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

ax + by + cz + d = 0,

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 vectors defining the plane.

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

• three points 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 P₀ = (x₀,y₀,z₀) and a vector , the plane equation is

ax + by + cz = ax₀ + by₀ + cz₀

for the plane passing through the point P₀ and perpendicular to the vector .

Plane after three points

The equation for the plane passing through three points P₁ = (x₁,y₁,z₁), P₂ = (x₂,y₂,z₂) and P₃ = (x₃,y₃,z₃) can be represented by the following determinant:

The distance from a point to a plane

For a point P₁ = (x₁,y₁,z₁) and a plane ax + by + cz + d = 0, the distance from P₁ to the plane is:

The angle between two planes

The angle between the planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is following

.