Direction (geometry)

In geometry, direction, also known as spatial direction, vector direction or relative direction, is the common characteristic of all rays which coincide when translated to share a common endpoint; equivalently, it is the common characteristic of vectors (such as the relative position between a pair of points) which can be made equal by scaling (by some positive scalar multiplier). Two vectors sharing the same direction are said to be codirectional or equidirectional. All codirectional line segments sharing the same size (length) are said to be equipollent. Two equipollent segments are not necessarily coincident; for example, a given direction can be evaluated at different starting positions, defining different unit directed line segments (as a bound vector instead of a free vector). Two colinear rays or oriented line segments (sharing the same supporting line) are not necessarily codirectional and vice versa. A direction is often represented as a unit vector, the result of dividing a vector by its length. A direction can alternately be represented by a point on a circle or sphere, the intersection between the sphere and a ray in that direction emanating from the sphere's center; the tips of unit vectors emanating from a common origin point lie on the unit sphere. A two-dimensional direction (2D direction), in 2D space, can be represented by its angle, measured from some reference direction, the angular component of polar coordinates (ignoring or normalizing the polar radius). A three-dimensional direction (3D direction), in 3D space, can be represented using two angles, the angular components of spherical coordinates: a polar angle relative to a fixed polar axis and an azimuthal angle about the polar axis. An arbitrary direction can also be specified in a Cartesian coordinate system, defined in terms of mutually orthogonal coordinate axes. Any arbitrary direction can be represented numerically by finding...