parallelism of two planes

Two planes π and ϱ in the 3-dimensional Euclidean spaceMathworldPlanetmath are parallelMathworldPlanetmathPlanetmath  iff they either have no common points or coincide, i.e. iff

πϱ=orπϱ=π. (1)

An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( condition of the parallelism is that the normal vectors of π and ϱ are parallel.
The parallelism of planes is an equivalence relation in any set of planes of the space.

If the planes have the equations

A1x+B1y+C1z+D1= 0andA2x+B2y+C2z+D2= 0, (2)

the parallelism means the proportionality ( of the coefficients of the variables:  there exists a k such that

A1=kA2,B1=kB2,C1=kC2. (3)

In this case, if also  D1=kD2,  then the planes coincide.

Using vectors, the condition (3) may be written

(A1B1C1)=k(A2B2C2) (4)

which equation utters the parallelism ( of the normal vectors.

Remark.  The shortest distanceMathworldPlanetmath of the parallel planes

Ax+By+Cz+D= 0andAx+By+Cz+E= 0

is obtained from the

d=|D-E|A2+B2+C2, (5)

as is easily shown by using Lagrange multipliers ( (see entry).

Title parallelism of two planes
Canonical name ParallelismOfTwoPlanes
Date of creation 2013-03-22 18:48:10
Last modified on 2013-03-22 18:48:10
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Topic
Classification msc 51N20
Classification msc 51M04
Classification msc 51A05
Synonym parallelism of planes
Synonym parallel planes
Related topic PlaneNormal
Related topic ParallelAndPerpendicularPlanes
Related topic ParallelityOfLineAndPlane
Related topic ExampleOfUsingLagrangeMultipliers
Related topic NormalOfPlane
Defines parallel
Defines parallelism