What's the opposite of a cross product? For example, $a \times b = c$
If you only know $a$ and $c$, what method can you use to find $b$?

The name "product" for the cross product is unfortunate. It really should not be thought of as a product in the ordinary sense; for example, it is not even associative. Thus one should not expect it to have properties analogous to the properties of ordinary multiplication.
What the cross product really is is a Lie bracket. 
