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? 

Write out the vectors as $a=(a_1,a_2,a_3)$, and so on, do the cross product, set the two sides equal, and you'll get a system of three linear equations in the three unknown components of $b$. 
