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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a position or position vector, also known as location vector or radius vector, is a
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
that represents the position of a
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
''P'' in
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
in relation to an arbitrary reference
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
''O''. Usually denoted x, r, or s, it corresponds to the straight line segment from ''O'' to ''P''. In other words, it is the
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
or
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
that maps the origin to ''P'': :\mathbf=\overrightarrow The term "position vector" is used mostly in the fields of
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
,
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
and occasionally
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
. Frequently this is used in
two-dimensional In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
or
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
, but can be easily generalized to
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s and
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
s of any
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
.Keller, F. J, Gettys, W. E. et al. (1993), p 28–29


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): :\Delta \mathbf=\mathbf - \mathbf=\overrightarrow where \mathbf=\overrightarrow. The relative direction between two points is their relative position normalized as a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
: :\Delta \mathbf=\Delta \mathbf / \Delta where the denominator is the distance between the two points, \Delta r = \, \Delta \mathbf \, .


Definition


Three dimensions

In
three dimensions Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
, 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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, or sometimes
spherical polar coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
, or
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
: : \begin \mathbf(t) & \equiv \mathbf(x,y,z) \equiv x(t)\mathbf_x + y(t)\mathbf_y + z(t)\mathbf_z \\ & \equiv \mathbf(r,\theta,\phi) \equiv r(t)\mathbf_r\big(\theta(t), \phi(t)\big) \\ & \equiv \mathbf(r,\phi,z) \equiv r(t)\mathbf_r\big(\phi(t)\big) + z(t)\mathbf_z, \\ \end where ''t'' is a
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
, owing to their rectangular or circular symmetry. These different coordinates and corresponding basis vectors represent the same position vector. More general
curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
could be used instead and are in contexts like
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
(in the latter case one needs an additional time coordinate).


''n'' dimensions

Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
allows for the abstraction of an ''n''-dimensional position vector. A position vector can be expressed as a linear combination of
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
vectors: :\mathbf = \sum_^n x_i \mathbf_i = x_1 \mathbf_1 + x_2 \mathbf_2 + \dotsb + x_n \mathbf_n. The
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of all position vectors forms
position space In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all '' position vectors'' r in space, and ...
(a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
whose elements are the position vectors), since positions can be added (
vector addition In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
) and scaled in length (
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by ...
) to obtain another position vector in the space. The notion of "space" is intuitive, since each ''xi'' (''i'' = 1, 2, …, ''n'') can have any value, the collection of values defines a point in space. The ''
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
'' of the position space is ''n'' (also denoted dim(''R'') = ''n''). The ''
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
'' of the vector r with respect to the basis vectors e''i'' are ''x''''i''. The vector of coordinates forms the
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
or ''n''-
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
(''x''1, ''x''2, …, ''xn''). Each coordinate ''xi'' may be parameterized a number of
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
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 In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
of a basis set ''B'' = equals the position space ''R'', denoted span(''B'') = ''R''.


Applications


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

In any
equation of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Verla ...
, 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 A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization (science philosophy), idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension (metaph ...
) – its location relative to a given
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
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 In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model (physics), lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approxi ...
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 : \mathbf \equiv x \equiv x(t), \quad r \equiv r(t), \quad s \equiv s(t).


Derivatives of position

For a position vector r that is a function of time ''t'', the
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
s can be computed with respect to ''t''. These derivatives have common utility in the study of
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
,
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
,
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
and other sciences. ;
Velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
:\mathbf = \frac, :where dr is an
infinitesimally In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
small
displacement (vector) In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a s ...
. ;
Acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
:\mathbf = \frac = \frac. ; Jerk :\mathbf = \frac = \frac = \frac. These names for the first, second and third derivative of position are commonly used in basic kinematics. 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 a sum of an infinite sequence, enabling several analytical techniques in engineering and physics.


See also

*
Affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
*
Coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
* Horizontal position *
Line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc l ...
*
Parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that oc ...
*
Position fixing Geopositioning, also known as geotracking, geolocalization, geolocating, geolocation, or geoposition fixing, is the process of determining or estimating the geographic position of an object. Geopositioning yields a set of geographic coordinates ...
*
Six degrees of freedom Six degrees of freedom (6DOF) refers to the six mechanical degrees of freedom of movement of a rigid body in three-dimensional space. Specifically, the body is free to change position as forward/backward (surge), up/down (heave), left/right ...
*
Vertical position Vertical position or vertical location, also known as vertical level or simply level, is a position along a vertical direction above or below a given vertical datum (reference level). Vertical distance or vertical separation is the distance betwee ...


Notes


References

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


External links

* {{Classical mechanics derived SI units * Kinematic properties