Past exam of the mathematical methods I course of the University of Oxford / 2013 resit / 10 / a / i / 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:It satisfies $R_z^T{R}_z=I$ and $\det R_z=1$. Stating the viewing convention avoids confusing this active rotation with a passive change of axes.