Plane (mathematics)

In mathematics, a plane is the fundamental two-dimensional object. Intuitively, it may be visualized as a flat infinite piece of paper.
Most of the fundamental work in geometry, trigonometry, and graphing is performed in two dimensions, or in other words, in a plane.

Given a plane, one can introduce a Cartesian coordinate system on it in order to label every point on the plane uniquely with two numbers, its coordinates.

In a three-dimensional x-y-z coordinate system, one can define a plane as the set of all solutions of an equation

where a, b, c and d are real numbers such that not all of a, b, c are zero. Alternatively, a plane may be described parametrically as the set of all points of the form u + s v + t w where s and t range over all real numbers, and u, v and w are given vectorss defining the plane.

A plane is uniquely determined by any of the following combinations:

three pointss not lying on a line
a line and a point not lying on the line
a point and a line, the normal to the plane
two lines which intersect in a single point or are parallel

In three-dimensional space, two different planes are either parallel or they intersect in a line. A line which is not parallel to a given plane intersects that plane in a single point.

Plane determined by a point and a normal vector

For a point and a vector
, the plane equation is

for the plane passing through the point and perpendicular to the vector .

Plane after three points

The equation for the plane passing through three points , and can be represented by the following determinant:

The distance from a point to a plane

For a point and a plane , the distance from to the plane is:

The angle between two planes

The angle between the planes and is following

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