How to isolate vector X in a vectorial equation with vectorial product?

Question

I have no idea how to start solving this equation... It seems that the way of solving it is very different of non-vectorial equations... does anybody knows how to do it?

$$(\vec x \times \vec a)\times(\vec x \times \vec b)=\vec c$$

Answer 1

You start with the LHS and use $\vec{a}\times(\vec{b}\times\vec{c})=\vec{b}(\vec{a}\cdot\vec{c})-\vec{c}(\vec{a}\cdot\vec{b})$ $$ {(\vec x \times \vec a)}\times(\vec x \times \vec b)=\vec x((\vec x \times \vec a)\cdot\vec b)-\vec b((\vec x \times \vec a)\cdot x) $$ The second term is zero as $(\vec x \times \vec a)$ and $\vec x$ are perpendicular. Then use that $\vec a\cdot(\vec b\times\vec c)=\vec b\cdot(\vec c\times\vec a)$ i.e. the triple product is cyclic. $$ \vec x((\vec x \times \vec a)\cdot\vec b)=\vec x(\vec x\cdot(\vec a\times \vec b))=\vec c $$ Now we see that $\vec x$ and $\vec c$ must be parallel as the second term on the LHS is just a scalar thus $\vec x=\lambda\vec c$. Inserting this we find $$ \lambda^2\vec c(\vec c\cdot(\vec a\times\vec b))=\vec c\quad\Rightarrow\quad \lambda^2(\vec c\cdot(\vec a\times\vec b))=1\quad\Rightarrow\quad \lambda=\frac{\pm 1}{\sqrt{\vec a\cdot(\vec b\times\vec c)}}\quad\Rightarrow\quad \vec x=\frac{\pm \vec c}{\sqrt{\vec a\cdot(\vec b\times\vec c)}} $$ Which holds when $\vec a\cdot(\vec b\times\vec c)\neq 0$.