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).