Definition:Vector Quantity

Definition
A vector is a mathematical entity which needs more than one component to specify it.

Formally, a vector is an element of a vector space, often the real vector space $\R^n$.

The usual intellectual frame of reference is to interpret a vector as having:
 * A magnitude
 * A direction.

This can be rendered on the page like so:


 * Vector.png

In a Euclidean $n$-space $\R^n$, it is implied that the arrow issues from the origin:
 * $O = \underbrace{\left({0, 0, \ldots, 0}\right)}_n$

Alternatively, and frequently more usefully, a vector can also expressed in terms of coordinates. In the above diagram, this would be the "head" of the vector.

It is important to note that there is no mathematical difference between interpreting a vector in $n$-space as "just the tip of the arrow" or "an arrow issuing from $O$ ending at the tip of the arrow". It is only a manner of connotation: both an arrow and a point have the same defining property of an ordered tuple.

In the contexts of physics and applied mathematics, it is a real-world physical quantity that needs for its model a mathematical object which contains more than one (usually numeric) component.

In this context it is frequently referred to as a vector quantity.

An example is a velocity.

The number of components in a vector is determined by the number of dimensions in the coordinate system of its frame of reference.

In more than three dimensions, the concepts of magnitude and direction are usually abandoned in favour of an ordered tuple of scalars.

A vector with $n$ components is sometimes called an $n$-vector.

Also see

 * Definition:Scalar Quantity
 * Definition:Scalar
 * Definition:Scalar Field