A vector is an element of a vector space. A vector space has a number of dimensions and in school the most common ones have two or three dimensions. They can be imagined as coordinate systems with a x- and y-axes (2 dimensions) or with x-, y- and z-axes (3 dimensions). Vectors can be represented as arrows in these spaces. An arrow has a direction and a length (magnitude). The same is true for vectors. Imagine an arrow in a two-dimensional coordinate system that starts at the origin and points at a point in the system. The location of the point will influence the direction of the arrow. Also, if the point is far from the origin, the arrow will be longer than if it was closer to it. Thus a vector can be defined using the coordinates of a point and the vector is like an arrow pointing to the point from the origin. It is important to note that the vector has no position. It merely describes a direction and magnitude but all arrows in a coordinate system are considered equal as long as their direction and magnitude are the same, regardless of their position.

Mathematically a vector called x is written as (or as but we use the arrow notation). In two dimensions it has two components and (which can be imagined as the coordinates of the point that it points to) and is written as (or as but we use the other notation). Similarly, for three dimensions we write .

For example, we show the two dimensional vector in a coordinate system. All shown vectors, drawn as arrows, are the same regardless of where they are positioned.