Complex Cross Product in Exponential Form

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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

$\blacksquare$


Sources