Unit vector

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

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

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

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

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

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