unit vector

A unit vectorMathworldPlanetmath 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 𝐮=1, where 𝐮 is the vector in question.

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


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


Consider 3 and the vector 𝐯=(1,2,3). The norm (length) is 14. Normalizing, we obtain the unit vector 𝐮 pointing in the same direction, namely 𝐮=(114,214,314).

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