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
Canonical name	UnitVector
Date of creation	2013-03-22 11:58:50
Last modified on	2013-03-22 11:58:50
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	16
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A03
Related topic	VectorNorm
Related topic	NormedVectorSpace
Defines	normalize