In mathematics and physics, a vector is an element of a vector space.
For many specific vector spaces, the vectors have received specific names, which are listed below.
Historically, vectors were introduced in geometry and physics (typically in mechanics) before the formalization of the concept of vector space. Therefore, one talks often of vectors without specifying the vector space to which they belong. Specifically, in a Euclidean space, one considers spatial vectors, also called Euclidean vectors which are used to represent quantities that have both magnitude and direction, and may be added and scaled (that is multiplied by a real number) for forming a vector space.
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Contents
Vectors in Euclidean geometry
In classical Euclidean geometry (that is in synthetic geometry), vectors were introduced (during 19th century) as equivalence classes, under equipollence, of ordered pairs of points; two pairs (A, B) and (C, D) being equipollent if the points A, B, D, C, in this order, form a parallelogram. Such an equivalence class is called a vector, more precisely, a Euclidean vector.^{[1]} The equivalence class of (A, B) is often denoted
A Euclidean vector, is thus an entity endowed with a magnitude (the length of the line segment (A, B)) and a direction (the direction from A to B). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars, which have no direction. For example, velocity, forces and acceleration are represented by vectors.
In modern geometry, Euclidean spaces are often defined from linear algebra. More precisely, a Euclidean space E is defined as a set to which is associated a inner product space of finite dimension over the reals and a group action of the additive group of which is free and transitive (See Affine space for details of this construction). The elements of are called translations.
It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations.
Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of equipped with the dot product. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product.
The Euclidean space is often presented as the Euclidean space of dimension n. This is motivated by the fact that every Euclidean space of dimension n is isomorphic to the Euclidean space More precisely, given such a Euclidean space, one may choose any point O as an origin. By Gram–Schmidt process, one may also find an orthonormal basis of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point P of the space, as the coordinates on this basis of the vector These choices define an isomorphism of the given Euclidean space onto by mapping any point to the ntuple of its Cartesian coordinates, and every vector to its coordinate vector.
Specific vectors in a vector space
 Zero vector (sometimes called null vector), the additive identity in a vector space. In a normed vector space, it is the unique vector of norm zero. In a Euclidean vector space, it is the unique vector of length zero.
 Basis vector, an element of a given basis of a vector space.
 Unit vector, a vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.
 Isotropic vector or null vector, in a vector space with a quadratic form, a nonzero vector for which the form is zero. If a null vector exists, the quadratic form is said an isotropic quadratic form.
Vectors in specific vector spaces
 Column vector, a matrix with only one column. The column vectors with a fixed number of rows form a vector space.
 Row vector, a matrix with only one row. The row vectors with a fixed number of columns form a vector space.
 Coordinate vector, the ntuple of the coordinates of a vector on a basis of n elements. For a vector space over a field F, these ntuples form the vector space (where the operation are pointwise addition and scalar multiplication).
 Displacement vector, 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.
 Position vector 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 or an affine space.
 Velocity vector, 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, also called axial vector, an element of the dual of a vector space. In a inner product space, the inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a pseudo vector from a vector. The distinction becomes apparent when one changes coordinates: the matrix used for a change of coordinates of pseudovectors is the transpose of that of vectors.
 Tangent vector, an element of the tangent space of a curve, a surface or, more generally, a differential manifold at a given point (these tangent spaces are naturally endowed with a structure of vector space)
 Normal vector 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. Normals are pseudovectors that belong to the dual of the tangent space.
 Gradient, the coordinates vector of the partial derivatives of a function of several real variables. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a scalar field. The gradient is a pseudo vector that is normal to a level curve.
 Fourvector, in the theory of relativity, a vector in a fourdimensional real vector space called Minkowski space
Tuples that are not really vectors
The set of tuples of n real numbers has a natural structure of vector space defined by componentwise addition and scalar multiplication. When such tuples are used for representing some data, it is common to call them vectors even if the vector addition does not mean anything for these data, which may make the terminology confusing. Similarly, some physical phenomena involve a direction and a magnitude. They are often represented by vectors, even if operations of vector spaces do not apply to them.
 Rotation vector, a Euclidean vector whose direction is that of the axis of a rotation and magnitude is the angle of the rotation.
 Darboux vector, the areal velocity vector of the Frenet frame of a space curve
 Burgers vector, a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice
 Laplace–Runge–Lenz vector, a vector used chiefly to describe the shape and orientation of the orbit of an astronomical body around another
 Interval vector, in musical set theory, an array that expresses the intervallic content of a pitchclass set
 Poynting vector, in physics, a vector representing the energy flux density of an electromagnetic field
 Probability vector, in statistics, a vector with nonnegative entries that sum to one.
 Random vector or multivariate random variable, in statistics, a set of realvalued random variables that may be correlated. However, a random vector may also refer to a random variable that takes its values in a vector space.
 Vector relation, a binary relation determined by a logical vector.
 Wave vector, a representation of the local phase evolution of a wave
Vectors in algebras
Every algebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called vectors, mainly for historical reasons.
 Vector quaternion, a quaternion with a zero real part
 Multivector or pvector, an element of the exterior algebra of a vector space.
 Spinors, 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 of rotation vectors is orientable, while the manifold of rotations is not. Spinors are elements of a vector subspace of some Clifford algebra.
 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 padic numbers.
See also
Look up vector in Wiktionary, the free dictionary. 
Vector spaces with more structure
 Graded vector space, a type of vector space that includes the extra structure of gradation
 Normed vector space, a vector space on which a norm is defined
 Hilbert space
 Ordered vector space, a vector space equipped with a partial order
 Super vector space, name for a Z_{2}graded vector space
 Symplectic vector space, a vector space V equipped with a nondegenerate, skewsymmetric, bilinear form
 Topological vector space, a blend of topological structure with the algebraic concept of a vector space
Vector fields
A vector field is a vectorvalued function that, generally, has a domain of the same dimension (as a manifold) as its codomain,
 Conservative vector field, a vector field that is the gradient of a scalar potential field
 Hamiltonian vector field, a vector field defined for any energy function or Hamiltonian
 Killing vector field, a vector field on a Riemannian manifold
 Solenoidal vector field, a vector field with zero divergence
 Vector potential, 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
 Vector Analysis, a textbook on vector calculus by Wilson, first published in 1901, which did much to standardize the notation and vocabulary of threedimensional 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
 Vector Laplacian, the vector Laplace operator, denoted by , 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 threedimensional Euclidean space, producing a third threedimensional Euclidean vector
 Vector projection, also known as vector resolute or vector component, a linear mapping producing a vector parallel to a second vector
 Vectorvalued function, a 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
 ^ In some old texts, the pair (A, B) is called a bound vector, and its equivalence class is called a free vector.