# 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 $z_2$.

## Proof

 $\displaystyle z_1 \times z_2$ $=$ $\displaystyle \map \Im {\overline {z_1} z_2}$ Definition 3 of Complex Cross Product $\displaystyle$ $=$ $\displaystyle \map \Im {r_1 e^{-i \theta_1} r_2 e^{i \theta_2} }$ Exponential Form of Complex Conjugate $\displaystyle$ $=$ $\displaystyle \map \Im {r_1 r_2 e^{i \paren {\theta_2 - \theta_1} } }$ Product of Complex Numbers in Exponential Form $\displaystyle$ $=$ $\displaystyle \map \Im {r_1 r_2 \paren {\map \cos {\theta_2 - \theta_1} + i \, \map \sin {\theta_2 - \theta_1} } }$ Definition of Polar Form of Complex Number $\displaystyle$ $=$ $\displaystyle r_1 r_2 \map \sin {\theta_2 - \theta_1}$ Definition of Imaginary Part

$\blacksquare$