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Concept
Position (geometry)
Summary
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin 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 or translation that maps the origin to P:
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 spaces and affine spaces of any dimension.
Directed line segment
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):
where .
The relative direction between two points is their relative position normalized as a unit vector:
where the denominator is the distance between the two points, .
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 coordinates:
where t is a 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).
Linear algebra allows for the abstraction of an n-dimensional position vector. A position vector can be expressed as a linear combination of basis vectors:
The 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.
Official source
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