Vector (mathematics and physics)

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Short description: Element of a vector space


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

Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers.

The term vector is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length.

Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called a coordinate vector space.

There are many vector spaces that are considered in mathematics, such as extension field, polynomial rings, algebras and function spaces. The term vector is generally not used for elements of these vectors spaces, and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces (for example, when discussing general properties of vector spaces).

Vectors in Euclidean geometry

Main page: Euclidean vector
A vector pointing from A to B

In mathematics, physics and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector[1] or spatial vector[2]) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B,[3] and denoted by [math]\displaystyle{ \overrightarrow{AB} }[/math] .

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".[4] It was first used by 18th century astronomers investigating planetary revolution around the Sun.[5] The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors,[6] operations which obey the familiar algebraic laws of commutativity, associativity, and 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 physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors.[7] Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.[8]

Vector spaces

Main page: Vector space
Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.

In mathematics, physics, and engineering, 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. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

Vector spaces generalize Euclidean vectors, which allow modeling physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations.

Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic). A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example, polynomial rings are 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 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 due to historical reasons.

  • Vector quaternion, a quaternion with a zero real part
  • Multivector or p-vector, 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 p-adic numbers.

Data represented by vectors

The set [math]\displaystyle{ \mathbb R^n }[/math] of tuples of n real numbers has a natural structure of vector space defined by component-wise addition and scalar multiplication. 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 whose direction is that of the axis of a 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.
  • Random vector or multivariate random variable, in statistics, a set of real-valued 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.
  • Logical vector, a vector of 0s and 1s (Booleans).

See also

  • Vector (disambiguation)

Vector spaces with more structure

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,

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 [math]\displaystyle{ \nabla }[/math]
  • Vector Laplacian, the vector Laplace operator, denoted by [math]\displaystyle{ \nabla^2 }[/math], 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

Notes

  1. Ivanov 2001
  2. Heinbockel 2001
  3. Itô 1993, p. 1678; Pedoe 1988
  4. Latin: vectus, perfect participle of vehere, "to carry"/ veho = "I carry". For historical development of the word vector, see vector n. (3rd ed.), Oxford University Press, September 2005, http://oed.com/search?searchType=dictionary&q=vector+%27%27n.%27%27  (Subscription or UK public library membership required.) and Jeff Miller. "Earliest Known Uses of Some of the Words of Mathematics". http://jeff560.tripod.com/v.html. 
  5. The Oxford english dictionary. (2nd. ed.). London: Claredon Press. 2001. ISBN 9780195219425. 
  6. "vector | Definition & Facts" (in en). https://www.britannica.com/science/vector-mathematics. 
  7. "Vectors". https://www.mathsisfun.com/algebra/vectors.html. 
  8. Weisstein, Eric W.. "Vector" (in en). https://mathworld.wolfram.com/Vector.html. 

References