Counterclockwise rotation through angle $\theta$ has matrix $R_\theta=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$. Its columns are the rotated coordinate vectors. It is orthogonal, has determinant one, and satisfies $R_\theta^{-1}=R_{-\theta}$ and $R_\theta R_\phi=R_{\theta+\phi}$.
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