation and we have preserved the angle between them, we get the same cross product.

Edit: As @Andrei points out in his comment, the rotation of $A$ and $B$ must be about $A\times B$ to preserve the cross product.