Editing Position (geometry)

{{short description|Vector representing the position of a point with respect to a fixed origin}}
[[File:Radius vector - position vector - ortsvektor - radijvektor.svg|thumb|Radius vector <math>\vec{r}</math> represents the position of a point <math>\mathrm{P}(x,y,z)</math> with respect to origin O. In Cartesian coordinate system <math>\vec{r}=x\,\hat{e}_x+y\,\hat{e}_y+z\,\hat{e}_z.</math>]]

In [[geometry]], a '''position''' or '''position vector''', also known as '''location vector''' or '''radius vector''', is a [[Euclidean vector]] that represents a [[Point (geometry)|point]] ''P'' in [[space]]. Its length represents the distance in relation to an arbitrary reference [[origin (mathematics)|origin]] ''O'', and its [[Direction (geometry)|direction]] represents the angular orientation with respect to given reference axes. Usually denoted '''x''', '''r''', or '''s''', it corresponds to the straight line segment from ''O'' to ''P''.
In other words, it is the [[displacement (vector)|displacement]] or [[translation (geometry)|translation]] that maps the origin to ''P'':<ref>The term ''displacement'' is mainly used in mechanics, while ''translation'' is used in geometry.</ref>

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

The term '''position vector''' is used mostly in the fields of [[differential geometry]], [[mechanics]] and occasionally [[vector calculus]].
Frequently this is used in [[two-dimensional]] or [[three-dimensional space]], but can be easily generalized to [[Euclidean space]]s and [[affine space]]s of any [[dimension]].<ref name="phys_keller">Keller, F. J., Gettys, W. E. et al. (1993), p.&nbsp;28–29.</ref>

==Relative position==

The '''relative position''' of a point ''Q'' with respect to point ''P'' is the Euclidean vector resulting from the subtraction of the two absolute position vectors (each with respect to the origin):
:<math>\Delta \mathbf{r}=\mathbf{s} - \mathbf{r}=\overrightarrow{PQ},</math>
where <math>\mathbf{s}=\overrightarrow{OQ}</math>.
The ''[[relative direction]]'' between two points is their relative position normalized as a [[unit vector]].

==Definition and representation{{anchor|Definition|Representation}}==

{{further|Coordinate system}}
===Three dimensions===

[[File:Space curve.svg|thumb|[[Space curve]] in 3D. The [[position vector]] '''r''' is parameterized 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} 
 \mathbf{r}(t) 
 & \equiv \mathbf{r}(x,y,z) \equiv x(t)\mathbf{\hat{e}}_x + y(t)\mathbf{\hat{e}}_y + z(t)\mathbf{\hat{e}}_z  \\
 & \equiv \mathbf{r}(r,\theta,\phi) \equiv r(t)\mathbf{\hat{e}}_r\big(\theta(t), \phi(t)\big) \\
 & \equiv \mathbf{r}(r,\phi,z) \equiv r(t)\mathbf{\hat{e}}_r\big(\phi(t)\big) + z(t)\mathbf{\hat{e}}_z, \\
\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 |url=https://archive.org/details/mathematicalmeth00rile |url-access=registration |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 |last1=Lipschutz |first1=S. |url=https://archive.org/details/linearalgebra0000lips_a2h3 |title=Linear Algebra |last2=Lipson |first2=M. |publisher=McGraw Hill |year=2009 |isbn=978-0-07-154352-1 |url-access=registration}}</ref>

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

The [[set (mathematics)|set]] of all position vectors forms [[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 ''x<sub>i</sub>'' (''i'' = 1, 2, …, ''n'') can have 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'''<sub>''i''</sub> are ''x''<sub>''i''</sub>. The vector of coordinates forms the [[coordinate vector]] or ''n''-[[tuple]] (''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x<sub>n</sub>'').

Each coordinate ''x<sub>i</sub>'' may be parameterized a number of [[parameter]]s ''t''. One parameter ''x<sub>i</sub>''(''t'') would describe a curved 1D path, two parameters ''x<sub>i</sub>''(''t''<sub>1</sub>, ''t''<sub>2</sub>) describes a curved 2D surface, three ''x<sub>i</sub>''(''t''<sub>1</sub>, ''t''<sub>2</sub>, ''t''<sub>3</sub>) describes a curved 3D volume of space, and so on.

The [[linear span]] of a basis set ''B'' = {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>,  …, '''e'''<sub>''n''</sub>} 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 parametrized by time; since each successive value of time corresponds to a sequence of successive spatial locations given by the coordinates, 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> \mathbf{x} \equiv x \equiv x(t), \quad r \equiv r(t), \quad s \equiv s(t).</math>

==Derivatives==
{{see also|Displacement (geometry)#Derivatives}}
[[File:Kinematics.svg|thumb|Kinematic quantities of a classical particle: mass&nbsp;''m'', position&nbsp;'''r''', velocity&nbsp;'''v''', acceleration&nbsp;'''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>\mathbf{v} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t},</math>
:where d'''r''' is an [[Differential (infinitesimal)|infinitesimally]] small [[displacement (vector)]].
;[[Acceleration]]
:<math>\mathbf{a} = \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = \frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2}.</math>
;[[Jerk (physics)|Jerk]]
:<math>\mathbf{j} = \frac{\mathrm{d}\mathbf{a}}{\mathrm{d}t} = \frac{\mathrm{d}^2\mathbf{v}}{\mathrm{d}t^2} = \frac{\mathrm{d}^3\mathbf{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
|author-link=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 sequence]], enabling several analytical techniques in engineering and physics.

==See also==
*[[Affine space]]
*[[Coordinate system]]
*[[Horizontal position]]
*[[Line element]]
*[[Parametric surface]]
*[[Position fixing]]
*[[Position four-vector]]
*[[Six degrees of freedom]]
*[[Vertical position]]

== Notes ==
<references />

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

==External links==
*{{Commonscatinline}}

{{Classical mechanics derived SI units}}

[[Category:Position|*]]
[[Category:Kinematic properties]]