= Solution
Use active <rotations> on column vectors and positive angles counter-clockwise when viewed from the positive axis toward the origin. Rotation about $z$ takes $\mathbf e_x$ to $(\cos\alpha,\sin\alpha,0)$, $\mathbf e_y$ to $(-\sin\alpha,\cos\alpha,0)$, and leaves $\mathbf e_z$ fixed. These images are the columns of the <rotation matrix>:
$$
\boxed{R_z(\alpha)=\begin{pmatrix}
\cos\alpha&-\sin\alpha&0\\
\sin\alpha&\cos\alpha&0\\
0&0&1
\end{pmatrix}.}
$$
It satisfies $R_z^TR_z=I$ and $\det R_z=1$. Stating the viewing convention avoids confusing this active <rotation> with a passive change of axes.
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