Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 1 / 7B / a / Solution

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.