relative direction

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'':

:$\mathbf=\overrightarrow$

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.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:
:$\Delta \mathbf=\Delta \mathbf / \Delta$
where the denominator is the distance between the two points, $\Delta r = \, \Delta \mathbf \,$.

Definition

Three dimensions

In three dimensions, 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:

:$\begin \mathbf(t) & \equiv \mathbf(x,y,z) \equiv x(t)\mathbf_x + y(t)\mathbf_y + z(t)\mathbf_z \\ & \equiv \mathbf(r,\theta,\phi) \equiv r(t)\mathbf_r\big(\theta(t), \phi(t)\big) \\ & \equiv \mathbf(r,\phi,z) \equiv r(t)\mathbf_r\big(\phi(t)\big) + z(t)\mathbf_z, \\ \end$

where ''t'' is a parameter, 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).

''n'' dimensions

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:

:$\mathbf = \sum_^n x_i \mathbf_i = x_1 \mathbf_1 + x_2 \mathbf_2 + \dotsb + x_n \mathbf_n.$

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. The notion of "space" is intuitive, since each ''x_i'' (''i'' = 1, 2, …, ''n'') can have any value, the collection of values defines a point in space.

The ''dimension'' of the position space is ''n'' (also denoted dim(''R'') = ''n''). The ''coordinates'' of the vector r with respect to the basis vectors e_''i'' are ''x''_''i''. The vector of coordinates forms the coordinate vector or ''n''-tuple (''x''₁, ''x''₂, …, ''x_n'').

Each coordinate ''x_i'' may be parameterized a number of parameters ''t''. One parameter ''x_i''(''t'') would describe a curved 1D path, two parameters ''x_i''(''t''₁, ''t''₂) describes a curved 2D surface, three ''x_i''(''t''₁, ''t''₂, ''t''₃) describes a curved 3D volume of space, and so on.

The linear span 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, the position vector r(''t'') is usually the most sought-after quantity because t