Q:

Question on vector cross product. 

Show that $\big((\mathbf{u} \times (\mathbf{u}\times \mathbf{v})) \times \mathbf{v}\big) \times (\mathbf{u} \times \mathbf{v})=0$.

From wolfram, it gives zero but there's no details.  How to prove this? 

A:

Using the identity
$$\textbf{A}\times (\textbf{B}\times \textbf{C})= (\textbf{A}.\textbf{C})\textbf{B}-(\textbf{A}.\textbf{B})\textbf{C}$$
we get,
$\begin{align}
((\textbf{u}\times(\textbf{u}\times \textbf{v}))\times \textbf{v})\times(\textbf{u}\times \textbf{v})\\&=(((\textbf{u}.\textbf{v})\textbf{u}-(\textbf{u}.\textbf{u})\textbf{v})\times\textbf{v})\times(\textbf{u}\times \textbf{v})\\&=((\textbf{u}.\textbf{v})(\textbf{u}\times\textbf{v})-(\textbf{u}.\textbf{u})(\textbf{v}\times\textbf{v}))\times(\textbf{u}\times \textbf{v})\\&=(\textbf{u}.\textbf{v})(\textbf{u}\times\textbf{v})\times(\textbf{u}\times \textbf{v})=0\\
\end{align}$

