For a unitary matrix $U$, $\langle Ux,Uy\rangle=(Ux)^*(Uy)=x^*U^*Uy=x^*y=\langle x,y\rangle$. Thus unitary transformations preserve inner products, norms, distances and orthogonality.
For real orthogonal matrices this also preserves ordinary angles. In complex spaces an angle convention must be specified; the inner-product preservation identity itself is unambiguous.
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