TheInfoList

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 ...
and
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 succession of eve ...

, a vector is an element of a
vector 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). ...
. For many specific vector spaces, the vectors have received specific names, which are listed below. In general, 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 with both length and direction (and so is frequently represented as a
ray Ray may refer to: Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive ...
). Such vectors can be added to each other or scaled using
vector algebraIn mathematics, vector algebra may mean: * Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space. * The algebraic operations in vector calculus Vector calculus, or vector anal ...
. Correspondingly, an ensemble of vectors is called a
vector 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). ...
. These objects are the subject of
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 through matrix (mat ...
and can be characterized by their
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 ...

. Historically, vectors were introduced 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 figures. A mat ...

and physics (typically in
mechanics Mechanics (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 is approximately 10.7 million ...

) before the formalization of the concept of a vector space. (In fact, the
Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

word ''vector'' means "carrier".) Therefore, one often talks about vectors without specifying the vector space to which they belong. Specifically, in a
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
, one considers ''s'', also called ''Euclidean vectors'' which are used to represent quantities that have both magnitude and direction, and may be , subtracted and (i.e. multiplied by a
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 ...
) for forming a vector space.

# Vector spaces

## Concepts related to vector spaces

;Specific vectors in a vector space *
Zero 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 ...
(sometimes also called ''null vector'' and denoted by $\mathbf$), the
additive identityIn 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 ...
in a vector space. In 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 ...
, it is the unique vector of norm zero. In a
Euclidean vector space 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 p ...
, it is the unique vector of length zero. *
Basis vector 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 ...
, an element of a given
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of a vector space. *
Unit vector 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 ...
, a vector in a normed vector space whose
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
is 1, or 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 ...
of length one. * Isotropic vector or
null vector 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 ...
, in a vector space with a
quadratic 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 ...
, a non-zero vector for which the form is zero. If a null vector exists, the quadratic form is said an
isotropic quadratic form In mathematics, a quadratic 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 (ma ...
. ;Vectors in specific vector spaces *
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 ...
, a matrix with only one column. The column vectors with a fixed number of rows form a vector space. *
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 ...
, a matrix with only one row. The row vectors with a fixed number of columns form a vector space. *
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 ...
, the -tuple of the
coordinates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

of a vector on a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of elements. For a vector space 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 ...
, these -tuples form the vector space $F^n$ (where the operation are pointwise addition and scalar multiplication). *
Displacement vector In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
, a vector that specifies the change in position of a point relative to a previous position. Displacement vectors belong to the vector space of
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
. *
Position vector In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

of a point, the displacement vector from a reference point (called the ''origin'') to the point. A position vector represents the position of a point in a
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
or an
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
. *
Velocity vector The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction o ...
, the derivative, with respect to time, of the position vector. It does not depend of the choice of the origin, and, thus belongs to the vector space of translations. *
Pseudovector 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. "Ph ...

, also called ''axial vector'' *
Covector 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 gene ...
, an element of the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
of a vector space. In an
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 ...
, the inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a covector from a vector. The distinction becomes apparent when one changes coordinates (non-orthogonally). *
Tangent vector :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
, an element of the
tangent space In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...
of a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

, a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
or, more generally, a
differential manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
at a given point (these tangent spaces are naturally endowed with a structure of vector space) *
Normal vector 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 ...

or simply ''normal'', in a Euclidean space or, more generally, in an inner product space, a vector that is perpendicular to a tangent space at a point. *
Gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

, the coordinates vector of the partial derivatives of a
function of several real variables In mathematical analysis, and applications in geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematic ...
. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a
scalar field 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). ...

. The gradient is a covector that is normal to a
level curve 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 ...
. *
Four-vector In special relativity 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 oth ...
, in the
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time ...
, a vector in a four-dimensional real vector space called
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclid ...

# Vectors in algebras

Every
algebra over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...
is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called ''vectors'', mainly due to historical reasons. * Vector quaternion, a
quaternion 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 an ...

with a zero real part *
Multivector In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded algebra, graded, associative algebra, associative and alternating algebra, alternating, and ...
or -vector, an element of the
exterior algebra 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 ...
of a vector space. *
Spinor In geometry and physics, spinors are elements of a 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 re ...
s, also called ''spin vectors'', have been introduced for extending the notion of rotation vector. In fact, rotation vectors represent well rotations ''locally'', but not globally, because a closed loop in the space of rotation vectors may induce a curve in the space of rotations that is not a loop. Also, the
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

of rotation vectors is
orientable is non-orientable 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 a ...
, while the manifold of rotations is not. Spinors are elements of a vector subspace of some
Clifford algebra In mathematics, a Clifford algebra is an algebra over a field, algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real nu ...
. * Witt vector, an infinite sequence of elements of a commutative ring, which belongs to an algebra over this ring, and has been introduced for handling carry propagation in the operations on
p-adic number group 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, analys ...
s.

# Data represented by vectors

The set $\mathbb R^n$ of
tuple 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). ...
s of real numbers has a natural structure of vector space defined by component-wise addition and
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 ...

. It is common to call these tuples ''vectors'', even in contexts where vector-space operations do not apply. More generally, when some data can be represented naturally by vectors, they are often called ''vectors'' even when addition and scalar multiplication of vectors are not valid operations on these data. Here are some examples. * Rotation vector, 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 ...
whose direction is that of the axis of a
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
and magnitude is the angle of the rotation. *
Burgers vectorIn materials science The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials science s ...
, a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice *
Interval vector Image:Interval vector C major chord.png, Interval vector: C major chord, set List of pitch-class sets, 3-11B, : 001110. File:Deep scale property of the diatonic scale.png, Diatonic scale in the chromatic circle with each interval vector a different ...
, in musical set theory, an array that expresses the intervallic content of a pitch-class set * Probability vector, in statistics, a vector with non-negative entries that sum to one. *
Random vector In probability Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...
or
multivariate random variable In probability Probability is the branch of 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 th ...
, in
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

, a set of
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
-valued
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s that may be
correlated In , correlation or dependence is any statistical relationship, whether or not, between two s or . In the broadest sense correlation is any statistical association, though it actually refers to the degree to which a pair of variables are rel ...
. However, a ''random vector'' may also refer to a
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
that takes its values in a vector space. * Logical vector, a vector of 0s and 1s ( Booleans).

* Vector (disambiguation)

## Vector spaces with more structure

*
Graded 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 and t ...
, a type of vector space that includes the extra structure of gradation *
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 ...
, a vector space on which a norm is defined *
Hilbert 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 and ...
*
Ordered vector space In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Definition Given a vector space ''X'' over the real numbers R and a preo ...
, a vector space equipped with a partial order *
Super vector spaceIn mathematics, a super vector space is a Quotient ring, \mathbb Z_2-graded vector space, that is, a vector space over a field (mathematics), field \mathbb K with a given direct sum, decomposition of subspaces of grade 0 and grade 1. The study of sup ...
, name for a Z2-graded vector space *
Symplectic vector spaceIn mathematics, a symplectic vector space is a vector space ''V'' over a Field (mathematics), field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a map (mathematics), mapping that is ...
, a vector space V equipped with a non-degenerate, skew-symmetric, bilinear form *
Topological vector 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). It ...
, a blend of topological structure with the algebraic concept of a vector space

## Vector fields

A
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

is a
vector-valued function A vector-valued function, also referred to as a vector function, is a function (mathematics), mathematical function of one or more variable (mathematics), variables whose range of a function, range is a set of multidimensional vector (mathematics ...

that, generally, has a domain of the same dimension (as a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

) as its codomain, *
Conservative vector field In vector calculus, a conservative vector field is a vector field that is the gradient of some function (mathematics), function. Conservative vector fields have the property that the line integral is path independent; the choice of any path betwee ...
, a vector field that is the gradient of a scalar potential field *
Hamiltonian vector fieldIn mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician William Rowan Hamilton, Sir William Rowan Hamilton, a Hamilt ...
, a vector field defined for any energy function or Hamiltonian *
Killing vector fieldIn 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 ...
, a vector field on a Riemannian manifold *
Solenoidal vector field In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes use ...
, a vector field with zero divergence *
Vector potential In vector calculus, a vector potential is a vector field whose Curl (mathematics), curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector ...
, a vector field whose curl is a given vector field * Vector flow, a set of closely related concepts of the flow determined by a vector field

## Miscellaneous

*
Ricci calculus 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 ...
* ''
Vector Analysis Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
,'' a textbook on vector calculus by Edwin Bidwell Wilson, Wilson, first published in 1901, which did much to standardize the notation and vocabulary of three-dimensional linear algebra and vector calculus * Vector bundle, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space * Vector calculus, a branch of mathematics concerned with differentiation and integration of vector fields * Vector differential, or ''del'', a vector differential operator represented by the nabla symbol $\nabla$ * Vector Laplacian, the vector Laplace operator, denoted by $\nabla^2$, is a differential operator defined over a vector field * Vector notation, common notation used when working with vectors * Vector operator, a type of differential operator used in vector calculus * Vector product, or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector * Vector projection, also known as ''vector resolute'' or ''vector component'', a linear mapping producing a vector parallel to a second vector * Vector-valued function, a function (mathematics), function that has a vector space as a codomain * Vectorization (mathematics), a linear transformation that converts a matrix into a column vector * Vector autoregression, an econometric model used to capture the evolution and the interdependencies between multiple time series * Vector boson, a boson with the spin quantum number equal to 1 * Vector measure, a function defined on a family of sets and taking vector values satisfying certain properties * Vector meson, a meson with total spin 1 and odd parity * Vector quantization, a quantization technique used in signal processing * Vector soliton, a solitary wave with multiple components coupled together that maintains its shape during propagation * Vector synthesis, a type of audio synthesis

# Notes

{{reflist Broad-concept articles Vectors (mathematics and physics),