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From Citizendium, the Citizens' Compendium

A vector is a widely used concept in mathematics, physics and all related sciences. Intuitively, it may be seen as a quantity which has both a magnitude and a direction. For example in elementary physics, velocity has both a magnitude and a direction, whereas speed is a scalar quantity with only a magnitude. A typical visualisation of a 2- or 3-dimensional vector is an arrow. More generally, a vector can be described as a n-tuple of numbers that transforms in a specific way by isometries of the frame of reference.

In mathematics, an abstract concept of vector space has been introduced. It describes in an axiomatic way the detailed properties one expects of objects that can be labelled as 'vectors'. Thus, a vector is defined as a member of any vector space. Typical vector spaces include the real line, the Euclidean plane or space, the set of continuous functions on the line (with the supremum norm taken to be the length of a vector).