Unitary matrices preserve inner products.

Theorem. Unitary matrices preserve the standard inner product on $\mathbb{C}^n$. For any vectors $x, y \in \mathbb{C}^n$ and unitary matrix $U$:
$$\langle Ux, Uy \rangle = \langle x, y \rangle$$

Proof.
$$\langle Ux, Uy \rangle = (Uy)^* (Ux) = y^* U^* U x = y^* x = \langle x, y \rangle$$

Consequences:

• Norm preservation: $\|Ux\| = \|x\|$ (unitary matrices are isometries)
• Distance preservation: $\|Ux - Uy\| = \|x - y\|$ (they are rigid motions)
• Angle preservation: the angle between $Ux$ and $Uy$ equals the angle between $x$ and $y$

This makes unitary transformations the natural notion of "rotation" in complex vector spaces.