With the introduction of the 3D coordinate system we find the concepts of skew, perpendicular and parallel lines in space. Skew lines are new, and are lines that are not parallel, yet never intersect. Perpendicular and parallel lines in space are very similar to those in 2D and finding if lines are perpendicular or parallel in space requires an understanding of the equations of lines in 3D.
Two lines in space either intersect or they don't intersect. Now if they do intersect they might just might intersect like this or they might actually be perpendicular. Now the test for perpendicularity is that the dot product of the direction vectors of the 2 lines has to be 0. Remember if the dot product of 2 vectors is 0 they're perpendicular. So if the direction vector is, of 2 lines are perpendicular then the lines are perpendicular.
Caution, you have to make sure that the lines actually intersect first, the dot product in direction of vectors could be 0 but if they don't intersect it doesn't mean they're perpendicular. Now if the lines don't intersect the lines could be parallel or they could be skew, remember skew lines are kind of like this they just never intersect one another and yet they are not parallel. How do you tell if lines are parallel well if you take their 2 direction vectors one direction vector will be a scalar multiple of the other that shows that they're parallel.
But be ware they could be the same lines so make sure that they're not the same line if you're testing for parallel lines.