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:

$\hat {\mathbf v} = \dfrac {\mathbf v} {\left|{\mathbf v}\right|}$

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

It can often be seen as:

$\hat {\mathbf v} = \dfrac {\mathbf v} {v}$

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.