The vector of coordinates forms the coordinate vector or n-tuple (x1, x2, …, xn). Each coordinate xi may be parameterized a number of parameters t. One parameter xi(t) would describe a curved 1D path, two parameters xi(t1, t2) describes a curved 2D surface, three xi(t1, t2, t3) describes a curved 3D volume of space, and so on. The linear span of a basis set B = {e1, e2, …, en} 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, 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 Derivatives of position 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. Velocity where dr is an infinitesimally small displacement (vector). Acceleration Jerk 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 Coordinate system Horizontal position Line element Parametric surface Position fixing Six degrees of freedom Vertical position Notes References Keller, F. J, Gettys, W. E. et al. (1993). "Physics: Classical and modern" 2nd ed. McGraw Hill Publishing.