Why do taking cross product in homogeneous coordinates of $2$ colinear points in cartesian coordiante give line at infinity?

Question

Lets say we have $2$ colinear points in cartesian coordinates $(1,2)$ and $(2,4)$. Converting this to homogeneous coordinates will give us $[1,2,1]$ and $[2,4,1]$. Taking the cross product of these $2$ points should give us a line connecting these $2$ points right? Well, the cross product of these $2$ points give us $[-2,1,0]$, which is a point (line) at infinity. Why is this the case?

Answer 1

Any two points are on a line. $(a:b:1)$ and $(c:d:1)$ cross to $(b-d:c-a:ad-bc)$ which is on the line at infinity precisely when $(a,b)$ and $(c,d)$ are linearly dependent, that is they lie on a line through the origin in the affine plane. One way to think about it, is that these points are the same point on the projective line: $(\frac12:1),$ one dimension lower, and what we are looking at is in the cone over this situation.