For a $90^\circ$ counterclockwise rotation, $R=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$. Thus $R(2,1)^T=(-1,2)^T$. Check $\|Rx\|=\|x\|$ and $R^4=I$. Its eigenvalues are $\pm i$ over $\mathbb C$; it has no real eigenvector.
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