VECTOR

A vector is a fundamental object used in mathematics and related disciplines to encode both magnitude and direction. In geometry a vector is often pictured as an arrow from an origin to a point, but formally it is an element of a vector space. In common practice, vectors are represented by ordered lists of numbers called components, relative to a chosen coordinate system, such as a vector in R^n.

Vectors support operations that mirror geometric intuition. Addition and subtraction combine vectors componentwise, and scalar multiplication stretches or shrinks a vector without changing its direction (unless the scalar is negative). The length or norm of a vector measures its magnitude, and a unit vector has length 1. Inner products (dot products) quantify angles and projections, while cross products (in three dimensions) yield a vector perpendicular to a pair of input vectors. Any vector can be scaled, added to others, or decomposed into a linear combination of basis vectors.

In linear algebra, a vector space is a set equipped with vector addition and scalar multiplication, satisfying axioms such as associativity, commutativity, and distributivity. Vectors can be expressed in coordinates (as column or row vectors) and manipulated via matrix operations. Concepts such as dimension, basis, and span describe the structure of vector spaces.

Beyond pure mathematics, vectors model physical quantities with magnitude and direction (displacement, velocity, force), enable geometric and graphical computations, and serve as data representations in science and engineering. In computer science, the term vector also denotes a dynamic array, a resizable container that stores elements of a single type.