Solving for vector in cross product

Question

Given $\vec{a} \times \vec{b} = \vec{n}$ where $\times$ is the cross product, is it possible to solve for $\vec{b}$ given $\vec{a}$ and $\vec{n}$? If so could you give an example?

Thanks so much!

Answer 1

No, it's not possible. If you manage to find some $\vec b$ which works, then so will $\vec b+t\vec a$ for any real number $t$. And none of those solutions stick out in any way as better or more natural than any of the others. So in the technical sense of the word "solve" it's impossible.

That being said, it's not too difficult to find a $\vec b$ which works. Just take $\frac{1}{\|\vec a\|^2}\cdot\vec n\times\vec a$.

Answer 2

Nope. You can figure out that $\vec{n}$ is normal to $\vec{a}$ and $\vec{b}$. This means given only $\vec{a}$ and $\vec{n}$ any vector normal to $\vec{n}$ with length $$\frac{|\vec{n}|}{|\vec{a}|\sin\theta}$$ can be a potential $\vec{b}$. $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.