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

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

P_0 = (x_0,y_0,z_0)

and a vector

\vec{n} = (a, b, c)

, the plane equation is

ax + by + cz = 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

ax + by + cz + 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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