Definition:Dot Product

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Definition

Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $n$ dimensions:

\(\ds \mathbf a\) \(=\) \(\ds \sum_{k \mathop = 1}^n a_k \mathbf e_k\)
\(\ds \mathbf b\) \(=\) \(\ds \sum_{k \mathop = 1}^n b_k \mathbf e_k\)

where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $\mathbf V$.


General Context

The dot product of $\mathbf a$ and $\mathbf b$ is defined as:

\(\ds \mathbf a \cdot \mathbf b\) \(:=\) \(\ds \sum_{k \mathop = 1}^n a_k b_k\)
\(\ds \) \(=\) \(\ds a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = \sum_{i \mathop = 1}^n a_i b_i\)


If the vectors are represented as column matrices:

$\mathbf a = \begin {bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end {bmatrix} , \mathbf b = \begin {bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end {bmatrix}$

we can express the dot product as:

$\mathbf a \cdot \mathbf b = \mathbf a^\intercal \mathbf b$

where:

$\mathbf a^\intercal = \begin {bmatrix} a_1 & a_2 & \cdots & a_n \end {bmatrix}$ is the transpose of $\mathbf a$
the operation between the matrices is the matrix product.


Real Euclidean Space

In the context of a real Euclidean space $\R^n$, the definition can be geometrical in nature:


Let $\mathbf a$ and $\mathbf b$ be vectors in real Euclidean space $\R^n$.

The dot product of $\mathbf a$ and $\mathbf b$ is defined as:

$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \cos \angle \mathbf a, \mathbf b$

where:

$\norm {\mathbf a}$ denotes the length of $\mathbf a$
$\angle \mathbf a, \mathbf b$ is the angle between $\mathbf a$ and $\mathbf b$, taken to be between $0$ and $\pi$.


Complex Numbers

The definition continues to hold when the vector space under consideration is the complex plane:


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.


Einstein Summation Convention

Let $\mathbf a$ and $\mathbf b$ be vector quantities.

The dot product of $\mathbf a$ and $\mathbf b$ can be expressed using the Einstein summation convention as:

\(\ds \mathbf a \cdot \mathbf b\) \(:=\) \(\ds a_i b_j \delta_{i j}\)
\(\ds \) \(=\) \(\ds a_i b_i\)
\(\ds \) \(=\) \(\ds a_j b_j\)

where $\delta_{i j}$ is the Kronecker delta.


Also known as

The dot product is also known as:

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

Arbitrary Example 1

Let:

\(\ds \mathbf A\) \(=\) \(\ds 6 \mathbf i + 4 \mathbf j + 3 \mathbf k\)
\(\ds \mathbf B\) \(=\) \(\ds 2 \mathbf i - 3 \mathbf j - 3 \mathbf k\)

Then:

$\mathbf A \cdot \mathbf B = -9$

Hence the angle between $\mathbf A$ and $\mathbf B$ is approximately $104.2 \degrees$.


Work Done

Let $\mathbf F$ represent a force acting on a body $B$.

Let $\mathbf d$ denote the displacement effected on $B$ by $\mathbf F$.

Then the work done by $\mathbf F$ on $B$ is given by:

$W = \mathbf F \cdot \mathbf d = \norm {\mathbf F} \norm {\mathbf d} \cos \theta$

where:

$\cdot$ denotes dot product
$\theta$ is the angle between the directions of $\mathbf F$ and $\mathbf d$.


Also see

  • Results about dot product can be found here.


Historical Note

During the course of development of vector analysis, various notations for the dot product were introduced, as follows:

Symbol Used by
$\mathbf a \cdot \mathbf b$ Josiah Willard Gibbs and Edwin Bidwell Wilson
$\mathbf a \mathbf b$ Oliver Heaviside
$\mathscr A \mathscr B$ Max Abraham
$\mathfrak A \mathfrak B$ Vladimir Sergeyevitch Ignatowski
$\paren {\mathbf A \cdot \mathbf B}$ Hendrik Antoon Lorentz
$\mathbf a \times \mathbf b$ Cesare Burali-Forti and Roberto Marcolongo


Sources

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