A unit vector is a vector with a magnitude of 1. The unit vector ${\displaystyle \mathbf{\hat{v}}}$ is defined by

${\displaystyle \mathbf{\hat{v}}=\frac{\mathbf{v}}{\|\mathbf{v}\|}}$

where ${\displaystyle \mathbf{v} }$ is a non-zero vector.

${\displaystyle \mathbf{\hat{i}}, \mathbf{\hat{j}}, \mathbf{\hat{k}}}$ are commonly used as the unit vectors forming a basis in the ${\displaystyle x, y, z}$ directions respectively. They are often used as notation for vectors or in vector functions:

${\displaystyle \vec{F}(t)=x(t)\mathbf{\hat{i}}+y(t)\mathbf{\hat{j}}+z(t)\mathbf{\hat{k}}}$

Another common set of unit vectors forming a basis is in the Frenet–Serret formulas, where ${\displaystyle \mathbf{\hat{T}},\mathbf{\hat{N}},\mathbf{\hat{B}}}$ represent the tangent, normal, and binormal vectors respectively.