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 $z_2$.

$\ds $

$\ds \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}$

$\ds $

$=$

$\ds \paren {z_1 \circ z_2} + i \frac {\overline {z_1} z_2 - z_1 \overline {z_2} } {2 i}$

$\ds $

$=$

$\ds \frac {\overline {z_1} z_2 + z_1 \overline {z_2} } 2 + \frac {\overline {z_1} z_2 - z_1 \overline {z_2} } 2$

Definition 4 of Dot Product

$\ds $

$=$

$\ds \frac {\overline {z_1} z_2 + z_1 \overline {z_2} + \overline {z_1} z_2 - z_1 \overline {z_2} } 2$

$\ds $

$=$

$\ds \overline {z_1} z_2$

$\blacksquare$