||This recipe is a tricky one to comment, especially without "Using Vector Operations" written yet. The overall impression is: we need to shorten this by removing unnecessary theory and add "Discussion" with examples of using this recipe.
More detailed comments (yes, quite a lot of them):
1. Cookbooks don't use numbered lists within one recipe, only "Problem-Solution-Discussion" format.
2. We don't need definitions of line and segment in introduction, we assume that they are intuitive/already known. I think "Problem" should say something like "Now that we are comfortable with using vector operations for doing simple geometry things, let's see how they work for next popular group of problems - problems which deal with lines and segments on the plane". Then you can remove references to R^n from "Solution", since we've already stated that it's R^2.
3. I don't think we need the proof of finding A,B and C from P and Q, just give the formulas (by the way, currently you've got at least one typo in them).
4. When you're dealing with two lines, it's better to use (A1,B1,C1) and (A2,B2,C2), this doesn't require an effort to recall what E or F was.
5. Finding a line which is perpendicular to a given one and comparing/ordering lines looks like they are not exactly popular topics and thus belong to "Discussion".
6. Point-line distance is an important topic, so it does need implementation code (as well as the later one point-segment). I like the way it was given in http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=geometry1 .
7. As for segments, I somewhat dislike using the term "collinear points", as it might not be a standard definition of all educational systems (as a matter of fact, it's the first time I've heard it). I'd prefer a generalized problem of determining whether the point belongs to the segment, with one solution as a whole.
8. All examples should be TC problems. We've had enough geometry to be able to choose :-)