 Solution

Applying the rightmost rotation matrix first givesEquality with ${\mathbf{e}}_y$ requires $\cos \theta =0$, so $\theta =\pi /2$ in the specified range. Then $\sin \varphi =1$ and $\cos \varphi =0$ giveIn the reverse order, $R_z$ first fixes ${\mathbf{e}}_z$, so $R_2{\mathbf{e}}_z=R_y(\pi /2){\mathbf{e}}_z={\mathbf{e}}_x$. Therefore $\boxed{R_2\mathbf{v}\ne {\mathbf{v}}^{\prime }}$. This explicitly demonstrates that rotations about different axes do not generally commute.