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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 Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
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 of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinea ...
s, for example in a
pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q(x ...
.
is an
algebraic operation Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
that takes two equal-length sequences of numbers (usually coordinate vectors), 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 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 pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
for more). Algebraically, the dot product is the sum of the
products 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 * Produ ...
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 c ...
,
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 and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
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'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
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 of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of 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 Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, rather than a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
, 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 vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
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 is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
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, 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 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'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
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 is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
, 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: functions, vectors, mat ...
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 and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. 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 \ [] \cdot [] &= ( \times ) + (\times) + (\times) \\ &= 4 - 6 + 5 \\ &= 3 \end Likewise, the dot product of the vector with itself is: : \begin \ [] \cdot [] &= ( \times ) + (\times) + (\times) \\ &= 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 from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
:\mathbf \cdot \mathbf = \mathbf\mathbf^\mathsf T, where \mathbf^\mathsf T denotes the
transpose In linear algebra, the transpose of a 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 other notations). The tr ...
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 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 n matrix for some n, c ...
) is multiplied by a 3 × 1 matrix (
column vector In linear algebra, a column vector with m elements is an m \times 1 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 n matrix for some n, c ...
) 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, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
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 (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 In mathematics, the scalar projection of a vector \mathbf on (or onto) a vector \mathbf, also known as the scalar resolute of \mathbf in the direction of \mathbf, is given by: :s = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf, wher ...
(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 (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
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'', : ( \alpha \mathbf ) \cdot \mathbf b = \alpha ( \mathbf a \cdot \mathbf b ) = \mathbf a \cdot ( \alpha \mathbf b ) . 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 ( \mathbf b + \mathbf c ) = \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 '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
. Moreover, this bilinear form is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
, 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 :\begin \mathbf a &= _1 , \dots , a_n= \sum_i a_i \mathbf e_i \\ \mathbf b &= _1 , \dots , b_n= \sum_i b_i \mathbf e_i. \end 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 is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
, 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, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. 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 ( \mathbf a \cdot \mathbf e_i ) = \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 Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
. #
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 o ...
: #: \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 (\mathbf + \mathbf) = \mathbf \cdot \mathbf + \mathbf \cdot \mathbf . # Bilinear: #: \mathbf \cdot ( r \mathbf + \mathbf ) = r ( \mathbf \cdot \mathbf ) + ( \mathbf \cdot \mathbf ) . #
Scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
: #: ( c_1 \mathbf ) \cdot ( c_2 \mathbf ) = c_1 c_2 ( \mathbf \cdot \mathbf ) . # Not associative 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: #: 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 In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . A ...
: #: 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, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
: #: If a and b are (vector-valued)
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
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 & = ( \mathbf - \mathbf) \cdot ( \mathbf - \mathbf ) \\ & = \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 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 law of cosines states ...
.


Triple product

There are two
ternary operation In mathematics, a ternary operation is an ''n''-ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''. In computer science, a ternary operator i ...
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 vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
. The scalar triple product of three vectors is defined as : \mathbf \cdot ( \mathbf \times \mathbf ) = \mathbf \cdot ( \mathbf \times \mathbf )=\mathbf \cdot ( \mathbf \times \mathbf ). Its value is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the matrix whose columns are the Cartesian coordinates 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). Th ...
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 Euclidea ...
defined by the three vectors, and is isomorphic to the three-dimensional special case of the
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
of three vectors. The vector triple product is defined by : \mathbf \times ( \mathbf \times \mathbf ) = ( \mathbf \cdot \mathbf )\, \mathbf - ( \mathbf \cdot \mathbf )\, \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 fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
.


Physics

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, vector magnitude is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
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 * Produ ...
of a
numerical value A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
and a
physical unit A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multip ...
, 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. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stre ...
is the dot product of
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
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 Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
is the dot product of
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
and
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
.


Generalizations


Complex vectors

For vectors with
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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 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 n matrix for some n, c ...
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 from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
involving a
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
, 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 In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows ...
rather than bilinear, as it is
conjugate linear In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y \ ...
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, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
s and general
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
s, which are widely used in mathematics and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. 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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
; it is a vector generalization of the ''
absolute square In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power  2, and is denoted by a superscript 2; for instance, the squar ...
'' of a complex scalar (see also:
squared Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...
).


Inner product

The inner product generalizes the dot product to abstract vector spaces over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of scalars, being either the field of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 ...
s \Complex . It is usually denoted using
angular brackets A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
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 In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows ...
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 or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
, 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 function In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s: 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 * Do ...
, 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 such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in 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(x) v(x) d x Generalized further to
complex function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
s and , by analogy with the complex inner product above, gives : \left\langle \psi , \chi \right\rangle = \int_a^b \psi(x) \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(x) and v(x) with respect to the weight function r(x)>0 is : \left\langle u , v \right\rangle = \int_a^b r(x) u(x) v(x) d x.


Dyadics and matrices

A double-dot product for
matrices Matrix most commonly refers to: * ''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 ''The Matrix'' (franchis ...
is the
Frobenius inner product In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise inner product of two matrices as though ...
, 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 ( \mathbf^\mathsf \mathbf ) = \operatorname ( \mathbf \mathbf^\mathsf ) . : \mathbf : \mathbf = \sum_i \sum_j A_ B_ = \operatorname ( \mathbf^\mathsf \mathbf ) = \operatorname ( \mathbf \mathbf^\mathsf ) = \operatorname ( \mathbf^\mathsf \mathbf ) = \operatorname ( \mathbf \mathbf^\mathsf ) . (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 algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
of order ''n'' and a tensor of order ''m'' is a tensor of order , see
Tensor contraction In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tens ...
for details.


Computation


Algorithms

The straightforward algorithm for calculating a floating-point dot product of vectors can suffer from catastrophic cancellation. 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 matrix ...
level 1 real SDOT, DDOT; complex CDOTU, ZDOTU = X^T * Y, CDOTC ZDOTC = X^H * Y *
Julia Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g ...
as    *
Matlab MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
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)


See also

*
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
*
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 vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
* 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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
, 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 from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
* Metric tensor *
Multiplication of vectors In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles: * Dot product – also known as the "scalar product", a binary operation that takes two vecto ...
* Outer product


Notes


References


External links

*
Explanation of dot product including with complex vectors

"Dot Product"
by Bruce Torrence,
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
, 2007. {{tensors Articles containing proofs Bilinear forms Operations on vectors Analytic geometry Tensors Scalars