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What is a case in which the statement, "Two planes parallel to the same line are parallel" be false?

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    $\begingroup$ simply because for a given line there are infinite planes parallel with this line. A way to visualize has been pointed out by @IvoTerek $\endgroup$
    – Ant
    May 10, 2015 at 23:26
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    $\begingroup$ @Ant: Sure, but given any line, there are infinite lines parallel to that line too, and yet parallelness is transitive for lines. $\endgroup$ May 11, 2015 at 5:38
  • $\begingroup$ @user2357112 you're right, totally overlooked that :) $\endgroup$
    – Ant
    May 11, 2015 at 9:18
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    $\begingroup$ Pavement. Wall. Yellow line in the middle of the street. $\endgroup$
    – Did
    May 11, 2015 at 13:33
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    $\begingroup$ Two planes that intersect are both parallel to the line of intersection. $\endgroup$ May 11, 2015 at 17:39

9 Answers 9

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Another example: take a right cylinder. Every tangent plane to the cylinder is parallel to the cylinder's axis.

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    $\begingroup$ I like this one! $\endgroup$
    – Martigan
    May 11, 2015 at 7:28
  • $\begingroup$ This is what I was thinking about, but couldn't have articulated it this nicely. $\endgroup$
    – Cruncher
    May 12, 2015 at 19:37
  • $\begingroup$ I wish that you were my teacher in school. $\endgroup$
    – Ejaz
    May 12, 2015 at 23:11
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enter image description here

Think in the floor of your room as the gray plane and think in the wall of your room as the blue plane. the blue line will be the line between the wall in front of the wall blue and the ceiling.

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    $\begingroup$ if and only if, you have a squeare-room. $\endgroup$
    – L F
    May 11, 2015 at 11:07
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    $\begingroup$ Judging by the dashed line, the line is not parallel to those planes, as the length of those differs greatly. $\endgroup$
    – Vlasec
    May 11, 2015 at 12:37
  • $\begingroup$ The comment before the draw improves the imagination ;) $\endgroup$
    – L F
    May 11, 2015 at 15:12
  • $\begingroup$ This was the example I'd have given. In a "perfect" house, the wall and the floor will both be parallel to the line where they intersect, but they will be perpendicular to each other. $\endgroup$
    – TecBrat
    May 12, 2015 at 4:13
  • $\begingroup$ About this kind of propositions always I seen the room where I am and try to find a relation with things around me $\endgroup$
    – L F
    May 12, 2015 at 4:15
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A very simple example is two different planes containing the same line: both are parallel to that line and to any other line parallel to it.

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    $\begingroup$ And since all pairs of non-parallel planes intersect, this exists for all of them. $\endgroup$
    – Random832
    May 11, 2015 at 4:39
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    $\begingroup$ As real world examples, any two planes connected by a hinge, like the keyboard and monitor of a laptop, a door and a wall... $\endgroup$
    – null
    May 11, 2015 at 12:13
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In $\mathbb{R}^3$, the planes $y = 1$ and $z=1$ are both parallel to the $x$-axis.

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If you take any two intersecting planes they will both be parallel to the line formed by their intersection, but they can't be parallel to each other because they intersect.

enter image description here

In the image above both planes are parallel to the line defined by points A and B.

More examples can be found on Google

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  • $\begingroup$ What did you use to make that diagram? $\endgroup$
    – Bob Happ
    Jul 22, 2019 at 21:31
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    $\begingroup$ I wish I was a good enough artist to make this. I found it on Google and I suspect it was scanned from an old geometry textbook. It doesn't look like I credited my source, which seems inappropriate in retrospect. Edit: it came from G.A. Wentworth's "Plane And Solid Geometry". etc.usf.edu/clipart/42100/42108/planelines_42108.htm $\endgroup$
    – dougcosine
    Jul 23, 2019 at 22:26
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Because in three-dimensional space, the planes that are anchored to a line still have one more degree of freedom. They can rotate around the line. So the two planes that are parallel to the same line could be at an infinite number of angles to each other.

Making lines parallel in 3D is simpler than making planes parallel, because planes have extra dimension. You can think of a plane as an intersection of two lines.

So as you can see there is one more line involved. Omitting that extra line is what makes your statement incomplete. Not necessarily false. It may still be true in one case out of infinity.

If you had two lines crossing each other (not necessarily at the right angle) and you had two planes that were parallel to both of the lines, then you could guarantee that the two planes are parallel, because then you would have anchored both of the plane's dimensions, not just one.

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  • $\begingroup$ To make the planes parallel you'd then have to make their both lines parallel. That's not true. You can rotate these intersecting lines around normals of their containing planes, and the planes will still remain parallel. $\endgroup$
    – Ruslan
    May 13, 2015 at 6:41
  • $\begingroup$ Hmmm... You sir are right. $\endgroup$ May 13, 2015 at 13:06
  • $\begingroup$ Fixed by removing the offending statement. $\endgroup$ May 13, 2015 at 21:22
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If both planes are infinite, and not parallel, their intersection will be a line, and any line parallel to that line will also be parallel to both planes. If the planes are parallel they of course won't have an intersection (Thanks @vlasic).

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    $\begingroup$ Provided that they have an intersection :) They might actually be parallel. $\endgroup$
    – Vlasec
    May 11, 2015 at 15:26
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All the given solutions are particular case which can be generalized in the following context:

consider that in a three dimensional space, a given plane has infinite parallels planes; the order of such an infinite is one (infinite of the first order, i.e. you can fix a single parameter, for instance the distance from the line and the plane, to identify the plane).

A straight line, in the same three dimensional space, has infinite planes parallel to itself; but the order of such an infinite is two (infinite of the second order: you need two parameters (distance and angle in a given space of coordinates).

So, we simply cannot compare the two situations: the only case in which the straight line and the two planes are parallel, happens when the angle is fixed in an opportune way. So, you remove a degree of freedom and finally obtain infinite^1 plans parallel to the line and each other.

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The answer to Why are two planes parallel to the same line not necessarily parallel? is: because for every two planes there is a line parallel to both of them, so if the two planes parallel to the line had to be parallel, every two planes would be parallel, which implies all planes would be parallel. Which is false.

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