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

a x + b y + c z + 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.

1 Plane determined by a point and a normal vector
2 Plane after three points
3 The distance from a point to a plane
4 The angle between two planes

Plane determined by a point and a normal vector

For a point $P 0 = (x 0, y 0, z 0)$ and a vector $\vec{n} = (a, b, c)$ , the plane equation is

a x + b y + c z = a x 0 + b y 0 + c z 0

for the plane passing through the point $P 0$ and perpendicular to the vector $\vec{n}$ .

Plane after three points

The equation for the plane passing through three points $P 1 = (x 1, y 1, z 1)$ , $P 2 = (x 2, y 2, z 2)$ and $P 3 = (x 3, y 3, z 3)$ can be represented by the following determinant:

$\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1& z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0$

The distance from a point to a plane

For a point $P 1 = (x 1, y 1, z 1)$ and a plane $a x + b y + c z + d = 0$ , the distance from $P 1$ to the plane is:

$D = \frac{\left | a x_1 + b y_1 + c z_1+d \right |}{\sqrt{a^2+b^2+c^2}}$

The angle between two planes

The angle between the planes $a 1 x + b 1 y + c 1 z + d 1 = 0$ and $a 2 x + b 2 y + c 2 z + d 2 = 0$ is following

$\cos {(\alpha)} = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$ .

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Plane (mathematics)

Contents

Plane determined by a point and a normal vector

Plane after three points

The distance from a point to a plane

The angle between two planes