Complex Cross Product in Exponential Form

Theorem
Let $z_1 := r_1 e^{i \theta_1}, z_2 := r_2 e^{i \theta_2} \in \C$ be complex numbers expressed in exponential form.

Then:
 * $z_1 \times z_2 = r_1 r_2 \map \sin {\theta_2 - \theta_1}$

where $z_1 \times z_2$ denotes the dot product of $z_1$ and $z_2$.