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:

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

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

Proof

From Vector Quantity as Scalar Product of Unit Vector Quantity:

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

whence the result.

$\blacksquare$

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 $\mathbf v$.

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.1$ Electric Charge
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): unit vector