page_title: Plane (geometry)
section_title: Line of intersection between two planes
hierarchical_section_title: Plane (geometry) / Operations / Line of intersection between two planes
context_page_description: In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.
  When working exclusively in two-dimensional Euclidean space, the definite article is used, so the plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, or, in other words, in the plane.
context_section_description: The line of intersection between two planes  and  where  are normalized is given by
  where
  This is found by noticing that the line must be perpendicular to both plane normals, and so parallel to their cross product  (this cross product is zero if and only if the planes are parallel, and are therefore non-intersecting or entirely coincident).
  The remainder of the expression is arrived at by finding an arbitrary point on the line. To do so, consider that any point in space may be written as , since  is a basis. We wish to find a point which is on both planes (i.e. on their intersection), so insert this equation into each of the equations of the planes to get two simultaneous equations which can be solved for  and .
  If we further assume that  and  are orthonormal then the closest point on the line of intersection to the origin is . If that is not the case, then a more complex procedure must be used.
caption: Two intersecting planes in three-dimensional space