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. <math alt="a-hat">\mathbf{\hat{v}}</math> is the unit vector of <math alt="a-hat">\mathbf{v}</math>.)

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

Standard basis vectors

Three common unit vectors are <math alt="i-hat">\mathbf{\hat{i}}</math>, <math alt="j-hat">\mathbf{\hat{j}}</math> and <math alt="k-hat">\mathbf{\hat{k}}</math>, 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: <math alt="Components i,j and k">\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>

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

Related pages

• Euclidean vector