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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, 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.

Euclidean geometry

Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry.[1] He selected a small core of undefined terms (called common notions) and postulates (or axioms) which he then used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions.[2] Euclid never used numbers to measure length, angle, or area. In this way the Euclidean plane is not quite the same as the Cartesian plane.

Three parallel planes.

A plane is a ruled surface.

Representation

This section is solely concerned with planes embedded in three dimensions: specifically, in R3.

Determination by contained points and lines

This section is solely concerned with planes embedded in three dimensions: specifically, in R3.

Determination by contained points and lines

In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following:

  • Three non-collinear points (points not on a single line).
  • A line and a point not on that line.
  • Two distinct but intersecting lines.
  • Two distinct but parallel lines.

Properties

The following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues:

  • Two distinct planes are either parallel or they intersect in a line.
  • A line is either parallel to a plane, intersects it at a single point, or is contained in the plane.
  • Two distinct lines perpendicular to the same plane must be parallel to each other.
  • Two distinct planes perpendicular to the same line must be parallel to each other.

Point-normal form and general form of the equation of a plane

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

Specifically, let r0 be the position vector of some point P0 = (x0, y0, z0), and let n = (a, b, c) be a nonzero vector. The plane determined by the point P0 and the vector ruled surface.

Representation

This section is solely concerned with planes embedded in three dimensions: specifically, in R3.

Determination by contained points and lines x 0 ) + b ( y y 0 ) + c ( z z 0 ) = 0 , {\displaystyle a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0,}

which is the point-normal form of the equation of a plane.[3] This is just a linear equation

where

This plane can also be described by the "point and a normal vector" prescription above. A suitable normal vector is given by the cross product

and the point r0 can be taken to be any of the given points p1,p2 or p3[6] (or any other point in the plane).

Operations

Distance from a point to a plane

For a plane and a point not necessarily lying on the plane, the shortest distance from to the plane is