# 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).

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 |