Prove unitary matrices do not alter the magnitude of a column vector with complex elements

Question

Prove the following:

1. Given $r'=Ur$ with $U$ a unitary matrix and $r$ a (column) vector with complex elements, show that the magnitude of $r$ is the same as the magnitude of $r'$.

2. The matrix $U$ transforms any column vector $r$ with complex elements into $r'$, leaving the magnitude invariant: $r^\dagger r= r’^\dagger r'$. Show that $U$ is unitary.

Both questions are in my text of mathematical methods for physics (Arfken), but I haven't a solution text.

Answer 1

A. Form r′†r′ = (Ur)†Ur = r†U†Ur = r†r

B. If for all r, r′†r = r†U†Ur, then it must be U†U = 1

Answer 2

For 1, use the fact that the norm of a vector is $xx^T$, along with $UU^T=I$, that should do it.