Direction (geometry)

In geometry, direction, also known as spatial direction or vector 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).

Three line segments with the same direction

Examples of two 2D direction vectors

Two airplanes in parallel (and opposite) directions

Two vectors sharing the same direction are said to be codirectional or equidirectional.^[1] 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 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 can be represented using a polar angle relative to a fixed polar axis and an azimuthal angle about the polar axis: the angular components of spherical coordinates.

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 the direction cosines (a list of cosines of the angles), which are equivalent to the Cartesian coordinates of the associated unit vector.

A direction is used to represent linear objects such as axes of rotation and normal vectors. A direction may be used as part of the representation of a more complicated object's orientation in physical space (e.g., axis–angle representation).

Two directions are said to be opposite if the unit vectors representing them are additive inverses, or if the points on a sphere representing them are antipodal, at the two opposite ends of a common diameter. Two directions are parallel (as in parallel lines) if they can be brought to lie on the same straight line without rotations; parallel directions are either codirectional or opposite.^[1][a]

Two directions are obtuse or acute if they form, respectively, an obtuse angle (greater than a right angle) or acute angle (smaller than a right angle); equivalently, obtuse directions and acute directions have, respectively, negative and positive scalar product (or scalar projection).

Non-oriented straight lines or line segments can also be considered to have a "direction", the common characteristic of all parallel lines, which can be made to coincide by translation to pass through a common point. The direction of a non-oriented line in a two-dimensional plane can be represented numerically by its slope with respect to a reference axis. However, the "direction" of a non-oriented line corresponds to two opposite directions of coincident oriented lines.

See also

• Body relative direction
• Euclidean vector
• Tangent direction

Notes

1. Sometimes, parallel and antiparallel are used as synonyms of codirectional and opposite, respectively.