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A rotation through an angle of $\pi$. In three dimensions, a half-turn about a straight line with unit vector $\mathbf{n}$ fixes its parallel component and negates its orthogonal complement, giving the rotation matrix $2\mathbf{n}{\mathbf{n}}^T-I$ by orthogonal projection.

The straight line $2x=y=z$ has direction vector $(1,2,2)$, so take the unit vector $\mathbf{n}=(1,2,2{)}^T/3$. A half-turn has $\cos \pi =-1$ and $\sin \pi =0$. The preceding rotation formula becomesThus the rotation matrix in the standard basis isIt fixes $(1,2,2{)}^T$ and negates every vector in its orthogonal complement, which also checks the half-turn geometrically.