Unit vector

Definition — Unit vector

The unit vector of a vector $\vec v\ne\vec 0$ is $\hat v = \frac{\vec v}{|\vec v|},$ that is, the vector of modulus $1$ with the same direction and sense as $\vec v$.

Dividing a vector by its modulus “normalises” it: it preserves its direction but brings its length to $1$.

Example

Let $\vec v = (2;3;-4)$. Then $|\vec v|=\sqrt{4+9+16}=\sqrt{29}\approx 5{,}39$, and $\hat v = \frac{1}{\sqrt{29}}(2;3;-4)\approx (0{,}371;\; 0{,}557;\; -0{,}742).$

Links

Topics: Analytic geometry in space
Concepts: Unit vector · Vector
Skills: Analytic geometry · Using formulae