Complex Multiplication as Geometrical Transformation/Corollary

Corollary to Complex Multiplication as Geometrical Transformation
Let $z = \polar {r, \theta}$ be a complex number expressed in polar form.

Let $z$ be represented on the complex plane $\C$ in vector form.

The effect of multiplying $z$ by $e^{i \alpha}$ is to rotate it about the origin of $\C$ by $\alpha$ in the positive direction

Proof
From Complex Multiplication as Geometrical Transformation, the effect of multiplying a complex number by $r e^{i \alpha}$ is:
 * to rotate it about the origin of $\C$ by $\alpha$ in the positive direction
 * to multiply its modulus by $r$.

In this instance $r = 1$.

Hence the result.