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]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/76/Like_or_Parallel_vector_.jpg/220px-Like_or_Parallel_vector_.jpg)
For certain vector physical quantities having a definite starting point, more specifically for bound vectors attached to a particular point of application, the corresponding (relative) direction is a directed line segment of unit length.[2]
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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/2D_Direction_Vectors.svg/220px-2D_Direction_Vectors.svg.png)
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).
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/6b/QF_A380_and_NZ_A320_SODPROPS_Sydney_Airport.jpg/220px-QF_A380_and_NZ_A320_SODPROPS_Sydney_Airport.jpg)
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
editNotes
edit- ^ Sometimes, parallel and antiparallel are used as synonyms of codirectional and opposite, respectively.
References
edit- ^ a b Harris, John W.; Stöcker, Horst (1998). Handbook of mathematics and computational science. Birkhäuser. Chapter 6, p. 332. ISBN 0-387-94746-9.
- ^ Toh, C. S. (2016-03-28). A-Level Study Guide Mathematics Ed H2.2. Step-by-Step International Pte. Ltd.