Unit vector

A unit vector is any vector that is one unit in length.

Unit vectors are often notated the same way as normal vectors, but with a mark over the letter (e.g. [math]\displaystyle{ \mathbf{\hat{a}} }[/math] is the unit vector of a.)

To make a vector into a unit vector, divide it by its length: [math]\displaystyle{ \widehat{u} = u / \lVert u \rVert }[/math]

In component form

Three common unit vectors used in component form are [math]\displaystyle{ \mathbf{\hat{i}} }[/math], [math]\displaystyle{ \mathbf{\hat{j}} }[/math] and [math]\displaystyle{ \mathbf{\hat{k}} }[/math], referring to the unit vectors for the x-, y- and z-axes respectively. They are commonly just notated as i, j and k.

They can be written as the following: [math]\displaystyle{ \mathbf{\hat{i}} = \begin{bmatrix}1 & 0 & 0\end{bmatrix}, \,\, \mathbf{\hat{j}} = \begin{bmatrix}0 & 1 & 0\end{bmatrix}, \,\, \mathbf{\hat{k}} = \begin{bmatrix}0 & 0 & 1\end{bmatrix} }[/math]