# 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**.

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

- Results about
**complex dot product**can be found**here**.