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
\(\ds z_1 \times z_2\) | \(=\) | \(\ds \map \Im {\overline {z_1} z_2}\) | Definition 3 of Complex Cross Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {r_1 e^{-i \theta_1} r_2 e^{i \theta_2} }\) | Exponential Form of Complex Conjugate | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {r_1 r_2 e^{i \paren {\theta_2 - \theta_1} } }\) | Product of Complex Numbers in Exponential Form | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {r_1 r_2 \paren {\map \cos {\theta_2 - \theta_1} + i \map \sin {\theta_2 - \theta_1} } }\) | Definition of Polar Form of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds r_1 r_2 \map \sin {\theta_2 - \theta_1}\) | Definition of Imaginary Part |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: The Dot and Cross Product: $112 \ \text{(b)}$