Vector (mathematics and physics)

In mathematics and physics, vector is a term that refers informally to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.

Historically, vectors were introduced in

This section is an excerpt from Euclidean vector.[edit]

In

directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B,^[3]

and denoted by ${\textstyle {\stackrel {\longrightarrow }{AB}}.}$

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".

distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space

.

Vectors play an important role in

physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.^[8]

This section is an excerpt from Vector space.[edit]

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector space and complex vector space are kinds of vector spaces based on different kinds of scalars: real coordinate space or complex coordinate space.

Vector spaces generalize

systems of linear equations

.

Vector spaces are characterized by their

countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum

as a dimension.

Many vector spaces that are considered in mathematics are also endowed with other structures. This is the case of algebras, which include field extensions, polynomial rings, associative algebras and Lie algebras. This is also the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces.

Vectors in algebra

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 due to historical reasons.

• Vector quaternion, a quaternion
  with a zero real part
• p-vector, an element of the exterior algebra
  of a vector space.
• orientable, while the manifold of rotations is not. Spinors are elements of a vector subspace of some Clifford algebra
  .
• algebra over this ring, and has been introduced for handling carry propagation in the operations on p-adic numbers
  .

Data represented by vectors

See also:

Vector (data type)

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The set ${\displaystyle \mathbb {R} ^{n}}$ of

• rotation
  and magnitude is the angle of the rotation.
• Burgers vector, a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice
• Interval vector, 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.
• correlated. However, a random vector may also refer to a random variable
  that takes its values in a vector space.
• Logical vector, a vector of 0s and 1s (Booleans).

Vectors in calculus

Calculus serves as a foundational mathematical tool in the realm of vectors, offering a framework for the analysis and manipulation of vector quantities in diverse scientific disciplines, notably physics and engineering. Vector-valued functions, where the output is a vector, are scrutinized using calculus to derive essential insights into motion within three-dimensional space. Vector calculus extends traditional calculus principles to vector fields, introducing operations like gradient, divergence, and curl, which find applications in physics and engineering contexts. Line integrals, crucial for calculating work along a path within force fields, and surface integrals, employed to determine quantities like flux, illustrate the practical utility of calculus in vector analysis. Volume integrals, essential for computations involving scalar or vector fields over three-dimensional regions, contribute to understanding mass distribution, charge density, and fluid flow rates.^{[citation needed]}

See also

Look up vector in Wiktionary, the free dictionary.

• Vector (disambiguation)

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 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

Vector fields

A vector field is a vector-valued 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 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 ${\displaystyle \nabla }$
• Vector Laplacian
  , the vector Laplace operator, denoted by ${\displaystyle \nabla ^{2}}$, 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 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
• Phase vector

Notes