Let $V$ be a normedvector space. A unit vector in $V$ is a vector with norm of 1. Given a non-zero vector $v \in V$, we can normalize it to a unit vector $\frac{v}{|v|}$ in the same direction.

In quantum mechanics

Classically, the space of states of a quantum-mechanical system is given by a Hilbert space$H$. Often, it is desirable to consider only states of unit norm. One can view this as requiring that probabilities sum up to 1.