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Is perpendicularity transitive?

  1. Apr 2, 2013 #1

    PhysicoRaj

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    Gold Member

    Is perpendicularity transitive??

    Is the relation 'perpendicularity' transitive on the set of straight lines? I've been taught 'NO', but I think yes because three mutually perpendicular lines prove the same.
    Thanks..
     
  2. jcsd
  3. Apr 2, 2013 #2
    If perpendicularity is transitive this means that if line a is perpendicular to b and b is perpendicular to c, that this implies a is perpendicular to c for ALL lines a,b,c not only for some lines.
     
  4. Apr 2, 2013 #3

    PhysicoRaj

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    So, mutual perpendicularity is the only case which holds good, and hence the relation is not transitive on the whole. I get it. But is there any condition such as coplanarity mentioned in the rule?
     
  5. Apr 2, 2013 #4

    Mark44

    Staff: Mentor

    What does "three mutually perpendicular lines prove the same." mean?

    If we're talking about lines in the plane (R2), then perpendicularity is NOT transitive. Suppose L1 ##\perp## L2, and that L2 ##\perp## L3. Then clearly, L1 || L3, so transitivity fails.
     
  6. Apr 6, 2013 #5
    Lets consider vectors to represent our lines. (in three dimensions)

    Let A = [1,0,0], B = [0,1,0], and C = [1,0,0].

    Since the dot product of A and B gives us 0, we know that A and B are perpendicular. [1]
    Similarly, B and C are perpendicular.
    However, the dot product of A and C gives us 1, hence, A and C are not perpendicular.

    This counterexample allows us to conclude that the proposition "Perpendicularity is transitive" is false.

    [1] http://en.wikipedia.org/wiki/Dot_product#Properties (property 5)


    This is precisely the same argument that Mark44 used except we fixed the third component at zero. We can further generalize this to n dimensions by fixing components three to n at zero.
     
    Last edited: Apr 6, 2013
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