I know the following about lines:
If two lines are parallel to a third line, then they are parallel to each other.
If two lines are perpendicular to a third line, then they are parallel to each other.
But do the same theorems apply to vectors? I've tried relating the vector equation of the lines to their direction vectors in an attempt to understand, but I can't make the proper deductions. My intuition says no, because
<1, 0, 1> $\cdot$ <0, 1, 0> = 0 and <1, 0, 0> $\cdot$ <0, 1, 0> = 0, but there is no scalar c such that c<1, 0, 1> = <1, 0, 0>.
But I do know that a $\times$ b is perpendicular to a and b, and for some reason this implies that a $\times$ b is parallel to another vector that is perpendicular to a and b. Why is this so?
Edit: I should note I am referring only to vectors in 3D space.