# 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 equal to $1$, and write $\|\mathbf{u}\|=1$, where $\mathbf{u}$ is the vector in question.

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

## Note:

The concept of a unit vector and normalization makes sense in any vector space equipped with a real or complex norm. Thus, in quantum mechanics one represents states as unit vectors belonging to a (possibly) infinite-dimensional Hilbert space. To obtain an expression for such states one normalizes the results of a calculation.

## Example:

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

Title unit vector UnitVector 2013-03-22 11:58:50 2013-03-22 11:58:50 rmilson (146) rmilson (146) 16 rmilson (146) Definition msc 15A03 VectorNorm NormedVectorSpace normalize