Definition:Vector Cross Product/Complex

Definition

Let $z_1 := x_1 + i y_1$ and $z_2 := x_2 + i y_2$ be complex numbers.

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.

Examples

Example: $\paren {3 - 4 i} \times \paren {-4 + 3 i}$

Let:

$z_1 = 3 - 4 i$

$z_2 = -4 + 3 i$

Then:

$z_1 \times z_2 = -7$

where $\times$ denotes (complex) cross product.

Also see

• Equivalence of Definitions of Complex Cross Product

• Results about Complex Cross Product can be found here.