If a and b are perpendicular, then a( a( a( a b))) is equal to:

Question

If $\vec a$ and $\vec b$ are perpendicular, then $\vec a	imes\big(\vec a	imes(\vec a	imes(\vec a	imes\vec b))\big)$ is equal to:

Accepted Answer

Using $\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c$ repeatedly, and $\vec a\cdot\vec b=0$:

$$\vec a\times(\vec a\times\vec b) = (\vec a\cdot\vec b)\vec a-(\vec a\cdot\vec a)\vec b = -|\vec a|^2\vec b$$

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