PlanetMath: parallelism of two planes

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (47)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Copyright
About

Entry average rating: No information on entry rating

parallelism of two planes

(Topic)

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

$\displaystyle \pi\cap\varrho \;=\; \varnothing$ or $\displaystyle \quad \pi\cap\varrho\;=\; \pi.$

(1)

An equivalent 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

$\displaystyle A_1x+B_1y+C_1z+D_1 = 0$ and $\displaystyle \quad A_2x+B_2y+C_2z+D_2 = 0,$

(2)

the parallelism means the proportionality of the coefficients of the variables: there exists a constant $k$ such that

$\displaystyle A_1 \;=\; kA_2, \quad B_1 \;=\; kB_2, \quad C_1 \;=\; kC_2.$

(3)

In this case, if also $D_1 \,=\, kD_2$ , then the planes coincide.

Using vectors, the condition (3) may be written

$\displaystyle \left(\!\begin{array}{c}A_1\\ B_1\\ C_1\end{array}!\right) \;=\; k\left(\!\begin{array}{c}A_2\\ B_2\\ C_2\end{array}!\right)$

(4)

which equation utters the parallelism of the normal vectors.

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 formula

$\displaystyle d \;=\; \frac{\vert D\!-\!E\vert}{\sqrt{A^2\!+\!B^2\!+\!C^2}},$

(5)

as is easily shown by using Lagrange multipliers (see this entry).

"parallelism of two planes" is owned by pahio.

(view preamble | get metadata)

Other names:

parallelism of planes, parallel planes

Also defines:

parallel, parallelism

Keywords:

parallel plane

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: distance, vectors, variables, coefficients, equations, equivalence relation, normal vectors, points, iff, Euclidean space, planes
There are 73 references to this entry.

This is version 11 of parallelism of two planes, born on 2009-02-05, modified 2009-02-12.
Object id is 11603, canonical name is ParallelismOfTwoPlanes.
Accessed 1790 times total.

Classification:

AMS MSC:	51A05 (Geometry :: Linear incidence geometry :: General theory and projective geometries)
	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact

post | correct | update request | add example | add (any)