Vector (mathematics and physics)

In mathematics and physics, a vector is a physical quantity that cannot be expressed by a single number. The term may also be used to refer to elements of some vector spaces, and in some contexts, is used for tuples, which are finite sequences of a fixed length.
Historically, vectors were introduced in geometry and physics 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.
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 the 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.
Many vector spaces are considered in mathematics, such as extension fields, polynomial rings, algebras and function spaces. The term vector is generally not used for elements of these vector spaces, and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces.

Vectors in Euclidean geometry

Vector quantities

Vector 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
Multivector or -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 of tuples of 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.
Vectors in 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.

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 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 associated with a symmetry
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 (mathematics and physics)

Vectors in Euclidean geometry

Vector quantities

Vector spaces

Vectors in algebra

Data represented by vectors

Vectors in calculus

Vector spaces with more structure

Vector fields