Unit Vector

linear-algebra

Definition

Unit Vector

A unit vector is a vector whose norm is equal to $1$. Equivalently, a vector $v$ in a normed vector space is a unit vector if

$$\Vert v\Vert =1.$$

In an inner product space, this is equivalent to

$$⟨v,v⟩=1.$$

Example: The vector

$$v=\left(\begin{array}{c}\frac{3}{5}\\ \frac{4}{5}\end{array}\right)$$

is a unit vector in ${\mathbb{R}}^2$, since

$$\Vert v\Vert =\sqrt{{\left(\frac{3}{5}\right)}^2+{\left(\frac{4}{5}\right)}^2}=1.$$