Solution
= Solution
With the same right-handed active <rotation> convention as $R_z$, the $y$ direction is fixed, $\mathbf e_z$ rotates towards $\mathbf e_x$, and $\mathbf e_x$ towards $-\mathbf e_z$. Therefore
$$
\boxed{R_y(\theta)=\begin{pmatrix}\cos\theta&0&\sin\theta\\0&1&0\\-\sin\theta&0&\cos\theta\end{pmatrix}.}
$$
Its columns give the images of the three Cartesian unit vectors, which also makes the signs of this <rotation matrix> explicit.