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. $\mathbf {\hat {v}}$ is the unit vector of $\mathbf {v}$.)

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

Standard basis vectors

Three common unit vectors are $\mathbf {\hat {i}}$, $\mathbf {\hat {j}}$ and $\mathbf {\hat {k}}$, 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: $\mathbf {\hat {i}} ={\begin{bmatrix}1&0&0\end{bmatrix}},\,\,\mathbf {\hat {j}} ={\begin{bmatrix}0&1&0\end{bmatrix}},\,\,\mathbf {\hat {k}} ={\begin{bmatrix}0&0&1\end{bmatrix}}$

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