en>Yobot: /* External links */WP:CHECKWIKI error fixes (22) using AWB (9842)

2014-01-08T13:56:15Z

External links: WP:CHECKWIKI error fixes (22) using AWB (9842)

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[[File:3D Vector.svg|right|thumb|300px|Versors '''i''', '''j''', '''k''' of the Cartesian axes ''x'', ''y'', ''z'' for a three-dimensional [[Euclidean space]]. Every vector '''a''' in that space is a [[linear combination]] of these versors.]]
{{about|a unit vector codirectional with an axis or with another vector|other uses of the term versor|Versor (disambiguation)}}
{{unreferenced|date=August 2013}}

In [[geometry]] and [[physics]], the '''versor''' of an axis or of a vector is a [[unit vector]] indicating its [[Direction (geometry)|direction]].

The versor of a [[Cartesian coordinate system|Cartesian axis]] is also known as a '''standard basis vector'''. The versor of a vector is also known as a '''normalized vector'''.

== Versors of a Cartesian coordinate system ==

The versors of the axes of a [[Cartesian coordinate system]] are the unit vectors codirectional with the axes of that system.
Every [[Euclidean vector]] '''a''' in a ''n''-dimensional [[Euclidean space]] ({{math|'''R'''''n''}}) can be represented as a [[linear combination]] of the ''n'' versors of the corresponding Cartesian coordinate system. For instance, in a three-dimensional space ({{math|'''R'''}}3), there are three versors:
:<math>\mathbf{i} = (1,0,0),</math>
:<math>\mathbf{j} = (0,1,0),</math>
:<math>\mathbf{k} = (0,0,1).</math>

They indicate the direction of the Cartesian axes ''x'', ''y'', and ''z'', respectively. In terms of these, any vector '''a''' can be represented as

:<math>\mathbf{a} = \mathbf{a}_x + \mathbf{a}_y + \mathbf{a}_z = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k},</math>

where '''a'''''x'', '''a'''''y'', '''a'''''z'' are called the [[vector component]]s (or vector projections) of '''a''' on the Cartesian axes ''x'', ''y'', and ''z'' (see figure), while ''a''''x'', ''a''''y'', ''a''''z'' are the respective [[scalar component]]s (or scalar projections).

In [[linear algebra]], the set formed by these ''n'' versors is typically referred to as the [[standard basis]] of the corresponding [[Euclidean space]], and each of them is commonly called a '''standard basis vector'''.

=== Notation ===
A [[Circumflex#Mathematics|hat]] above the symbol of a versor is sometimes used to emphasize its status as a [[unit vector]] (e.g., <math alt= "unit vector i">\hat{\bold{\imath}}</math>).

In most contexts it can be assumed that '''i''', '''j''', and '''k''', (or <math alt="vector i">\vec{\imath},</math> <math alt= "vector j">\vec{\jmath},</math> and <math alt= "vector k"> \vec{k}</math>) are versors of a 3-D Cartesian coordinate system. The notations <math alt="x-hat, y-hat, z-hat">(\hat{\bold{x}}, \hat{\bold{y}}, \hat{\bold{z}})</math>, <math alt="x-hat sub 1, x-hat sub 2, x-hat sub 3">(\hat{\bold{x}}_1, \hat{\bold{x}}_2, \hat{\bold{x}}_3)</math>, <math alt="e-hat sub x, e-hat sub y, e-hat sub z">(\hat{\bold{e}}_x, \hat{\bold{e}}_y, \hat{\bold{e}}_z)</math>, or <math alt= "e-hat sub 1, e-hat sub 2, e-hat sub 3">(\hat{\bold{e}}_1, \hat{\bold{e}}_2, \hat{\bold{e}}_3)</math>, with or without [[Circumflex#Mathematics|hat]], are also used, particularly in contexts where '''i''', '''j''', '''k''' might lead to confusion with another quantity. This is recommended, for instance, when [[Indexed family|index]] symbols such as ''i'', ''j'', ''k'' are used to identify an element of a set of variables.

== Versor of a non-zero vector ==
The versor (or '''normalized vector''') <math>\hat{\mathbf{u}}</math> of a non-zero vector <math>\mathbf{u}</math> is the unit vector codirectional with <math>\mathbf{u}</math>:

:<math>\hat{\mathbf{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}.</math>

where <math>\|\mathbf{u}\|</math> is the [[norm (mathematics)|norm]] (or length) of <math>\mathbf{u}</math>.

[[Category:Vectors]]
[[Category:Concepts in physics]]
[[Category:Geometry]]
[[Category:Elementary mathematics]]

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en>Yobot: /* External links */WP:CHECKWIKI error fixes (22) using AWB (9842)