Equation on cross product I have to solve the equation $\vec x\times\vec a=\vec b-\vec x$  ,   where $\vec x, \vec a,\vec b$ are vectors and the last two are known.
I have proven that $\vec a\cdot \vec x=\vec a\cdot\vec b$, but that is all.
Thank you in advance !
<< Symbol : $\vec a \vec b = \vec a \cdot \vec b$ >>

If $\vec{x}=\vec{y},\,\vec{x}=\vec{y}+\vec{z}$ both work, $\vec{z}\times\vec{a}=-\vec{z}$, so $\vec{z}$ is self-orthogonal and vanishes, and the solution $\vec{x}$ is unique. Now try the Ansatz $\vec{x}=A\vec{a}+B\vec{b}+C\vec{a}\times\vec{b}$. As I noted in a comment on @thecandyman's answer, you should find$$\vec{x}=\frac{(\vec{a}\cdot\vec{b})\vec{a}+\vec{b}+\vec{a}\times\vec{b}}{a^2+1}.$$
