Category:Examples of Complex Dot Product

This category contains examples of Complex Dot Product.

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

Definition 1

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

$z_1 \circ z_2 = x_1 x_2 + y_1 y_2$

Definition 2

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

$z_1 \circ z_2 = \cmod {z_1} \, \cmod{z_2} \cos \theta$

where:

$\cmod {z_1}$ denotes the complex modulus of $z_1$

$\theta$ denotes the angle between $z_1$ and $z_2$.

Definition 3

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

$z_1 \circ z_2 := \map \Re {\overline {z_1} z_2}$

where:

$\map \Re z$ denotes the real 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 dot product of $z_1$ and $z_2$ is defined as:

$z_1 \circ z_2 := \dfrac {\overline {z_1} z_2 + z_1 \overline {z_2} } 2$

where:

$\overline {z_1}$ denotes the complex conjugate of $z_1$

$\overline {z_1} z_2$ denotes complex multiplication.