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 called a circumflex over the letter (e.g. $\widehat{𝐯}$ is the unit vector of $𝐯$.)

To make a vector into a unit vector, one just needs to divide it by its length: $\widehat{𝐯}=𝐯/\Vert 𝐯\Vert$. The resulting unit vector will be in the same direction as the original vector.

Standard basis vectors

Three common unit vectors are $\widehat{𝐢}$, $\widehat{𝐣}$ and $\widehat{𝐤}$, referring to the three-dimensional unit vectors for the x-, y- and z-axes, respectively. These vectors are called the standard basis vectors of a 3-dimensional Cartesian coordinate system. They are commonly just notated as i, j and k.

They can be written as follows: $\widehat{𝐢}=\left[\begin{array}{ccc}1& 0& 0\end{array}\right],\widehat{𝐣}=\left[\begin{array}{ccc}0& 1& 0\end{array}\right],\widehat{𝐤}=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]$

For the $i$-th standard basis vector of a vector space, the symbol $e_i$ (or ${\hat{e}}_i$) may be used. This refers to the vector with 1 in the $i$-th component, and 0 elsewhere.

Related pages

• Euclidean vector