Position (geometry)

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:[1] The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.

The relative direction between two points is their relative position normalized as a unit vector 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.

The linear span of a basis set B = {e1, e2, …, en} equals the position space R, denoted span(B) = R. 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.

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.

Equivalent notations include For a position vector r that is a function of time t, the time derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, control theory, engineering and other sciences.

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.