Plane (mathematics)

From Exampleproblems

In mathematics, a plane is a fundamental two-dimensional object. Intuitively, it may be visualized as a flat infinite sheet of paper. There are several definitions of the plane, equivalent in the sense of Euclidean geometry, but which can be extended in different ways to define object in other areas of mathematics.

In some areas of mathematics, such as plane geometry or 2D computer graphics, the whole space in which the work is carried out is a single plane. In such situations the definite article is used: the plane. Many fundamental tasks in geometry, trigonometry, and graphing are performed in the two dimensional space, or in other words, in the plane.

1 Euclidean geometry
2 Planes embedded in R³
3 See also
4 External links

Euclidean geometry

A plane is a surface such that, given any two points on the surface, the surface also contains the straight line that passes through the two points. One can introduce a Cartesian coordinate system on a given plane in order to label every point on it uniquely with two numbers, the point's coordinates.

Within any Euclidean space, a plane is uniquely determined by any of the following combinations:

three non-collinear points (not lying on the same line)
a line and a point not on the line
two different lines which intersect
two different lines which are parallel

Planes embedded in R³

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

Properties

In three-dimensional space, we may exploit the following facts that do not hold in higher dimensions:

Two planes are either parallel or they intersect in a line.
A line is either parallel to a plane or they intersect at a single point.
Two lines normal to the same plane must be parallel to each other.
Two planes normal to the same line must be parallel to each other.

Point and a normal vector

In a three-dimensional ambient space, there is another important way of defining a plane:

a point and a line, which is normal (perpendicular) to the plane

We can explicitly describe the resulting plane; let $\vec p$ be the point we wish to lie in the plane, and let $\vec n$ be a nonzero vector parallel to the line we wish to be normal to the plane. The desired plane is the set of all points $\vec r$ such that

$\vec n\cdot(\vec r-\vec p)=0.$

If we write $\vec n = (a, b, c)$ , $\vec r = (x, y, z)$ , and $\vec n\cdot\vec p=-d$ , then the plane is determined by the condition

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

where a, b, c and d could be any 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

$\vec{u} + s\vec{v} + t\vec{w},$

where s and t range over all real numbers, and $\vec{u}$ , $\vec{v}$ and $\vec{w}$ are given vectors defining the plane.

Plane through three points

The plane passing through three points $\vec p_1 = (x_1,y_1,z_1)$ , $\vec p_2 = (x_2,y_2,z_2)$ and $\vec p_3 = (x_3,y_3,z_3)$ can be determined by the following determinant equations:

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

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 $\vec n = ( \vec p_2 - \vec p_1 ) \times ( \vec p_3 - \vec p_1 ),$ and the point $\vec p$ can be taken to be $\vec p_1$ .

The distance from a point to a plane

For a plane $a x + b y + c z + d = 0$ and a point $\vec p_1 = (x_1,y_1,z_1)$ not necessarily lying on the plane, the distance from $\vec p_1$ 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 line of intersection between two planes

Given intersecting planes described by $\vec n_1\cdot \vec r = h_1$ and $\vec n_2\cdot \vec r = h_2$ , the line of intersection is perpendicular to both $\vec n_1$ and $\vec n_2$ and thus parallel to

$\vec n_1 \times \vec n_2$ .

If we further assume that $\vec n_1$ and $\vec n_2$ are orthonormal then the closest point on the line of intersection to the origin is

$\vec r_0 = h_1\vec n_1 + h_2\vec n_2$ .

The dihedral angle

Given two intersecting planes described by $a 1 x + b 1 y + c 1 z + d 1 = 0$ and $a 2 x + b 2 y + c 2 z + d 2 = 0$ , the dihedral angle between them is defined to be the angle $α$ between their normal directions:

$\cos\alpha = \hat n_1\cdot \hat n_2 = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.$

External links

Mathworld: Plane af:Vlak

bg:Равнина (математика) ca:Pla cs:Rovina da:Plan (matematik) de:Ebene (Mathematik) es:Plano (geometría) fr:Plan (mathématiques) io:Plano he:מישור (גאומטריה) nl:Vlak ja:平面 pl:Płaszczyzna ru:Плоскость fi:Taso sv:Plan zh:平面

Retrieved from "http://www.exampleproblems.com/wiki/index.php/Plane_%28mathematics%29"

Categories: Euclidean geometry | Surfaces

Plane (mathematics)

From Exampleproblems

Contents

Euclidean geometry

Planes embedded in R³

Properties

Point and a normal vector

Plane through three points

The distance from a point to a plane

The line of intersection between two planes

The dihedral angle

See also

External links

Views

Personal tools

Example problems .com

Search

Toolbox

Plane (mathematics)

From Exampleproblems

Contents

Euclidean geometry

Planes embedded in R3

Properties

Point and a normal vector

Plane through three points

The distance from a point to a plane

The line of intersection between two planes

The dihedral angle

See also

External links

Views

Personal tools

Example problems .com

Search

Toolbox

Planes embedded in R³