Answer
See solution
Work Step by Step
a) $||Ux||^2=(Ux)\cdot (Ux)=(Ux)^T(Ux)=x^TU^TUx=x^Tx=||x||^2$
Taking the square root of both sides, $||Ux||=||x||$
b) $(Ux)\cdot (Uy)=(Ux)^T(Uy)=x^TU^TUy=x^Ty=x\cdot y$
c) Because $x\cdot y$ and $(Ux)\cdot (Uy)$ are equal, they are both 0 or are both nonzero.