unit vector

A unit vector is a unit-length element of Euclidean space.Equivalently, one may say that the norm of a unit vector is equalto $1$, and write $\parallel 𝐮\parallel =1$, where $𝐮$ is the vector inquestion.

Let $𝐯$ be a non-zero vector. To normalize $𝐯$ is to findthe unique unit vector with the same direction as $𝐯$. This is doneby multiplying $𝐯$ by the reciprocal of its length; thecorresponding unit vector is given by $𝐮=\frac{𝐯}{\parallel 𝐯\parallel }$.

Note:

The concept of a unit vector and normalization makessense in any vector space equipped with a real or complex norm.Thus, in quantum mechanics one represents states as unit vectorsbelonging to a (possibly) infinite-dimensional Hilbert space. Toobtain an expression for such states one normalizesthe results of a calculation.

Example:

Consider ${ℝ}^3$ and the vector$𝐯=(1,2,3)$. The norm (length) is $\sqrt{14}$. Normalizing, we obtainthe unit vector $𝐮$ pointing in the same direction, namely$𝐮=(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}})$.