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'parallelism of two planes'
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| Title of object: |
parallelism of two planes |
| Canonical Name: |
ParallelismOfTwoPlanes |
| Type: |
Topic |
| Created on: |
2009-02-05 07:33:25 |
| Modified on: |
2009-02-09 07:57:42 |
| Classification: |
msc:51A05, msc:51M04, msc:51N20 |
| Keywords: |
parallel plane |
| Defines: |
parallel, parallelism |
| Synonyms: |
parallelism of two planes=parallelism of planes
parallelism of two planes=parallel planes |
Revision comment (for changes between this and next version):
| distance formula corrected |
Preamble:
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\usepackage{amssymb}
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%\usepackage{psfrag}
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\newtheorem*{thmplain}{Theorem}
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Content:
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 (2) is obtained from the \PMlinkescapetext{formula}
\begin{align}
d \;=\; \frac{|D_1\!-\!D_2|}{\sqrt{A_1^2\!+\!B_1^2\!+\!C_1^2}},
\end{align}
as is easily shown by using \PMlinkname{Lagrange multipliers}{LagrangeMultiplierMethod} (see \PMlinkid{this entry}{11604}).
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