Q:

Solve : $p\vec x+\vec{x}(\vec{x}\cdot \vec{b})=\vec{a}\times \vec{b}+\vec{c}$.

Prob: Solve : $p\vec x+\vec{x}(\vec{x}\cdot \vec{b})=\vec{a}\times \vec{b}+\vec{c}$.  
I could not understand how to calculate it to solve the equation for $\vec x$.

A:

First multiply by $b$ and notice that $(a\times b)\cdot b=0$.  You'll get a quadratic equation for $x\cdot b$. Plug the solution in the original equation and solve for $x$.

