Product of Complex Number with Conjugate by Dot and Cross Product

Theorem
Let $z_1$ and $z_2$ be complex numbers.

Then:
 * $\overline {z_1} z_2 = \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}$

where:
 * $\overline {z_1}$ denotes the complex conjugate of $z_1$
 * $z_1 \circ z_2$ denotes the complex dot product of $z_1$ with $z_2$
 * $z_1 \times z_2$ denotes the complex cross product of $z_1$ with $z_2$.