Question

How do I know if two lines are perpendicular in three-dimensional space?

Answer 1

First you have to prove the lines intersect at a point. Then you need to show the angle between them is 90°.

Depending on what form your lines' equations take on, you need to find an (x, y, z) point that's on both lines. If the lines are in parametric form, x=a+bt, y=c+dt, z=e+ft for line 1 and x=g+hu, y=i+ju, z=k+mu for line 2, then set a+bt=g+hu, etc, and you have 3 equations in 2 unknowns, t and u. These may or may not have a solution; if they do then the lines intersect at that (x, y, z).

The next step is to find two vectors starting from the point of intersection: let (p,q,r) be the intersection point, and on line 1 use the equation to find any other point, say (a, c, e) with t=0.

Then v = < a-p, c-q, e-r > is a vector in line 1 direction; do the same for line 2 to get vector w. Then v is perpendicular to w if and only if the dot product v · w is 0. You can do the dot product coordinate-wise, using the formula < a, b, c > · < d, e, f > = ad + be + cf.

Hope this helps! Try and work out an example and then you'll understand it better, says dansmath!