shasadou wrote:
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
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