Vector (mathematics and physics)

In mathematics and physics, a vector is a generalization of a single number. It may denote a vector quantity, i.e., physical quantity that cannot be expressed by a single scalar quantity. The term may also be used to refer to elements of vector spaces, that can be added together and multiplied ("scaled") by scalars. In some contexts, vectors are tuples, which are finite sequences (of numbers or other objects) of a fixed length.

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.

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).