Question on vector cross product.

Question

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

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

Answer 1

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,

$\begin{align} ((\textbf{u}\times(\textbf{u}\times \textbf{v}))\times \textbf{v})\times(\textbf{u}\times \textbf{v})\\&=(((\textbf{u}.\textbf{v})\textbf{u}-(\textbf{u}.\textbf{u})\textbf{v})\times\textbf{v})\times(\textbf{u}\times \textbf{v})\\&=((\textbf{u}.\textbf{v})(\textbf{u}\times\textbf{v})-(\textbf{u}.\textbf{u})(\textbf{v}\times\textbf{v}))\times(\textbf{u}\times \textbf{v})\\&=(\textbf{u}.\textbf{v})(\textbf{u}\times\textbf{v})\times(\textbf{u}\times \textbf{v})=0\\ \end{align}$

Answer 2

Hint:

$a\times b=0$ if and only if $a$ and $b$ are parallel.