Is perpendicularity transitive?

[PhysicoRaj]

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..

 

[willem2]

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.

 

[PhysicoRaj]

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?

 

[Mark44]

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.

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.

 

[DeeAytch]

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.