Direction (geometry)

Three line segments with the same direction

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).

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).

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 Cartesian coordinate system is defined in terms of several oriented reference lines, called coordinate axes; any arbitrary direction can be represented numerically by finding the direction cosines (a list of cosines of the angles) between the given direction and the directions of the axes; the direction cosines are the coordinates of the associated unit vector.

A two-dimensional direction can also be represented by its angle, measured from some reference direction, the angular component of polar coordinates (ignoring or normalizing the radial component). 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.

Non-oriented straight lines 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, given a Cartesian coordinate system, can be represented numerically by its slope.

Examples of two 2D direction vectors

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 airplanes in parallel (and opposite) directions.

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).

See also

Notes

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

References

  1. ^ a b Harris, John W.; Stöcker, Horst (1998). Handbook of mathematics and computational science. Birkhäuser. Chapter 6, p. 332. ISBN 0-387-94746-9.

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