Direction (geometry)

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, direction, also known as spatial direction or vector direction, is the common characteristic of all <a href="/facts/Ray_(geometry)/gJqsr4Gj">rays</a> which coincide when <a href="/facts/Translation_(geometry)/KlpWmnfX">translated</a> to share a common endpoint; equivalently, it is the common characteristic of <a href="/facts/Vector_(geometry)/bCLrdksy">vectors</a> (such as the <a href="/facts/Relative_position/jkmEfS3C">relative position</a> between a pair of points) which can be made equal by <a href="/facts/Scaling_(geometry)/pFm6fvpu">scaling</a> (by some positive <a href="/facts/Scalar_multiplication/jroFDfbM">scalar multiplier</a>). 
Two vectors sharing the same direction are said to be codirectional or equidirectional. 
All co<a href="/facts/Directional_line_segment/V8E0M4qk">directional line segments</a> sharing the same size (length) are said to be <a href="/facts/Equipollent_(geometry)/9G4LtQ4w">equipollent</a>. Two equipollent segments are not necessarily coincident; for example, a given direction can be evaluated at different starting <a href="/facts/Position_(geometry)/Tpk5d9f2">positions</a>, defining different unit directed line segments (as a <a href="/facts/Bound_vector/bCLrdksy">bound vector</a> instead of a <a href="/facts/Free_vector/bCLrdksy">free vector</a>). 
A direction is often represented as a <a href="/facts/Unit_vector/GyaXneDn">unit vector</a>, the result of dividing a vector by its length. A direction can alternately be represented by a <a href="/facts/Point_(geometry)/6YPAvX2a">point</a> on a <a href="/facts/Circle/9fy5ggKn">circle</a> or <a href="/facts/Sphere/jywYLNM2">sphere</a>, the <a href="/facts/Intersection_(geometry)/RwH57yPq">intersection</a> between the sphere and a ray in that direction emanating from the sphere's center; the tips of unit vectors emanating from a common <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a> point lie on the <a href="/facts/Unit_sphere/5UeoTcug">unit sphere</a>.
A <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian coordinate system</a> is defined in terms of several oriented reference lines, called coordinate axes; any arbitrary direction can be represented numerically by finding the <a href="/facts/Direction_cosine/6YbPD687">direction cosines</a> (a list of <a href="/facts/Cosine/czej7ur0">cosines</a> 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 <a href="/facts/Angle_(geometry)/gFUTEQYq">angle</a>, measured from some reference direction, the angular component of <a href="/facts/Polar_coordinates/HBko4YoN">polar coordinates</a> (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 <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a>.
Non-oriented straight lines can also be considered to have a direction, the common characteristic of all <a href="/facts/Parallel_lines/16r5yq7D">parallel lines</a>, 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 <a href="/facts/Slope_(geometry)/a2L3k0j5">slope</a>.

A direction is used to represent linear objects such as <a href="/facts/Axis_of_rotation/PGcg3IY5">axes of rotation</a> and <a href="/facts/Normal_(geometry)/jhtLIhcu">normal vectors</a>. A direction may be used as part of the representation of a more complicated <a href="/facts/Rigid_body/THezBlq2">object</a>'s <a href="/facts/Orientation_(geometry)/Tj24mTJ8">orientation</a> in <a href="/facts/Physical_space/tq6LUM3C">physical space</a> (e.g., <a href="/facts/Axis%25E2%2580%2593angle_representation/WDfmToVP">axis–angle representation</a>).

Two directions are said to be opposite if the unit vectors representing them are <a href="/facts/Additive_inverse/n8gKUmCf">additive inverses</a>, or if the points on a sphere representing them are <a href="/facts/Antipodal_point/2V3228jA">antipodal</a>, at the two opposite ends of a common diameter. Two directions are parallel (as in <a href="/facts/Parallel_lines/16r5yq7D">parallel lines</a>) if they can be brought to lie on the same straight line without rotations; parallel directions are either codirectional or opposite.
Two directions are obtuse or acute if they form, respectively, an <a href="/facts/Obtuse_angle/gFUTEQYq">obtuse angle</a> (greater than a right angle) or <a href="/facts/Acute_angle/gFUTEQYq">acute angle</a> (smaller than a right angle);
equivalently, obtuse directions and acute directions have, respectively, negative and positive <a href="/facts/Scalar_product/tNz8MLjT">scalar product</a> (or <a href="/facts/Scalar_projection/SmKRBlwm">scalar projection</a>).

Direction (geometry) open-in-new

Direction (geometry)