# 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

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 22$: Fundamental Definitions: $4.$