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parallelism of two planes

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\begin{document}
Two planes $\pi$ and $\varrho$ in the 3-dimensional Euclidean space are {\em parallel}\, iff they either have no common points or coincide, i.e. iff
\begin{align}
\pi\cap\varrho \;=\; \varnothing \quad \mbox{or} \quad \pi\cap\varrho\;=\; \pi.
\end{align}
An \PMlinkname{equivalent}{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
\begin{align}
A_1x\!+\!B_1y\!+\!C_1z\!+\!D_1 \;=\; 0 \quad \mbox{and} \quad A_2x\!+\!B_2y\!+\!C_2z\!+\!D_2 \;=\; 0,
\end{align}
the parallelism means the \PMlinkname{proportionality}{Variation} of the coefficients of the variables:\, there exists a \PMlinkescapetext{constant} $k$ such that
\begin{align}
A_1 \;=\; kA_2, \quad B_1 \;=\; kB_2, \quad C_1 \;=\; kC_2.
\end{align}
In this case, if also\, $D_1 \,=\, kD_2$,\, then the planes coincide.

Using vectors, the condition (3) may be written
\begin{align}
       \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)
\end{align}
which equation utters the \PMlinkname{parallelism}{MutualPositionsOfVectors} of the normal vectors.\\


\textbf{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 \PMlinkescapetext{formula}
\begin{align}
d \;=\; \frac{|D\!-\!E|}{\sqrt{A^2\!+\!B^2\!+\!C^2}},
\end{align}
as is easily shown by using \PMlinkname{Lagrange multipliers}{LagrangeMultiplierMethod} (see \PMlinkid{this entry}{11604}).

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