Equivalence of Definitions of Complex Cross Product

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Theorem

The following definitions of the concept of Complex Cross Product are equivalent:

Definition 1

The cross product of $z_1$ and $z_2$ is defined as:

$z_1 \times z_2 = x_1 y_2 - y_1 x_2$

Definition 2

The cross product of $z_1$ and $z_2$ is defined as:

$z_1 \times z_2 = \cmod {z_1} \, \cmod {z_2} \sin \theta$

where:

$\cmod {z_1}$ denotes the complex modulus of $z_1$
$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction.

Definition 3

The cross product of $z_1$ and $z_2$ is defined as:

$z_1 \times z_2 := \map \Im {\overline {z_1} z_2}$

where:

$\map \Im z$ denotes the imaginary part of a complex number $z$
$\overline {z_1}$ denotes the complex conjugate of $z_1$
$\overline {z_1} z_2$ denotes complex multiplication.

Definition 4

The cross product of $z_1$ and $z_2$ is defined as:

$z_1 \times z_2 := \dfrac {\overline {z_1} z_2 - z_1 \overline {z_2}} {2 i}$

where:

$\overline {z_1}$ denotes the complex conjugate of $z_1$
$\overline {z_1} z_2$ denotes complex multiplication.


Proof

Definition 1 equivalent to Definition 3

\(\ds \) \(\) \(\ds \map \Im {\overline {z_1} z_2}\) Definition 3 of Vector Cross Product
\(\ds \) \(=\) \(\ds \map \Im {\paren {x_1 - i y_1} \paren {x_2 + i y_2} }\) Definition of Complex Conjugate
\(\ds \) \(=\) \(\ds \map \Im {\paren {x_1 x_2 + y_1 y_2} + i \paren {x_1 y_2 - x_2 y_1} }\) Definition of Complex Multiplication
\(\ds \) \(=\) \(\ds x_1 y_2 - x_2 y_1\) Definition of Imaginary Part

$\Box$


Definition 2 equivalent to Definition 3

\(\ds \) \(\) \(\ds \map \Im {\overline {z_1} z_2}\) Definition 3 of Vector Cross Product
\(\ds \) \(=\) \(\ds r_1 r_2 \sin \paren {\theta_2 - \theta_1}\) Complex Cross Product in Exponential Form
\(\ds \) \(=\) \(\ds \cmod {z_1} \, \cmod {z_2} \map \sin {\theta_2 - \theta_1}\) Definition of Polar Form of Complex Number
\(\ds \) \(=\) \(\ds \cmod {z_1} \, \cmod {z_2} \sin \theta\) where $\theta = \theta_2 - \theta_1$ is the angle between $z_1$ and $z_2$

$\Box$


Definition 1 equivalent to Definition 4

\(\ds \) \(\) \(\ds \frac {\overline {z_1} z_2 - z_1 \overline {z_2} } {2 i}\) Definition 4 of Complex Dot Product
\(\ds \) \(=\) \(\ds \frac {\paren {x_1 - i y_1} \paren {x_2 + i y_2} - \paren {x_1 + i y_1} \paren {x_2 - i y_2} } {2 i}\) Definition of Complex Conjugate
\(\ds \) \(=\) \(\ds \frac {\paren {\paren {x_1 x_2 + y_1 y_2} + i \paren {x_1 y_2 - x_2 y_1} } - \paren {\paren {x_1 x_2 + y_1 y_2} + i \paren {-x_1 y_2 + x_2 y_1} } } {2 i}\) Definition of Complex Multiplication
\(\ds \) \(=\) \(\ds x_1 y_2 - x_2 y_1\) after algebra

$\blacksquare$


Sources