# Equivalence of Definitions of Complex Cross Product

## 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

$\blacksquare$