Reordering vector product

Question

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?

Answer 1

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 $b$.

Answer 2

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 $a$: $$ b_2\times c = (b+c)\times c = b\times c + c\times c = b\times c = b_1\times c. $$

Answer 3

To elaborate on @Ralph Tandetzky's answer: Interpreting the cross product geometrically, assuming $a\ne 0$, $b$ must be in the plane orthogonal to $a$, and the signed area of the parallelogram spanned by $b$ and $c$ must be $\|a\|$. Thus, there is a line — parallel to $c$ — of solutions.