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$\vec{u},\vec{w},\vec{v}\in\mathbb{R^3}$ $(\vec{u}\times\vec{w})\times \vec{v}=0$ if an only if $(\vec{u}\times\vec{v})\times \vec{w}= \vec{u}\times(\vec{v}\times \vec{w})$ is it always true and how to prove it?

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$\newcommand{\t}{\times}$ Note that the expression

$(u\t w)\t v + (w\t v)\t u + (v\t u)\t w$

can be expanded using the vector triple product and shown to be equal to zero.

The result you are asked to prove follows easily.