In

_{2}-graded vector space
*

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.
Vectors in Euclidean geometry

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 $\backslash mathbf$), 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

Everyalgebra 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 $\backslash mathbb\; R^n$ oftuple
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).
See also

* 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 ZSymplectic 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

Avector 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 $\backslash nabla$
* Vector Laplacian, the vector Laplace operator, denoted by $\backslash nabla^2$, 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),