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 consider *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.

## Contents

- 1 Vectors in Euclidean geometry
- 2 Specific vectors in a vector space
- 3 Vectors in specific vector spaces
- 4 Vectors representing physical (or mathematical) quantities
- 5 Others
- 6 Vector fields
- 7 Vector spaces
- 8 Manipulation of vectors, fields, and spaces
- 9 Other uses in mathematics and physics
- 10 See also

## Vectors in Euclidean geometry

- Euclidean vector, a geometric entity endowed with magnitude and direction and a positive-definite inner product; an element of a Euclidean vector space. In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, such as force, in contrast to scalar quantities, which have no direction.

## Specific vectors in a vector space

- 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.

## 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 rows form a vector space.
- Coordinate vector, the n-uple of the coordinates of a vector on a basis of n elements. For a vector space over a field F, these n-uples 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.
- 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.
- Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space

## Vectors representing physical (or mathematical) quantities

Vector are often used to represent physical quantities that have both magnitude and direction. In this case, the vector space to which they belong are not always explicitly defined.

- Velocity vector
- 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

## Others

- Vector product, or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector
- Interval vector, in musical set theory, an array that expresses the intervallic content of a pitch-class set
- P-vector, the tensor obtained by taking linear combinations of the wedge product of p tangent vectors
- Position vector, a vector representing the position of a point in an affine space in relation to a reference point
- Poynting vector, in physics, a vector representing the energy flux density of an electromagnetic field
- Probability vector, in statistics, a vector with non-negative entries that sum to one
- Random vector or multivariate random variable, in statistics, a set of real-valued random variables that may be correlated
- Spin vector, or
*spinor*, is an element of a complex vector space introduced to expand the notion of spatial vector - The vector part of a quaternion, a mathematical entity that is one possible generalisation of a vector
- Tuple, an ordered list of numbers, sometimes used to represent a vector
- Wave vector, a vector representation of the local phase evolution of a wave

## Vector fields

- Vector field, a construction in vector calculus that associates a vector to every point in a subset of Euclidean space
- 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

## Vector spaces

- Vector space, a mathematical structure made up of vectors, objects that may be added with another vector or multiplied by a scalar value
- Dual vector space, a vector space consisting of all linear functionals on another, given vector space
- Euclidean vector space, an
*n*-dimensional space with notions of distance and angle that obey the Euclidean relationships - 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
- 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 non-degenerate, skew-symmetric, bilinear form
- Topological vector space, a blend of topological structure with the algebraic concept of a vector space

## Manipulation of vectors, fields, and spaces

*Vector Analysis,*a textbook on vector calculus by 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 - 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 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 mathematical function that maps real numbers to vectors
- Vectorization (mathematics), a linear transformation that converts a matrix into a column vector

## Other uses in mathematics and physics

- 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
- Witt vector, an infinite sequence of elements of a commutative ring

## See also

Look up in Wiktionary, the free dictionary.vector |

This article includes a list of related items that share the same name (or similar names). If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article. |