parallelism of two planes

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

 $\displaystyle\pi\cap\varrho\;=\;\varnothing\quad\mbox{or}\quad\pi\cap\varrho\;% =\;\pi.$ (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

 $\displaystyle A_{1}x\!+\!B_{1}y\!+\!C_{1}z\!+\!D_{1}\;=\;0\quad\mbox{and}\quad A% _{2}x\!+\!B_{2}y\!+\!C_{2}z\!+\!D_{2}\;=\;0,$ (2)

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

 $\displaystyle A_{1}\;=\;kA_{2},\quad B_{1}\;=\;kB_{2},\quad C_{1}\;=\;kC_{2}.$ (3)

In this case, if also  $D_{1}\,=\,kD_{2}$,  then the planes coincide.

Using vectors, the condition (3) may be written

 $\displaystyle\left(\!\begin{array}[]{c}A_{1}\\ B_{1}\\ C_{1}\end{array}\!\right)\;=\;k\left(\!\begin{array}[]{c}A_{2}\\ B_{2}\\ C_{2}\end{array}\!\right)$ (4)

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

Remark.  The shortest distance of the parallel planes

 $Ax\!+\!By\!+\!Cz\!+\!D\;=\;0\quad\mbox{and}\quad Ax\!+\!By\!+\!Cz\!+\!E\;=\;0$

is obtained from the

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