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.