Solution
= Solution
The columns of $R_z$ are an <orthonormal basis>, since $\cos^2\phi+\sin^2\phi=1$ and the first two columns have zero <dot product>. Thus $R_z^TR_z=I$: it is an <orthogonal matrix>. Consequently
$$
|\mathbf v'|^2=(R_z\mathbf v)^T(R_z\mathbf v)=\mathbf v^TR_z^TR_z\mathbf v=\mathbf v^T\mathbf v,
$$
and hence $\boxed{|\mathbf v'|=|\mathbf v|}$. This is the preservation of the <Euclidean norm> by a <rotation matrix>.