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The operation $\vec{a}\times\left(\vec{b}\times\vec{c}\right)$ can be simplified to $\vec{b}\left(\vec{a}\cdot\vec{c}\right) - \vec{c}\left(\vec{a}\cdot\vec{b}\right)$ and can easily be remembered by the mnemonic "BAC minus CAB".

However what does the operation $\left(\vec{a}\times\vec{b}\right)\times\vec{c}$ lead to? And can it be simplified? Cross product is generally not associative.

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    $\begingroup$ The cross product is anticommutative, $\vec{x}\times \vec{y} = -\vec{y}\times\vec{x}$. Play with that. $\endgroup$ – Daniel Fischer Jan 28 '14 at 19:08
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$$(\vec a\times \vec b)\times\vec c=-\vec c\times (\vec a\times \vec b)=\vec c\times(\vec b \times \vec a)=\vec b(\vec c\cdot\vec a)-\vec a(\vec c\cdot \vec b).$$

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