:Direction (geometry)

{{Short description|Property shared by codirectional lines}}

File:Like or Parallel vector .jpg

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.{{cite book |last1=Harris |first1=John W. |last2=Stöcker |first2=Horst |year=1998 |title=Handbook of mathematics and computational science |publisher=Birkhäuser |isbn=0-387-94746-9 |at=Chapter 6, p. 332 |url=https://books.google.com/books?id=DnKLkOb_YfIC&pg=PA332 }}

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.

File:2D Direction Vectors.svg

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).

File: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.{{efn|Sometimes, parallel and antiparallel are used as synonyms of codirectional and opposite, respectively.}}

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).