PlanetMath: unit vector

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unit vector

(Definition)

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 , and write $\Vert \mathbf{u}\Vert=1$ , 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 $\mathbf{u}=\frac{\mathbf{v}}{\Vert \mathbf{v}\Vert}$ .

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 $\mathbf{u}=\left(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\right)$ .

"unit vector" is owned by rmilson. [ full author list (2) | owner history (1) ]

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See Also: vector norm, normed vector space

Also defines:

normalize

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Cross-references: normalizing, expression, Hilbert space, infinite-dimensional, represents, complex, real, vector space, length, reciprocal, non-zero vector, vector, norm, Euclidean space
There are 34 references to this entry.

This is version 12 of unit vector, born on 2001-11-15, modified 2005-07-26.
Object id is 865, canonical name is UnitVector.
Accessed 19074 times total.

Classification:

AMS MSC:

15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)

Pending Errata and Addenda

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