Vector (mathematics and physics)

Specific vectors in a vector space

• Zero vector (sometimes also called null vector and denoted by ${\displaystyle \mathbf {0} }$), 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.[10]
• 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.[10]
• 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
• Covector, 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 covector from a vector. The distinction becomes apparent when one changes coordinates (non-orthogonally).
• 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.
• 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 covector 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

• 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

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

• 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