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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 $\Vert \bu\Vert=1$ where $\bu$ is the vector in question.
Let $\bv$ be a non-zero vector. To normalize $\bv$ is to find the unique unit vector with the same direction as $\bv$ This is done by multiplying $\bv$ by the reciprocal of its length; the corresponding unit vector is given by $\bu=\frac{\bv}{\Vert \bv\Vert}$
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.
Consider $\mathbb{R}^3$ and the vector $\bv=(1,2,3)$ The norm (length) is $\sqrt{14}$ Normalizing, we obtain the unit vector $\bu$ pointing in the same direction, namely $\bu=\left(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\right)$
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"unit vector" is owned by rmilson. [ full author list (2) | owner history (1) ]
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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 36 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 24223 times total.
Classification:
| AMS MSC: | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) |
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Pending Errata and Addenda
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