dot product

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OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field (mathematics), field of Scalar (mathematics), scalars such that the order of the two vectors does not affect the value of ...
s, for example in a
pseudo-Euclidean space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
.
is an
algebraic operation Algebraic may refer to any subject related to algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in fo ...
that takes two equal-length sequences of numbers (usually
coordinate vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...
s), and returns a single number. In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, the dot product of the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry ...
of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see
Inner product space In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
,
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s are often defined by using
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s. In this case, the dot product is used for defining lengths (the length of a vector is the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square (algebra), square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots o ...
of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to a ...
of their dot product by the product of their lengths). The name "dot product" is derived from the centered dot " · " that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a
vector Vector most often refers to: *Euclidean vector In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities ...
, as is the case for the
vector product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two Euclidean vector, vectors in a three-dimensional Orientation (vector space), oriented E ...
in three-dimensional space.

# Definition

The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude) of vectors. The equivalence of these two definitions relies on having a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distance ...
for Euclidean space. In modern presentations of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, the points of space are defined in terms of their
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry ...
, and
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
itself is commonly identified with the
real coordinate space In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the tuple, -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real v ...
R''n''. In such a presentation, the notions of length and angles are defined by means of the dot product. The length of a vector is defined as the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square (algebra), square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots o ...
of the dot product of the vector by itself, and the cosine of the (non oriented) angle between two vectors of length one is defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry.

## Coordinate definition

The dot product of two vectors and specified with respect to an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit ...
, is defined as: :$\mathbf\cdot\mathbf=\sum_^n _i_i=_1_1+_2_2+\cdots+_n_n$ where Σ denotes
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: function (mathematics), fu ...
and ''n'' is the
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of the
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
. For instance, in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
, the dot product of vectors and is: :$\begin \ \left[\right] \cdot \left[\right] &= \left( \times \right) + \left(\times\right) + \left(\times\right) \\ &= 4 - 6 + 5 \\ &= 3 \end$ Likewise, the dot product of the vector with itself is: :$\begin \ \left[\right] \cdot \left[\right] &= \left( \times \right) + \left(\times\right) + \left(\times\right) \\ &= 1 + 9 + 25 \\ &= 35 \end$ If vectors are identified with row matrices, the dot product can also be written as a
matrix product In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
:$\mathbf \cdot \mathbf = \mathbf\mathbf^\mathsf T,$ where $\mathbf^\mathsf T$ denotes the
transpose In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among othe ...
of $\mathbf$. Expressing the above example in this way, a 1 × 3 matrix (
row vector In linear algebra, a column vector with m elements is an m \times 1 Matrix_(mathematics), matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times ...
) is multiplied by a 3 × 1 matrix (
column vector In linear algebra, a column vector with m elements is an m \times 1 Matrix_(mathematics), matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times ...
) to get a 1 × 1 matrix that is identified with its unique entry: :$\begin \color1 & \color3 & \color-5 \end \begin \color4 \\ \color-2 \\ \color-1 \end = \color3$.

## Geometric definition

In
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, a
Euclidean vector In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction to which the arrow points. The magnitude of a vector a is denoted by $\left\, \mathbf \right\,$. The dot product of two Euclidean vectors a and b is defined by :$\mathbf\cdot\mathbf=\, \mathbf\, \ \, \mathbf\, \cos\theta ,$ where is the
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
between and . In particular, if the vectors and are
orthogonal In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
(i.e., their angle is or 90°), then $\cos \frac \pi 2 = 0$, which implies that :$\mathbf a \cdot \mathbf b = 0 .$ At the other extreme, if they are codirectional, then the angle between them is zero with $\cos 0 = 1$ and :$\mathbf a \cdot \mathbf b = \left\, \mathbf a \right\, \, \left\, \mathbf b \right\,$ This implies that the dot product of a vector a with itself is :$\mathbf a \cdot \mathbf a = \left\, \mathbf a \right\, ^2 ,$ which gives : $\left\, \mathbf a \right\, = \sqrt ,$ the formula for the Euclidean length of the vector.

## Scalar projection and first properties

The scalar projection (or scalar component) of a Euclidean vector a in the direction of a Euclidean vector b is given by :$a_b = \left\, \mathbf a \right\, \cos \theta ,$ where is the angle between a and b. In terms of the geometric definition of the dot product, this can be rewritten :$a_b = \mathbf a \cdot \widehat ,$ where $\widehat = \mathbf b / \left\, \mathbf b \right\,$ is the
unit vector In mathematics, a unit vector in a normed vector space is a Vector_(mathematics_and_physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
in the direction of b. The dot product is thus characterized geometrically by :$\mathbf a \cdot \mathbf b = a_b \left\, \mathbf \right\, = b_a \left\, \mathbf \right\, .$ The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar ''α'', :$\left( \alpha \mathbf \right) \cdot \mathbf b = \alpha \left( \mathbf a \cdot \mathbf b \right) = \mathbf a \cdot \left( \alpha \mathbf b \right) .$ It also satisfies a
distributive law In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
, meaning that :$\mathbf a \cdot \left( \mathbf b + \mathbf c \right) = \mathbf a \cdot \mathbf b + \mathbf a \cdot \mathbf c .$ These properties may be summarized by saying that the dot product is a
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called ''Vector (mathematics), vectors'') over a Field (mathematics), field ''K'' (the elements of which are called ''scalar (mathematics), scalars''). ...
. Moreover, this bilinear form is positive definite, which means that $\mathbf a \cdot \mathbf a$ is never negative, and is zero if and only if $\mathbf a = \mathbf 0$—the zero vector. The dot product is thus equivalent to multiplying the norm (length) of b by the norm of the projection of a over b.

## Equivalence of the definitions

If e1, ..., e''n'' are the standard basis vectors in R''n'', then we may write : The vectors e''i'' are an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit ...
, which means that they have unit length and are at right angles to each other. Hence since these vectors have unit length :$\mathbf e_i \cdot \mathbf e_i = 1$ and since they form right angles with each other, if , :$\mathbf e_i \cdot \mathbf e_j = 0 .$ Thus in general, we can say that: :$\mathbf e_i \cdot \mathbf e_j = \delta_ .$ Where δ ij is the
Kronecker delta In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
. Also, by the geometric definition, for any vector e''i'' and a vector a, we note :$\mathbf a \cdot \mathbf e_i = \left\, \mathbf a \right\, \, \left\, \mathbf e_i \right\, \cos \theta_i = \left\, \mathbf a \right\, \cos \theta_i = a_i ,$ where ''a''''i'' is the component of vector a in the direction of e''i''. The last step in the equality can be seen from the figure. Now applying the distributivity of the geometric version of the dot product gives :$\mathbf a \cdot \mathbf b = \mathbf a \cdot \sum_i b_i \mathbf e_i = \sum_i b_i \left( \mathbf a \cdot \mathbf e_i \right) = \sum_i b_i a_i= \sum_i a_i b_i ,$ which is precisely the algebraic definition of the dot product. So the geometric dot product equals the algebraic dot product.

# Properties

The dot product fulfills the following properties if a, b, and c are real vectors and ''r'' is a scalar. #
Commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
: #: $\mathbf \cdot \mathbf = \mathbf \cdot \mathbf ,$ #: which follows from the definition (''θ'' is the angle between a and b): #: $\mathbf \cdot \mathbf = \left\, \mathbf \right\, \left\, \mathbf \right\, \cos \theta = \left\, \mathbf \right\, \left\, \mathbf \right\, \cos \theta = \mathbf \cdot \mathbf .$ # Distributive over vector addition: #: $\mathbf \cdot \left(\mathbf + \mathbf\right) = \mathbf \cdot \mathbf + \mathbf \cdot \mathbf .$ # Bilinear: #: $\mathbf \cdot \left( r \mathbf + \mathbf \right) = r \left( \mathbf \cdot \mathbf \right) + \left( \mathbf \cdot \mathbf \right) .$ #
Scalar multiplication In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
: #: $\left( c_1 \mathbf \right) \cdot \left( c_2 \mathbf \right) = c_1 c_2 \left( \mathbf \cdot \mathbf \right) .$ # Not
associative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
because the dot product between a scalar (a ⋅ b) and a vector (c) is not defined, which means that the expressions involved in the associative property, (a ⋅ b) ⋅ c or a ⋅ (b ⋅ c) are both ill-defined. Note however that the previously mentioned scalar multiplication property is sometimes called the "associative law for scalar and dot product" or one can say that "the dot product is associative with respect to scalar multiplication" because ''c'' (a ⋅ b) = (''c'' a) ⋅ b = a ⋅ (''c'' b). #
Orthogonal In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
: #: Two non-zero vectors a and b are ''orthogonal''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
. # No cancellation: #: Unlike multiplication of ordinary numbers, where if , then ''b'' always equals ''c'' unless ''a'' is zero, the dot product does not obey the cancellation law: #: If and , then we can write: by the
distributive law In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
; the result above says this just means that a is perpendicular to , which still allows , and therefore allows . #
Product rule In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the stu ...
: #: If a and b are (vector-valued)
differentiable function In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...
s, then the derivative ( denoted by a prime ) of is given by the rule .

## Application to the law of cosines

Given two vectors a and b separated by angle ''θ'' (see image right), they form a triangle with a third side . The dot product of this with itself is: :$\begin \mathbf \cdot \mathbf & = \left( \mathbf - \mathbf\right) \cdot \left( \mathbf - \mathbf \right) \\ & = \mathbf \cdot \mathbf - \mathbf \cdot \mathbf - \mathbf \cdot \mathbf + \mathbf \cdot \mathbf \\ & = \mathbf^2 - \mathbf \cdot \mathbf - \mathbf \cdot \mathbf + \mathbf^2 \\ & = \mathbf^2 - 2 \mathbf \cdot \mathbf + \mathbf^2 \\ \mathbf^2 & = \mathbf^2 + \mathbf^2 - 2 \mathbf \mathbf \cos \mathbf \\ \end$ which is the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or Jamshīd al-Kāshī, al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the l ...
.

# Triple product

There are two
ternary operation In mathematics, a ternary operation is an ''n''-arity, ary Operation (mathematics), operation with ''n'' = 3. A ternary operation on a set (mathematics), set ''A'' takes any given three elements of ''A'' and combines them to form a single element o ...
s involving dot product and
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two Euclidean vector, vectors in a three-dimensional Orientation (vector space), oriented E ...
. The scalar triple product of three vectors is defined as :$\mathbf \cdot \left( \mathbf \times \mathbf \right) = \mathbf \cdot \left( \mathbf \times \mathbf \right)=\mathbf \cdot \left( \mathbf \times \mathbf \right).$ Its value is the
determinant In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics), function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In p ...
of the matrix whose columns are the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry ...
of the three vectors. It is the signed
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of the
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean ...
defined by the three vectors, and is isomorphic to the three-dimensional special case of the
exterior product In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point ...
of three vectors. The vector triple product is defined by :$\mathbf \times \left( \mathbf \times \mathbf \right) = \left( \mathbf \cdot \mathbf \right)\, \mathbf - \left( \mathbf \cdot \mathbf \right)\, \mathbf .$ This identity, also known as ''Lagrange's formula'', may be remembered as "ACB minus ABC", keeping in mind which vectors are dotted together. This formula has applications in simplifying vector calculations in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
.

# Physics

In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
, vector magnitude is a scalar in the physical sense (i.e., a
physical quantity A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
independent of the coordinate system), expressed as the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Pr ...
of a numerical value and a
physical unit A unit of measurement is a definite magnitude of a quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", ...
, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. For example: *
Mechanical work In physics, work is the energy transferred to or from an object via the application of force along a Displacement (vector), displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the pro ...
is the dot product of
force In physics, a force is an influence that can change the motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to accelerate. Force can ...
and
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
vectors, * Power is the dot product of
force In physics, a force is an influence that can change the motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to accelerate. Force can ...
and
velocity Velocity is the directional derivative, directional speed of an physical object, object in motion as an indication of its time derivative, rate of change in position (vector), position as observed from a particular frame of reference and as m ...
.

# Generalizations

## Complex vectors

For vectors with complex entries, using the given definition of the dot product would lead to quite different properties. For instance, the dot product of a vector with itself could be zero without the vector being the zero vector (e.g. this would happen with the vector a = i. This in turn would have consequences for notions like length and angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up the symmetric and bilinear properties of the dot product, through the alternative definition :$\mathbf \cdot \mathbf = \sum_i ,$ where $\overline$ is the
complex conjugate In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
of $b_i$. When vectors are represented by
column vector In linear algebra, a column vector with m elements is an m \times 1 Matrix_(mathematics), matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times ...
s, the dot product can be expressed as a
matrix product In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
involving a
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n Complex number, complex matrix (mathematics), matrix \boldsymbol is an n \times m matrix obtained by transpose, transposing \boldsymbol and applying ...
, denoted with the superscript H: :$\mathbf \cdot \mathbf = \mathbf^\mathsf \mathbf .$ In the case of vectors with real components, this definition is the same as in the real case. The dot product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, the complex dot product is sesquilinear rather than bilinear, as it is conjugate linear and not linear in a. The dot product is not symmetric, since :$\mathbf \cdot \mathbf = \overline .$ The angle between two complex vectors is then given by :$\cos \theta = \frac .$ The complex dot product leads to the notions of
Hermitian form In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
s and general
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
s, which are widely used in mathematics and
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
. The self dot product of a complex vector $\mathbf \cdot \mathbf = \mathbf^\mathsf \mathbf$, involving the conjugate transpose of a row vector, is also known as the norm squared, $\mathbf \cdot \mathbf = \, \mathbf\, ^2$, after the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
; it is a vector generalization of the '' absolute square'' of a complex scalar (see also: squared Euclidean distance).

## Inner product

The inner product generalizes the dot product to abstract vector spaces over a field of scalars, being either the field of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s $\R$ or the field of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
s $\Complex$. It is usually denoted using angular brackets by $\left\langle \mathbf \, , \mathbf \right\rangle$. The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is sesquilinear instead of bilinear. An inner product space is a
normed vector space In mathematics, a normed vector space or normed space is a vector space over the Real number, real or Complex number, complex numbers, on which a Norm (mathematics), norm is defined. A norm is the formalization and the generalization to real ve ...
, and the inner product of a vector with itself is real and positive-definite.

## Functions

The dot product is defined for vectors that have a finite number of entries. Thus these vectors can be regarded as discrete functions: a length- vector is, then, a function with
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Dom ...
, and is a notation for the image of by the function/vector . This notion can be generalized to
continuous function In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...
s: just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some interval (also denoted ): :$\left\langle u , v \right\rangle = \int_a^b u\left(x\right) v\left(x\right) d x$ Generalized further to complex functions and , by analogy with the complex inner product above, gives :$\left\langle \psi , \chi \right\rangle = \int_a^b \psi\left(x\right) \overline d x .$

## Weight function

Inner products can have a
weight function A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
(i.e., a function which weights each term of the inner product with a value). Explicitly, the inner product of functions $u\left(x\right)$ and $v\left(x\right)$ with respect to the weight function $r\left(x\right)>0$ is :$\left\langle u , v \right\rangle = \int_a^b r\left(x\right) u\left(x\right) v\left(x\right) d x.$

A double-dot product for
matrices Matrix most commonly refers to: * The Matrix (franchise), ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within Th ...
is the
Frobenius inner product In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
, which is analogous to the dot product on vectors. It is defined as the sum of the products of the corresponding components of two matrices A and B of the same size: :$\mathbf : \mathbf = \sum_i \sum_j A_ \overline = \operatorname \left( \mathbf^\mathsf \mathbf \right) = \operatorname \left( \mathbf \mathbf^\mathsf \right) .$ :$\mathbf : \mathbf = \sum_i \sum_j A_ B_ = \operatorname \left( \mathbf^\mathsf \mathbf \right) = \operatorname \left( \mathbf \mathbf^\mathsf \right) = \operatorname \left( \mathbf^\mathsf \mathbf \right) = \operatorname \left( \mathbf \mathbf^\mathsf \right) .$ (For real matrices) Writing a matrix as a dyadic, we can define a different double-dot product (see ,) however it is not an inner product.

## Tensors

The inner product between a
tensor In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vect ...
of order ''n'' and a tensor of order ''m'' is a tensor of order , see
Tensor contraction In multilinear algebra Multilinear algebra is a subfield of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and qua ...
for details.

# Computation

## Algorithms

The straightforward algorithm for calculating a floating-point dot product of vectors can suffer from
catastrophic cancellation In numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathemati ...
. To avoid this, approaches such as the Kahan summation algorithm are used.

## Libraries

A dot product function is included in: *
BLAS Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and m ...
level 1 real SDOT, DDOT; complex CDOTU, ZDOTU = X^T * Y, CDOTC ZDOTC = X^H * Y * Julia as    *
Matlab MATLAB (an abbreviation of "MATrix LABoratory") is a Proprietary software, proprietary multi-paradigm programming language, multi-paradigm programming language and numerical analysis, numeric computing environment developed by MathWorks. MATLA ...
as    or    or    * GNU Octave as    * Intel oneAPI Math Kernel Library real p?dot dot = sub(x)'*sub(y); complex p?dotc dotc = conjg(sub(x)')*sub(y)

*
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used Inequality (mathematics), inequalities in mathematics. The inequality for sums was published by . The c ...
*
Cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two Euclidean vector, vectors in a three-dimensional Orientation (vector space), oriented E ...
* Dot product representation of a graph *
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, the square-root of the self dot product *
Matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
*
Metric tensor In the mathematical field of differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of d ...
*
Multiplication of vectors In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
* Outer product