Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 1 / 7B (Algebra and Geometry)

Take positive angles counterclockwise. The rotation sends the standard basis vectors to $(\cos \alpha ,\sin \alpha )$ and $(-\sin \alpha ,\cos \alpha )$. These images form the columns of its matrix, soThe columns are an orthonormal basis and the determinant is one, as required for a plane rotation.

Orient the axis towards $n=(3/5,4/5,0)$ and measure positive angles by the right-hand rule about $n$. The vectors $n$, $m=(-4/5,3/5,0)$ and $e_3=(0,0,1)$ form a positively oriented orthonormal basis, since $n\times m=e_3$. Its change-of-basis matrix isRelative to this basis, the rotation fixes the first coordinate and rotates the last two. Writing $c=\cos \alpha$ and $s=\sin \alpha$, its matrix in the standard basis is thereforeIn particular $Rn=n$, $Rm=cm+s{e}_3$, and $R{e}_3=-sm+c{e}_3$, verifying the axis and the direction of rotation. Reversing the chosen orientation of the axis replaces $\alpha$ by $-\alpha$.