Any vector of a magnitude of is called a unit vector, . In general, a unit vector doesn’t have to point in a particular direction. As long as the vector is one unit long, it’s a unit vector.

Every vector in space will have a corresponding unit vector. It will be the vector that points in exactly the same direction as , but is only one unit long.

Oftentimes the unit vector is written as rather than with typical vector notation, .

The unit vector that points to a vector could be found by using:

||\vec{v}|| = \sqrt{v_1^2+v_2^2+v_3^2 +\dots+v_n^2}$$

Example: If we are to find the unit vector , first find the length of .

\vec{u} = \frac{1}{||\vec{v}||}\cdot \vec{v} = \frac{1}{\sqrt{21}}\cdot \begin{bmatrix} 1 \ 4 \ -2 \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{21}} \ \frac{4}{\sqrt{21}} \ -\frac{2}{\sqrt{21}} \end{bmatrix}