Unit vector
High school
A unit vector is a vector of length $1$, such as $(1, 0)$ or $\left(\frac{3}{5}, \frac{4}{5}\right)$ in the plane.
Undergraduate (normed vector spaces)
Consider a vector space $V$ with a norm $\norm{\cdot}$. A unit vector is a vector $u \in V$ such that
\[\norm{u} = 1.\]Properties
- Dividing a nonzero vector $v$ by its norm gives the unit vector $v / \norm{v}$ that points in the same direction as $v$; this is called normalizing $v$.
- Whether a vector is a unit vector depends on the norm: $(1, 1)$ is a unit vector in the max norm, because $\max(\abs{1}, \abs{1}) = 1$, but $(1, 1)$ is not a unit vector in the Euclidean norm, because $\sqrt{1^2 + 1^2} = \sqrt{2}$.
- An orthogonal list of unit vectors is called orthonormal.