Unit Vector in Direction of Vector

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

The unit vector $\mathbf {\hat v}$ in the direction of $\mathbf v$ is:
 * $\mathbf {\hat v} = \dfrac {\mathbf v} {\size {\mathbf v} }$

where $\size {\mathbf v}$ is the magnitude of $\mathbf v$.

Proof
From Vector Quantity as Scalar Product of Unit Vector Quantity:


 * $\mathbf v = \size {\mathbf v} \mathbf {\hat v}$

whence the result.

Also presented as
This result is often seen presented as:
 * $\mathbf {\hat v} = \dfrac {\mathbf v} v$

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