In three-dimensional space, if each of the two lines L1 and L2 is perp

Question

In three-dimensional space, if each of the two lines L1 and L2 is perpendicular to line L3, which of the following must be true?

(1) L1 is parallel to L2.
(2) L1 is perpendicular to L2.
(3) L1 and L2 lie on the same plane.

A. I only
B. I and II
C. II and III
D. III only
E. none of the above

OptimusPrepJanielle · Answer

Let us try to solve this problem by taking one statement at a time:
We are told that both L1 and L2 are perpendicular to L3.
The first thing that comes to our mind is that L1 and L2 are parallel.
 But please keep in mind that the lines may be in different planes too

(1) L1 is parallel to L2.
May or may not be. L1 and L2 can be in different planes and can both be perpendicular to L3.
Consider the example of x, y and z axes

(2) L1 is perpendicular to L2.
Again, in the cases of different planes, this might be true.
But if the lines are all in the same plane, then this is false

(3) L1 and L2 lie on the same plane.
Same reasoning as the above.

Hence none of the statements are always true.
Option E

rhine29388 · Answer

L1 may or may not be be parallel to L2 as the lines can lie in same planes or in different planes
L1 may or may not be perpendicular to L2 as the lines can lie in same planes or in different planes
L1 and L2 may or may not lie on the same plane as the lines can lie in same planes or in different planes
So none of the statements are true
correct answer - E

rohit8865 · Answer

can anyone illustrate with fig.??? As per me line L1 & L2 have same slope and thus are parallel..

Thanks

jahidhassan · Answer

Thanks for the solution. I understand reasoning 2 and reasoning 3. But, i could not understand the reasoning 1. If two lines (line X and line Y) are perpendicular to a third line (line Z), initial two lines must be parallel to each other.

Could you please explain me in details when this case might not happen.

rohit8865 · Answer

Please refer attached fig for Case 1: where lines X & Y being perpendicular to line Z are not parallel

Nunuboy1994 · Answer

A basic principle within the scope of the GMAT is that a line cannot be both perpendicular and parallel to another line. Within the concept of this question- the space doesn't necessarily HAVE to be three dimensional but this rather stated because it opens up the possibility for counter examples. What exactly do I mean? Well... if you look at the diagram below- we have two designs - the design of a deck's fence and a railroad supported by a pole with another railroad intersecting it. The two rails of the fence are perpendicular to the edge of the fence but the two rails are connected to the house so they never touch and are thus parallel. 
 
 Now, if we look at the bridge running to Calcutta- we can see the underneath bridge is a pole that supports and is thus perpendicular; however, there is another railroad to Mumbai that intersects the railroad to Calcutta at a 90 degree angle. Yes, the railroad has a finite length but it (the railroad to Mumbai) would still never touch the particular pole in the diagram that supports the right side of the railroad to Mumbai. The railroad diagram disproves statement 3. Statement 1 and 2 cannot both be true- so B gets thrown away. And because either statement 1 or statement 2  could  be true to say that 1 or 2 alone be  must true is clearly false.

Thus 
 "E"

LakerFan24 · Answer

rohit8865  i don't understand your diagram. for lines to be perpendicular, they must produce a 90 degree angle. from an eye test, it doesn't look like there are any 90 degree angles in your illustration nor is there the box that indicates a right angle.

chesstitans · Answer

remember, "must be true" means true all the time regardless what the situation is.

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In three-dimensional space, if each of the two lines L1 and L2 is perp

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In three-dimensional space, if each of the two lines L1 and L2 is perp

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