Reordering vector product If I have vectors $a, b, c \in \mathbb{R}^3$, and if we have e.g. $a = b\times c$, is there any way to express $b$ in terms of the other two? 

No, it's not possible, since infinitely many values of $b$ lead to the same value of $a$. This is obvious for $c=0$. For $c\neq0$ both $b_1=b$ and $b_2=b+c$ give the same $a$:
$$ b_2\times c = (b+c)\times c = b\times c + c\times c = b\times c = b_1\times c. $$
