= Rodrigues rotation formula
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For a real <angle> $\alpha$ and a unit <vector> $\mathbf n$, the <rotation> about $\mathbf n$ sends
$$
\mathbf x\longmapsto\cos\alpha\,\mathbf x+\sin\alpha\,(\mathbf n\times\mathbf x)+(1-\cos\alpha)\mathbf n(\mathbf n\cdot\mathbf x).
$$
The <vector triple product identity> gives the equivalent expression $\mathbf x+\sin\alpha\,\mathbf n\times\mathbf x+(1-\cos\alpha)\mathbf n\times(\mathbf n\times\mathbf x)$. To see why it is a <rotation>, decompose $\mathbf x$ into its parallel and perpendicular components relative to $\mathbf n$. The parallel component is fixed, while the perpendicular component becomes $\cos\alpha\,\mathbf x_\perp+\sin\alpha\,\mathbf n\times\mathbf x_\perp$, the usual plane <rotation>. It is also the <matrix exponential> of the <linear operator> $H\mathbf x=\alpha\mathbf n\times\mathbf x$: the identity $H^3=-\alpha^2H$ reduces its <exponential series> to these three terms.
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