Vector (mathematics and physics)

Updated on Nov 26, 2024

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When used without any further description, vector refers either to:

Most generally, an element of a vector space

In physics and geometry, a Euclidean vector, used to represent physical quantities that have both magnitude and direction

Vector can also have a variety of different meanings depending on context.

Vectors

An element of a vector space

An element of the real coordinate space Rⁿ

Basis vector, one of a set of vectors (a "basis") that, in linear combination, can represent every vector in a given vector space

Column vector or row vector, a one-dimensional matrix often representing the solution of a system of linear equations

Coordinate vector, in linear algebra, an explicit representation of an element of any abstract vector space

Axial vector, or pseudovector, a quantity that transforms like a vector under proper rotation but not generally under reflection

Darboux vector, the areal velocity vector of the Frenet frame of a space curve

Displacement vector, a vector that specifies the change in position of a point relative to a previous position

Euclidean vector, a geometric entity endowed with magnitude and direction as well as 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.

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 one astronomical body around another

Normal vector, or surface normal, a vector that is perpendicular to a (hyper)surface at a point

Vector product, or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector

Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space

Gradient vector, the vector giving the magnitude and direction of maximum increase of a scalar field

Gyrovector, a hyperbolic geometry version of a vector

Interval vector, in musical set theory, an array that expresses the intervallic content of a pitch-class set

Null vector, a vector whose magnitude is zero

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

Tangent vector, an element of the tangent space of a curve, a surface or, more generally, a differential manifold at a given point.

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

Unit vector, a vector in a normed vector space whose length is 1

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 Z₂-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 decomposition, a technique to convert a vector of Rⁿ into several vectors, each linearly independent

Vector differential, or del, a vector differential operator represented by the nabla symbol ∇

Vector Laplacian, the vector Laplace operator, denoted by ∇ 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 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

References

Vector (mathematics and physics) Wikipedia

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