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Position (geometry)

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Vector representing the position of a point with respect to a fixed origin

Radius vector ${\vec {r}}$ represents the position of a point $\mathrm {P} (x,y,z)$ with respect to origin O. In Cartesian coordinate system ${\vec {r}}=x\,{\hat {e}}_{x}+y\,{\hat {e}}_{y}+z\,{\hat {e}}_{z}.$

{\displaystyle {\vec {r}}} — Radius vector ${\vec {r}}$ represents the position of a point $\mathrm {P} (x,y,z)$ with respect to origin O. In Cartesian coordinate system ${\vec {r}}=x\,{\hat {e}}_{x}+y\,{\hat {e}}_{y}+z\,{\hat {e}}_{z}.$

Ingeometry, aposition orposition vector, also known aslocation vector orradius vector, is aEuclidean vector that represents apointP inspace. Its length represents the distance in relation to an arbitrary referenceoriginO, and itsdirection represents the angular orientation with respect to given reference axes. Usually denotedx,r, ors, it corresponds to the straight line segment fromO toP.In other words, it is thedisplacement ortranslation that maps the origin toP:^[1]

\mathbf {r} ={\overrightarrow {OP}}.

The termposition vector is used mostly in the fields ofdifferential geometry,mechanics and occasionallyvector calculus.Frequently this is used intwo-dimensional orthree-dimensional space, but can be easily generalized toEuclidean spaces andaffine spaces of anydimension.^[2]

Relative position

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Therelative position of a pointQ with respect to pointP is the Euclidean vector resulting from the subtraction of the two absolute position vectors (each with respect to the origin):

\Delta \mathbf {r} =\mathbf {s} -\mathbf {r} ={\overrightarrow {PQ}},

where $\mathbf {s} ={\overrightarrow {OQ}}$ .Therelative direction between two points is their relative position normalized as aunit vector.

Definition and representation

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Further information:Coordinate system

Three dimensions

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Space curve in 3D. Theposition vectorr is parameterized by a scalart. Atr =a the red line is the tangent to the curve, and the blue plane is normal to the curve.

Inthree 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 familiarCartesian coordinate system, or sometimesspherical polar coordinates, orcylindrical coordinates:

{\begin{aligned}\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{aligned}}

wheret is aparameter, owing to their rectangular or circular symmetry. These different coordinates and corresponding basis vectors represent the same position vector. More generalcurvilinear coordinates could be used instead and are in contexts likecontinuum mechanics andgeneral relativity (in the latter case one needs an additional time coordinate).

n dimensions

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Linear algebra allows for the abstraction of ann-dimensional position vector. A position vector can be expressed as a linear combination ofbasis vectors:^[3]^[4]

\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}.

Theset of all position vectors formsposition space (avector 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 eachx_i (i = 1, 2, …,n) can have any value, the collection of values defines a point in space.

Thedimension of the position space isn (also denoted dim(R) =n). Thecoordinates of the vectorr with respect to the basis vectorse_i arex_i. The vector of coordinates forms thecoordinate vector orn-tuple (x₁,x₂, …,x_n).

Each coordinatex_i may be parameterized a number ofparameterst. One parameterx_i(t) would describe a curved 1D path, two parametersx_i(t₁,t₂) describes a curved 2D surface, threex_i(t₁,t₂,t₃) describes a curved 3D volume of space, and so on.

Thelinear span of a basis setB = {e₁,e₂, …,e_n} equals the position spaceR, denoted span(B) =R.

Applications

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Differential geometry

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Main article: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

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Main articles:Newtonian mechanics,Analytical mechanics, andEquation of motion

In anyequation of motion, the position vectorr(t) is usually the most sought-after quantity because this function defines the motion of a particle (i.e. apoint mass) – its location relative to a given coordinate system at some timet.

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, thecontinuum 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 thex direction, or the radialr direction. Equivalent notations include

\mathbf {x} \equiv x\equiv x(t),\quad r\equiv r(t),\quad s\equiv s(t).

Derivatives

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Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a

For a position vectorr that is a function of timet, thetime derivatives can be computed with respect tot. These derivatives have common utility in the study ofkinematics,control theory,engineering and other sciences.

Velocity: $\mathbf {v} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}},$; where dr is aninfinitesimally smalldisplacement (vector).
Acceleration: $\mathbf {a} ={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}.$
Jerk: $\mathbf {j} ={\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\mathbf {v} }{\mathrm {d} t^{2}}}={\frac {\mathrm {d} ^{3}\mathbf {r} }{\mathrm {d} t^{3}}}.$

These names for the first, second and third derivative of position are commonly used in basic kinematics.^[5] 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 asa sum of an infinite sequence, enabling several analytical techniques in engineering and physics.

Notes

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^The termdisplacement is mainly used in mechanics, whiletranslation is used in geometry.
^Keller, F. J., Gettys, W. E. et al. (1993), p. 28–29.
^Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010).Mathematical methods for physics and engineering. Cambridge University Press.ISBN 978-0-521-86153-3.
^Lipschutz, S.; Lipson, M. (2009).Linear Algebra. McGraw Hill.ISBN 978-0-07-154352-1.
^Stewart, James (2001). "§2.8. The Derivative As A Function".Calculus (2nd ed.). Brooks/Cole.ISBN 0-534-37718-1.

References

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

External links

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Media related toPosition (geometry) at Wikimedia Commons

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Classical mechanics SI units

Dimensions	1	L	L²	Dimensions	1	θ	θ²
Linear/translational quantities				Angular/rotational quantities
T	time:t s	absement:A m s		T	time:t s
1		distance:d,position:r,s,x,displacement m	area:A m²	1		angle:θ,angular displacement:θ rad	solid angle:Ω rad², sr
T⁻¹	frequency:f s⁻¹,Hz	speed:v,velocity:v m s⁻¹	kinematic viscosity:ν, specific angular momentum: h m² s⁻¹	T⁻¹	frequency:f,rotational speed:n,rotational velocity:n s⁻¹,Hz	angular speed:ω,angular velocity:ω rad s⁻¹
T⁻²		acceleration:a m s⁻²		T⁻²	rotational acceleration s⁻²	angular acceleration:α rad s⁻²
T⁻³		jerk:j m s⁻³		T⁻³		angular jerk:ζ rad s⁻³
M	mass:m kg	weighted position:M ⟨x⟩ = ∑mx	moment of inertia: I kg m²	ML
MT⁻¹	Mass flow rate: ${\dot {m}}$ kg s⁻¹	momentum:p,impulse:J kg m s⁻¹,N s	action:𝒮,actergy:ℵ kg m² s⁻¹,J s	MLT⁻¹		angular momentum:L,angular impulse:ΔL kg m rad s⁻¹
MT⁻²		force:F,weight:F_g kg m s⁻²,N	energy:E,work:W,Lagrangian:L kg m² s⁻²,J	MLT⁻²		torque:τ,moment:M kg m rad s⁻²,N m
MT⁻³		yank:Y kg m s⁻³, N s⁻¹	power:P kg m² s⁻³, W	MLT⁻³		rotatum:P kg m rad s⁻³, N m s⁻¹