two lines that are perpendicular to a third line do not have to be parallel.

Question

In 3-space, two lines that are perpendicular to the same plane must be parallel. However, two lines that are perpendicular to a third line do not have to be parallel. Explain why. I am having a very difficult time understanding this problem.

Answer 1

Given a line in $3D$ and a point on it, there is a plane that passes through that point and is perpendicular to that line. Now in a plane you can have infinitely many lines, choose two which pass through the point and lie in the plane and are NOT perpendicular to each other.

Think about the three axes. Two of them are perpendicular to the third one.

Answer 2

I think this graph will be enough to make you understand this.look at those planes(instead of lines) perpendicular to the xy plane.they are clearly parallel to each other.

Now look at the x,y and z axis.look x and y axis are perpendicular to z axis.but are they parallel?