parallelism of two planes

Two planes $\pi$ and $\varrho$ in the 3-dimensional Euclidean space are parallel  iff they either have no common points or coincide, i.e. iff

$\pi \cap \varrho =\mathrm{\varnothing }\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}\pi \cap \varrho =\pi .$ (1)

An equivalent (http://planetmath.org/Equivalent3) condition of the parallelism is that the normal vectors of $\pi$ and $\varrho$ are parallel.
The parallelism of planes is an equivalence relation in any set of planes of the space.

If the planes have the equations

$A_{1}x+B_{1}y+C_{1}z+D_1=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{A}_{2}x+B_{2}y+C_{2}z+D_2=\mathrm{\hspace{0.33em}0},$ (2)

the parallelism means the proportionality (http://planetmath.org/Variation) of the coefficients of the variables:  there exists a $k$ such that

$A_1=k{A}_2,B_1=k{B}_2,C_1=k{C}_2.$ (3)

In this case, if also  $D_1=k{D}_2$,  then the planes coincide.

Using vectors, the condition (3) may be written

$\left(\begin{array}{c}\hfill A_1\hfill \\ \hfill B_1\hfill \\ \hfill C_1\hfill \end{array}\right)=k\left(\begin{array}{c}\hfill A_2\hfill \\ \hfill B_2\hfill \\ \hfill C_2\hfill \end{array}\right)$ (4)

which equation utters the parallelism (http://planetmath.org/MutualPositionsOfVectors) of the normal vectors.

Remark.  The shortest distance of the parallel planes

$$Ax+By+Cz+D=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}Ax+By+Cz+E=\mathrm{\hspace{0.33em}0}$$

is obtained from the

$d={\displaystyle \frac{|D-E|}{\sqrt{A^2+B^2+C^2}}},$ (5)

as is easily shown by using Lagrange multipliers (http://planetmath.org/LagrangeMultiplierMethod) (see http://planetmath.org/node/11604this 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