Transition radiation detector

2013-06-09T02:51:51Z

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In [[geometry]], a '''position''' or '''position vector''', also known as '''location vector''' or '''radius vector''', is a [[Euclidean vector]] which represents the position of a point ''P'' in [[space#Classical_mechanics|space]] in relation to an arbitrary reference [[origin (mathematics)|origin]] ''O''. Usually denoted '''x''', '''r''' or '''s''', it corresponds to the [[displacement (vector)|displacement]] from O to P:<ref>{{cite book|title=University Physics|author=H.D. Young, R.A. Freedman|year=2008|edition=12th Edition|publisher=Addison-Wesley (Pearson International)|isbn=0-321-50130-6}}</ref>

:<math>\bold{r}=\overrightarrow{OP}.</math>

Frequently this is used in [[two-dimensional space|two-dimensional]] or [[three-dimensional space]], but can be easily generalized to [[Euclidean space]]s in any number of [[dimension]]s.<ref name="phys_keller">Keller, F. J, Gettys, W. E. et al. (1993), p 28–29</ref>

==Definition==

===Three dimensions===

[[File:Space curve.svg|width=200px|thumb|A [[space curve]] in 3d. The [[position vector]] '''r''' is parametrized by a scalar ''t''. At '''r''' = '''a''' the red line is the tangent to the curve, and the blue plane is normal to the curve.]]

In [[three dimensions]], any set of three dimensional coordinates and their corresponding basis vectors can be used to define the location of a point in space—whichever is the simplest for the task at hand may be used.

Commonly, one uses the familiar [[Cartesian coordinate system]], or sometimes [[spherical polar coordinates]], or [[cylindrical coordinate system|cylindrical coordinates]];

:<math> \begin{align}
\bold{r}(t)
& \equiv \bold{r}\left(x,y,z\right) \equiv x(t)\bold{\hat{e}}_x + y(t)\bold{\hat{e}}_y + z(t)\bold{\hat{e}}_z \\
& \equiv \bold{r}\left(r,\theta,\phi\right) \equiv r(t)\bold{\hat{e}}_r(\theta(t), \phi(t)) \\
& \equiv \bold{r}\left(r,\theta,z\right) \equiv r(t)\bold{\hat{e}}_r(\theta(t)) + z(t)\bold{\hat{e}}_z \\
& \,\!\cdots \\
\end{align}</math>

where ''t'' is a [[parametric equation|parameter]], owing to their rectangular or circular symmetry. These different coordinates and corresponding basis vectors represent the same position vector. More general [[curvilinear coordinates]] could be used instead, and are in contexts like [[continuum mechanics]] and [[general relativity]] (in the latter case one needs an additional time coordinate).

===''n'' dimensions===

[[Linear algebra]] allows for the abstraction of an ''n''-dimensional position vector. A position vector can be expressed as a linear combination of [[basis (linear algebra)|basis]] vectors:<ref>{{cite book |title=Mathematical methods for physics and engineering |first1=K.F. |last1=Riley |first2=M.P.|last2=Hobson |first3=S.J. |last3=Bence | publisher=Cambridge University Press |year=2010 |isbn=978-0-521-86153-3}}</ref><ref>{{cite book |title=Linear Algebra|first1=S. |last1=Lipschutz|first2=M.|last2=Lipson| publisher=McGraw Hill|year=2009|isbn=978-0-07-154352-1}}</ref>

:<math>\bold{r} = \sum_{i=1}^n x_i \bold{e}_i = x_1 \bold{e}_1 + x_2 \bold{e}_2 + \cdots x_n \bold{e}_n </math>

The [[set (mathematics)|set]] of all position vectors form [[position space]] (a [[vector space]] whose elements are the position vectors), since positions can be added ([[vector addition]]) and scaled in length ([[scalar multiplication]]) to obtain another position vector in the space. The notion of "space" is intuitive since each ''xi'' (''i'' = 1,2, ... ''n'') can be any value, the collection of values defines a point in space.

The ''[[dimension (linear algebra)|dimension]]'' of the position space is ''n'' (also denoted dim(''R'') = ''n''). The ''[[coordinates]]'' of the vector '''r''' with respect to the basis vectors '''e'''''i'' are ''x''''i''. The vector of coordinates forms the [[coordinate vector]] or ''n''-[[tuple]] (''x''1, ''x''2, ... ''xn'').

Each coordinate ''xi'' may be [[parametric equation|parameterized]] a number of [[parameter]]s ''t''. One parameter ''xi''(''t'') would describe a curved 1d path, two parameters ''xi''(''t''1, ''t''2) describes a curved 2d surface, three ''xi''(''t''1, ''t''2, ''t''3) describes a curved 3d volume of space, and so on.

The [[linear span]] of a basis set ''B'' = {'''e'''1, '''e'''2 ... '''e'''''n''} equals the position space ''R'', denoted span(''B'') = ''R''.

==Applications==

===Differential geometry===

{{main|Differential geometry}}

Position vector fields are used to describe continuous and differentiable space curves, in which case the independent parameter needs not be time, but can be (e.g.) arc length of the curve.

===Mechanics===

{{main|Newtonian mechanics|Analytical mechanics|Equation of motion}}

In any [[equation of motion]], the position vector '''r'''(''t'') is usually the most sought-after quantity because this function defines the motion of a particle (i.e. a [[point mass]]) - its location relative to a given [[coordinate system]] at some time ''t''.

To define motion in terms of position, each coordinate may be [[parameter|parametrized]] by time; since each successive value of time corresponds to a sequence of successive spatial locations given by the coordinates, so the continuum limit of many successive locations is a path the particle traces.

In the case of one dimension, the position has only one component, so it effectively degenerates to a scalar coordinate. It could be, say, a vector in the ''x''-direction, or the radial ''r''-direction. Equivalent notations include:

:<math> \bold{x} \equiv x \equiv x(t), \quad r \equiv r(t), \quad s \equiv s(t) \cdots \,\!</math>

==Derivatives of position==

[[File:Kinematics.svg|thumb|300px|Kinematic quantities of a classical particle: mass ''m'', position '''r''', velocity '''v''', acceleration '''a'''.]]

For a position vector '''r''' that is a function of time ''t'', the [[time derivative]]s can be computed with respect to ''t''. These derivatives have common utility in the study of [[kinematics]], [[control theory]], [[engineering]] and other sciences.

;[[Velocity]]
:<math>\bold{v}=\frac{\mathrm{d}\bold{r}}{\mathrm{d}t}</math>

where d'''r''' is an [[Differential (infinitesimal)|infinitesimally]] small [[displacement (vector)]].

;[[Acceleration]]
:<math>\bold{a}=\frac{\mathrm{d}\bold{v}}{\mathrm{d}t}=\frac{\mathrm{d}^2\bold{r}}{\mathrm{d}t^2}</math>
;[[Jerk (physics)|Jerk]]
:<math>\bold{j}=\frac{\mathrm{d}\bold{a}}{\mathrm{d}t}=\frac{\mathrm{d}^2\bold{v}}{\mathrm{d}t^2}=\frac{\mathrm{d}^3\bold{r}}{\mathrm{d}t^3}</math>

These names for the first, second and third derivative of position are commonly used in basic kinematics.<ref name='stewart'>{{cite book
|last= Stewart
|first= James
|authorlink=James Stewart (mathematician)
|title= [[Calculus]]
|publisher= Brooks/Cole
|year= 2001
|edition= 2nd
|isbn= 0-534-37718-1
|chapter= §2.8 - The Derivative As A Function
}}
</ref> By extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to accurately represent the displacement function as [[Taylor series|a sum of an infinite series]], enabling several analytical techniques in engineering and physics.

== Relationship to displacement vectors ==

A [[displacement vector]] can be defined as the ''action'' of uniformly translating spatial points in a given direction over a given distance. Thus the addition of displacement vectors expresses the composition of these displacement actions and scalar multiplication as scaling of the distance. With this in mind we may then define a ''position vector'' of a point in space as the displacement vector mapping a given origin to that point. Note thus position vectors depend on a choice of origin for the space, as well as displacement vectors depend on the choice of an initial point.

==See also==
*[[Affine space]]
*[[Curve]]
*[[Displacement vector]]
*[[Line element]]
*[[Parametric surface]]
*[[Point (geometry)]]

== Notes ==
<references />

== References ==
# Keller, F. J, Gettys, W. E. et al. (1993). "Physics: Classical and modern" 2nd ed. McGraw Hill Publishing

[[Category:Geometric measurement]]
[[Category:Vectors]]

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Transition radiation detector