Vector (mathematics and physics) explained

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 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 or other objects) 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 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 (for example, when discussing general properties of vector spaces).

Vectors in Euclidean geometry

See main article: Euclidean vector.

Vector spaces

See main article: Vector space.

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

See also: Vector (data type).

The set

R ^{n} 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 (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.

See also

• 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

\nabla

• Vector Laplacian, the vector Laplace operator, denoted by

\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 perpendicular to the original two
• 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

References

• Vectors - The Feynman Lectures on Physics
• Book: Heinbockel, J. H. . Introduction to Tensor Calculus and Continuum Mechanics . 2001 . Trafford Publishing . 1-55369-133-4.
• Book: Itô, Kiyosi . Encyclopedic Dictionary of Mathematics . 1993 . 2nd . . 978-0-262-59020-4.
  • Book: Pedoe, Daniel . Geometry: A comprehensive course . Dover . 1988 . 0-486-65812-0 . Daniel Pedoe . registration.