Vector (mathematics and physics)

In mathematics and physics, a vector is an element of a vector space.

For many specific vector spaces, the vectors have received specific names, which are listed below.

Historically, vectors were introduced in geometry and physics (typically in mechanics) before the formalization of the concept of vector space. Therefore, one often talks about vectors without specifying the vector space to which they belong. Specifically, in a Euclidean space, one considers spatial vectors, also called Euclidean vectors which are used to represent quantities that have both magnitude and direction, and may be added, subtracted and scaled (i.e. multiplied by a real number) for forming a vector space.[1]

Vectors in Euclidean geometry

In classical Euclidean geometry (i.e., synthetic geometry), vectors were introduced (during the 19th century) as equivalence classes under equipollence, of ordered pairs of points; two pairs (A, B) and (C, D) being equipollent if the points A, B, D, C, in this order, form a parallelogram. Such an equivalence class is called a vector, more precisely, a Euclidean vector.[2] The equivalence class of (A, B) is often denoted ${\displaystyle {\overrightarrow {AB}}.}$

A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment (A, B)) and same direction (e.g., the direction from A to B).[3] In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars, which have no direction.[4] For example, velocity, forces and acceleration are represented by vectors.

In modern geometry, Euclidean spaces are often defined from linear algebra. More precisely, a Euclidean space E is defined as a set to which is associated an inner product space of finite dimension over the reals ${\displaystyle {\overrightarrow {E}},}$ and a group action of the additive group of ${\displaystyle {\overrightarrow {E}},}$ which is free and transitive (See Affine space for details of this construction). The elements of ${\displaystyle {\overrightarrow {E}}}$ are called translations.

It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations.

Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of ${\displaystyle \mathbb {R} ^{n}}$ equipped with the dot product. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, ${\displaystyle \mathbb {R} ^{n}}$ is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product.

The Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is often presented as the Euclidean space of dimension n. This is motivated by the fact that every Euclidean space of dimension n is isomorphic to the Euclidean space ${\displaystyle \mathbb {R} ^{n}.}$ More precisely, given such a Euclidean space, one may choose any point O as an origin. By Gram–Schmidt process, one may also find an orthonormal basis of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point P of the space, as the coordinates on this basis of the vector ${\displaystyle {\overrightarrow {OP}}.}$ These choices define an isomorphism of the given Euclidean space onto ${\displaystyle \mathbb {R} ^{n},}$ by mapping any point to the n-tuple of its Cartesian coordinates, and every vector to its coordinate vector.

Specific vectors in a vector space

• Zero vector (sometimes also called null vector and denoted by ${\displaystyle \mathbf {0} }$[5]), the additive identity in a vector space. In a normed vector space, it is the unique vector of norm zero. In a Euclidean vector space, it is the unique vector of length zero.[6]
• Basis vector, an element of a given basis of a vector space.
• Unit vector, a vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.[6]
• Isotropic vector or null vector, in a vector space with a quadratic form, a non-zero vector for which the form is zero. If a null vector exists, the quadratic form is said an isotropic quadratic form.

Vectors in specific vector spaces

• Column vector, a matrix with only one column. The column vectors with a fixed number of rows form a vector space.
• Row vector, a matrix with only one row. The row vectors with a fixed number of columns form a vector space.
• Coordinate vector, the n-tuple of the coordinates of a vector on a basis of n elements. For a vector space over a field F, these n-tuples form the vector space ${\displaystyle F^{n}}$ (where the operation are pointwise addition and scalar multiplication).
• Displacement vector, a vector that specifies the change in position of a point relative to a previous position. Displacement vectors belong to the vector space of translations.
• Position vector of a point, the displacement vector from a reference point (called the origin) to the point. A position vector represents the position of a point in a Euclidean space or an affine space.
• Velocity vector, the derivative, with respect to time, of the position vector. It does not depend of the choice of the origin, and, thus belongs to the vector space of translations.
• Pseudovector, also called axial vector, an element of the dual of a vector space. In an inner product space, the inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a pseudo vector from a vector. The distinction becomes apparent when one changes coordinates: the matrix used for a change of coordinates of pseudovectors is the transpose of that of vectors.
• Tangent vector, an element of the tangent space of a curve, a surface or, more generally, a differential manifold at a given point (these tangent spaces are naturally endowed with a structure of vector space)
• Normal vector or simply normal, in a Euclidean space or, more generally, in an inner product space, a vector that is perpendicular to a tangent space at a point. Normals are pseudovectors that belong to the dual of the tangent space.
• Gradient, the coordinates vector of the partial derivatives of a function of several real variables. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a scalar field. The gradient is a pseudo vector that is normal to a level curve.
• Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space

Tuples that are not really vectors

The set ${\displaystyle \mathbb {R} ^{n}}$ of tuples of n real numbers has a natural structure of vector space defined by component-wise addition and scalar multiplication. When such tuples are used for representing some data, it is common to call them vectors, even if the vector addition does not mean anything for these data, which may make the terminology confusing. Similarly, some physical phenomena involve a direction and a magnitude. They are often represented by vectors, even if operations of vector spaces do not apply to them.

• 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.
• Vector relation, a binary relation determined by a logical vector.

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.

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 ${\displaystyle \nabla }$
• 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

Notes

1. "vector | Definition & Facts". Encyclopedia Britannica. Retrieved 2020-08-19.
2. In some old texts, the pair (A, B) is called a bound vector, and its equivalence class is called a free vector.
3. "1.1: Vectors". Mathematics LibreTexts. 2013-11-07. Retrieved 2020-08-19.
4. "Vectors". www.mathsisfun.com. Retrieved 2020-08-19.
5. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-19.
6. Weisstein, Eric W. "Vector". mathworld.wolfram.com. Retrieved 2020-08-19.