Definition:Unit Vector

Definition
Let $$\mathbf v$$ be a vector quantity.

The unit vector in the direction of $$\mathbf v$$ is defined and denoted as:
 * $$\hat {\mathbf v} = \frac {\mathbf v} {\left|{\mathbf v}\right|}$$

where $$\left|{\mathbf v}\right|$$ is the magnitude of $$\mathbf v$$.

It can often be seen as:
 * $$\hat {\mathbf v} = \frac {\mathbf v} {v}$$

as in this context $$v$$ is usually understood as being the magnitude of $$\mathbf v$$.

Dimension
The unit vector has no dimension.

This is because it consists of a quantity (of a dimension $$D$$) divided by another instance of that same quantity (also of dimension $$D$$), leaving no dimension.