Definition:Dot 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 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.

Also known as

The dot product is also known as:

• The scalar product (but this can be confused with multiplication by a scalar so is less recommended)
• The standard inner product.

Some sources refer to it as just the inner product, but this is a more general term of which the dot product is merely an example.

The symbol used for the dot is variously presented; another version is $\mathbf a \bullet \mathbf b$, which can be preferred if there is ambiguity between the dot product and standard multiplication.

In the complex plane, where it is commonplace to use $\cdot$ to denote complex multiplication, the symbol $\circ$ is often used to denote the dot product.

Examples

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

Let:

$z_1 = 3 - 4 i$

$z_2 = -4 + 3 i$

Then:

$z_1 \circ z_2 = -24$

where $\circ$ denotes (complex) dot product.

Acute Angle Between $\paren {3 - 4 i}$ and $\paren {-4 + 3 i}$

Consider:

$z_1 = 3 - 4 i$

$z_2 = -4 + 3 i$

expressed as vectors.

Then the acute angle between $z_1$ and $z_2$ is approximately $16 \degrees 16'$

Also see

• Equivalence of Definitions of Complex Dot Product

• Results about complex dot product can be found here.