# parallelism of two planes

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