Vector (mathematics and physics)

In mathematics and physics, vector can refer to:

Vectors

• Euclidean vector, a geometric entity endowed with both length and direction; an element of a Euclidean vector space. In physics, euclidean vectors are used to represent physical quantities which have both magnitude and direction, such as force, in contrast to scalar quantities, which have no direction.
  • 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 know as the vector resolute, a mapping of one vector onto another
  • The vector part of a quaternion, a term used in 19th century mathematical literature on quaternions
  • Burgers vector, a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice
  • Displacement vector, a vector that specifies the change in position of a point relative to a previous position
  • Gradient vector, one vector in a vector field
  • Normal vector, or surface normal, a vector which is perpendicular to a surface
  • Null vector, or zero vector, a vector whose components are all zero
  • Orbital state vectors, which define the state of an orbiting body
  • Position (vector), a vector which represents the position of an object in space in relation to an arbitrary reference point
  • Poynting vector, in physics, a vector representing the energy flux of an electromagnetic field
  • Tangent vector, a vector that follows the direction of a curve or a surface at a given point
  • Wave vector, a vector representation of a wave
• Axial vector, or pseudovector, a quantity that transforms like a vector under a proper rotation
• Basis vector, one of a set of vectors (a "basis") that, in linear combination, can represent every vector in a given vector space
• Coordinate vector, in linear algebra, an explicit representation of an element of any abstract vector space
• P-vector, the tensor obtained by taking linear combinations of the wedge product of p tangent vectors
• Probability vector, in statistics, a vector with non-negative entries that add up to one
• Row vector or column vector, a one-dimensional matrix often representing the solution of a system of linear equations
• Tuple, an ordered list of numbers, sometimes used to represent a vector

Vector fields

• Vector field, a construction in vector calculus which associates a vector to every point in a subset of Euclidean space
  • Conservative vector field, a vector field which 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 spaces

• Vector space, a mathematical structure made up of vectors, objects which may be added with another vector or multiplied by a scalar value
  • 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
  • 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

Manipulation of vectors, fields, and spaces

• Vector bundle, a topological construction which 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 notation, common notations used when working with vectors
• 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 know as the vector resolute, a mapping of one vector onto another
• Vector quantization, a quantization technique used in signal processing
• Vector-valued function, a mathematical function that maps real numbers to vectors
• Vectorization (mathematics), a linear transformation which converts a matrix into a column vector

Other uses in math and physics

• Vector boson, a boson with the spin quantum number equal to 1
• Vector meson, a meson with total spin 1 and odd parity
• Vector soliton, a solitary wave with multiple components coupled together that maintains its shape during propagation
• Vector synthesis, a type of audio synthesis
• Witt vector, an infinite sequence of elements of a commutative ring