Definition:Unit Vector

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Definition

Let $\mathbf v$ be a vector quantity.

The unit vector in the direction of $\mathbf v$ is defined and denoted as:

$\hat {\mathbf v} = \dfrac {\mathbf v} {\left\lvert{\mathbf v}\right\rvert}$

where $\left\lvert{\mathbf v}\right\rvert$ is 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.


Also presented as

The unit vector 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 $\mathbf v$.


Sources