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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a
symmetric bilinear formA 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 bilinear function B tha ...
, for example for a
pseudo-Euclidean spaceIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.
is an
algebraic operation In mathematics, a basic algebraic operation is any one of the common Operation (mathematics), operations of arithmetic, which include addition, subtraction, multiplication, Division (mathematics), division, raising to an integer exponentiation, powe ...
that takes two equal-length sequences of numbers (usually
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their a ...
s), and returns a single number. In
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, the dot product of the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

Cartesian coordinates
of two
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 * Produc ...
of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the
Euclidean magnitude
Euclidean magnitude
s of the two vectors and the
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

cosine
of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
,
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
s are often defined by using
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s. In this case, the dot product is used for defining lengths (the length of a vector is the
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
of the dot product of the vector by itself) and angles (the cosine of the angle of two vectors is the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
of their dot product by the product of their lengths). The name "dot product" is derived from the
centered dot An interpunct, , also known as an interpoint, middle dot, middot and centered dot or centred dot, is a punctuation mark consisting of a vertically centered dot used for interword separation in ancient Latin alphabet, Latin script. (Word-separatin ...

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 as ...
, rather than a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
, 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 three-dimensional space \mathbb^3, and is denoted by the sym ...

vector product
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 Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
for Euclidean space. In modern presentations of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, the points of space are defined in terms of their
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

Cartesian coordinates
, and
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
itself is commonly identified with the
real coordinate space of ordered pairs . Blue lines denote coordinate axes, horizontal green lines are integer , vertical cyan lines are integer , brown-orange lines show half-integer or , magenta and its tint show multiples of one tenth (best seen under magnification ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
of the dot product of the vector by itself, and the
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

cosine
of the (non oriented) angle of 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.


Algebraic definition

The dot product of two vectors and is defined as: :\mathbf\cdot\mathbf=\sum_^n _i_i=_1_1+_2_2+\cdots+_n_n where Σ denotes
summation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

summation
and ''n'' is the
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
of the
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. For instance, in
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
, the dot product of vectors and is: : \begin \ [] \cdot [] &= ( \times ) + (\times) + (\times) \\ &= 4 - 6 + 5 \\ &= 3 \end If vectors are identified with row matrix, row matrices, the dot product can also be written as a matrix multiplication, matrix product :\mathbf \cdot \mathbf = \mathbf\mathbf^\mathsf T, where \mathbf^\mathsf T denotes the
transpose 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 a ...

transpose
of \mathbf. Expressing the above example in this way, a 1 × 3 matrix (
row 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 and th ...
) is multiplied by a 3 × 1 matrix (
column 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 and ...
) 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 classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, a
Euclidean vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
between and . In particular, if the vectors and are
orthogonal In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bili ...
(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 In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occa ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Moreover, this bilinear form is
positive definiteIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, 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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. 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 Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
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 as ...
. #
Commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
: #: \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 (mathematics), module in abstract algebra). In common geometrical contexts, scalar multiplication of a re ...

Scalar multiplication
: #: ( c_1 \mathbf ) \cdot ( c_2 \mathbf ) = c_1 c_2 ( \mathbf \cdot \mathbf ) . # Not
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bili ...
: #: Two non-zero vectors a and b are ''orthogonal''
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
. # 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 (algebra), magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always imp ...
: #: If and , then we can write: by the
distributive law In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
; 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 (mathematics), functions. For two functions, it may be stated in Notation for differentiatio ...
: #: If a and b are (vector-valued)
differentiable function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

differentiable function
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 Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focu ...
.


Triple product

There are two
ternary operationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s involving dot product and
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

cross product
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
of the matrix whose columns are the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

Cartesian coordinates
of the three vectors. It is the signed
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
of the
parallelepiped In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

parallelepiped
defined by the three vectors, and is isomorphic to the three-dimensional special case of the
exterior product In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
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'', 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 (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
.


Physics

In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
, 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 as ...
in the physical sense (i.e., a
physical quantity A physical quantity is a physical property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ''numerical value'' ...
independent of the coordinate system), expressed as the product 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 b ...

numerical value
and a
physical unit A unit of measurement is a definite magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematic ...
, 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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
is the dot product of
force In physics, a force is an influence that can change the motion (physics), 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 acce ...

force
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 c ...
vectors, *
Power Power most often refers to: * Power (physics) In physics, power is the amount of energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, bu ...
is the dot product of
force In physics, a force is an influence that can change the motion (physics), 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 acce ...

force
and
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

velocity
.


Generalizations


Complex vectors

For vectors with
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

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 would be an arbitrary complex number, and could be zero without the vector being the zero vector (such vectors are called
isotropic Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by ...
); 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 scalar product, through the alternative definition : \mathbf \cdot \mathbf = \sum_i , where \overline is the
complex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of b_i. When vectors are represented by
row 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 and th ...
s, the dot product can be expressed as a matrix product involving a
conjugate transpose In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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 scalar product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, the complex scalar product is
sesquilinear In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
rather than bilinear, as it is conjugate linear and not linear in a. The scalar product is not symmetric, since : \mathbf \cdot \mathbf = \overline . The angle between two complex vectors is then given by : \cos \theta = \frac . The complex scalar 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 map, linear in each of its arguments, but a sesquilinear fo ...
s and general
inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s, which are widely used in mathematics and
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
. The self dot product of a complex vector \mathbf \cdot \mathbf = \mathbf^\mathsf \mathbf , involving the conjugate transpose of a row vector, is a generalization of the scalar ''
absolute square . Each block represents one unit, , and the entire square represents , or the area of the square. In mathematics, a square is the result of multiplication, multiplying a number by itself. The verb "to square" is used to denote this operation. Squa ...
'' of a complex number.


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 grassl ...
of scalars, being either the field of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s \R or the field of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
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 ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
instead of bilinear. An inner product space is a
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, 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 theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''elements'', or ''terms''). ...
s: a length- vector is, then, a function with
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
, and is a notation for the image of by the function/vector . This notion can be generalized to
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 of the function . Hue represents the Argument (complex analysis), argument, brightness the Absolute_value#Complex_numbers, magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of m ...
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 or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
is the Frobenius inner product, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

tensor
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 vector space, dual. In components, it is expressed as a sum of products of scalar compo ...
for details.


Computation


Algorithms

The straightforward algorithm for calculating a floating-point dot product of vectors can suffer from
catastrophic cancellation Catastrophe or catastrophic comes from the Greek κατά (''kata'') = down; στροφή (''strophē'') = turning ( el, καταστροφή). It may refer to: A general or specific event * Disaster, a devastating event * The Asia Minor A ...

catastrophic cancellation
. To avoid this, approaches such as the
Kahan summation algorithmIn numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite-decimal precision, precision floating-point numbers, compared to ...
are used.


Libraries

A dot product function is included in: *
BLASBlas is mainly a Spanish given name and surname, related to Blaise. It may refer to Places * Piz Blas, mountain in Switzerland * San Blas (disambiguation), many places - see separate article, also **Cape San Blas Light, lighthouse ** Church San Bl ...

BLAS
level 1 real SDOT, DDOT; complex CDOTU, ZDOTU = X^T * Y, CDOTC ZDOTC = X^H * Y *
Matlab MATLAB (an abbreviation of "MATrix LABoratory") is a and environment developed by . MATLAB allows manipulations, plotting of and data, implementation of s, creation of s, and interfacing with programs written in other languages. Althoug ...
as A' * B or conj(transpose(A)) * B or sum( conj(A) .* B) * GNU Octave as sum (conj (X) .* Y, dim) * 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
*
Cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

Cross product
* Dot product representation of a graph *
Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
, the square-root of the self dot product *
Matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

Matrix multiplication
*
Metric tensor In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
*
Multiplication of vectorsIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
*
Outer product 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 and t ...


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 hoste ...
, 2007. {{tensors Articles containing proofs Bilinear forms Operations on vectors Analytic geometry Tensors Scalars