Rotation about the $z$-axis in $\mathbb R^3$ has matrix $\begin{bmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{bmatrix}$. It rotates the $xy$-plane and fixes the $z$ coordinate. For $\theta=\pi/2$, $(1,2,3)$ becomes $(-2,1,3)$. The matrix is orthogonal with determinant one.
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