# Direction (geometry) > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Direction_(geometry) > Markdown URL: https://mediated.wiki/source/Direction_(geometry).md > Source: https://en.wikipedia.org/wiki/Direction_(geometry) > Source revision: 1356464950 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Property shared by codirectional lines}} thumb|Three line segments with the same direction thumb|Examples of two 2D direction vectors [[File:QF A380 and NZ A320 SODPROPS Sydney Airport.jpg|thumb|Two airplanes in [parallel (and opposite) directions](/source/simultaneous_opposite_direction_parallel_runway_operations)]] In [geometry](/source/geometry), '''direction''', also known as '''spatial direction''', '''vector direction''' or '''relative direction''', is the common characteristic of all [rays](/source/ray_(geometry)) which coincide when [translated](/source/translation_(geometry)) to share a common endpoint; equivalently, it is the common characteristic of [vectors](/source/vector_(geometry)) (such as the [relative position](/source/relative_position) between a pair of points) which can be made equal by [scaling](/source/scaling_(geometry)) (by some positive [scalar multiplier](/source/scalar_multiplication)). 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 co[directional line segment](/source/directional_line_segment)s sharing the same size (length) are said to be ''[equipollent](/source/equipollent_(geometry))''. Two equipollent segments are not necessarily coincident; for example, a given direction can be evaluated at different starting [positions](/source/Position_(geometry)), defining different unit directed line segments (as a [bound vector](/source/bound_vector) instead of a [free vector](/source/free_vector)). Two ''[colinear](/source/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](/source/unit_vector), the result of dividing a vector by its length. A direction can alternately be represented by a [point](/source/point_(geometry)) on a [circle](/source/circle) or [sphere](/source/sphere), the [intersection](/source/intersection_(geometry)) 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](/source/origin_(mathematics)) point lie on the [unit sphere](/source/unit_sphere). A ''two-dimensional direction'' (''2D direction''), in [2D space](/source/2D_space), can be represented by its [angle](/source/angle_(geometry)), measured from some reference direction, the angular component of [polar coordinates](/source/polar_coordinates) (ignoring or normalizing the polar radius). A ''three-dimensional direction'' (''3D direction''), in [3D space](/source/3D_space), can be represented using two angles, the angular components of [spherical coordinates](/source/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](/source/Cartesian_coordinate_system), defined in terms of mutually orthogonal [coordinate axes](/source/coordinate_axes). Any arbitrary direction can be represented numerically by finding the [direction cosines](/source/direction_cosines) (a list of [cosine](/source/cosine)s 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](/source/axis_of_rotation) and [normal vectors](/source/normal_(geometry)). A direction may be used as part of the representation of a more complicated [object](/source/rigid_body)'s [attitude](/source/attitude_(geometry)) or orientation in [physical space](/source/physical_space), which may be specified by means of three angles – two angles for specifying the 3D direction and a third one for a possible [axial rotation](/source/axial_rotation) – or other equivalent set of three parameters (see [Rotation formulations in three dimensions](/source/Rotation_formulations_in_three_dimensions)). Two directions are said to be '''opposite''' if the unit vectors representing them are [additive inverse](/source/additive_inverse)s, or if the points on a sphere representing them are [antipodal](/source/antipodal_point), at the two opposite ends of a common diameter. The combination of a given direction and its corresponding opposite direction forms an undirected ["line"](/source/line_(geometry)), sometimes called the '''direction line'''{{cite book | last1=Dorst | first1=Leo | last2=Fontijne | first2=Daniel | last3=Mann | first3=Stephen | title=Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry | publisher=Elsevier | date=2010-07-26 | isbn=978-0-08-055310-8 | url=https://books.google.com.br/books/about/Geometric_Algebra_for_Computer_Science.html?id=Sg_uR57ZktoC&redir_esc=y | access-date=2026-03-12 | page=}} or an '''orientation''',{{efn|Not strictly a line, as the direction "line" or "orientation" (not to be confused with an attitude) is a free vector.}} e.g., the east-west orientation supporting both east direction and west direction.{{cite journal | last1=Burigo | first1=Michele | last2=Schultheis | first2=Holger | title=The effects of direction and orientation of located objects on spatial language comprehension | journal=Language and Cognition | volume=10 | issue=2 | date=2018 | issn=1866-9808 | doi=10.1017/langcog.2018.3 | pages=298–328 | url=https://www.cambridge.org/core/product/identifier/S1866980818000030/type/journal_article | access-date=2026-03-12| url-access=subscription }} Two directions are '''parallel''' (as in [parallel lines](/source/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](/source/obtuse_angle) (greater than a right angle) or [acute angle](/source/acute_angle) (smaller than a right angle); equivalently, obtuse directions and acute directions have, respectively, negative and positive [scalar product](/source/scalar_product) (or [scalar projection](/source/scalar_projection)). Non-oriented straight lines or [line segment](/source/line_segment)s can also be considered to have a "direction", the common characteristic of all [parallel lines](/source/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](/source/slope_(geometry)) 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](/source/Body-relative_direction) *[Euclidean vector](/source/Euclidean_vector) *[Tangent direction](/source/Tangent_direction) ==Notes== {{Notelist}} ==References== {{reflist}} Category:Orientation (geometry) Category:Elementary mathematics Category:Euclidean geometry --- Adapted from the Wikipedia article [Direction (geometry)](https://en.wikipedia.org/wiki/Direction_(geometry)) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Direction_(geometry)?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.