I am dealing with projective plane and I am missing some basics. I would like to ask a specific question, but any resources and hints that might make me get the general idea are welcome. I quite get what is going on, but sometimes in my lectures I see some formulas that I do not know from where they come from and I cannot figure them out by myself and get very pissed.
So here is the question:
- First the statement in my lectures:
- We have a line a with the common equation: Ax+By+C=0
- We have a vector p(q,s)
- In order the vector p to be co-linear to line a the following must be true:
q*A+s*B=0
2 The actual question
From where comes this dependence? Why is C irrelevant in this case? How can I find this dependence by myself? Where I can find information about such dependencies, I could not find the answer with google?
Thank you in advance!
The relation $\;qA+sB=0$ means the vector $\vec p$ is orthogonal to vector $\vec n(A, B)$, which is normal to the line with equation $Ax+By+C=0$.
All lines with such an equation share this normal vector $\vec n$, hence they're all parallel, whatever $C$.
B.t.w.: what has this question to do with projective space? Everything happens in the affine plane.