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I was reading this blog about how to compute intersections in the real projective plane and I'm struggling with this sentence:

Notice that the POINT (a, b, c) is contained in the LINE [x, y, z] exactly if ax + by + cz = 0

Can you explain this a bit Better by example, please?

Thanks in advance

EDIT: I'll try to explain Better my concern. Usually a point defines a line in 3d Euclidean space. Here that line will be perpendicular to the Plane ax + by + cz = 0, which the 3d representation of the [a, b, c] line. So I can't see how the point can lie on that plane, since being part of a 3dline perpendicular to It (modulo being the origin)

riccardoventrella
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  • That's the definition of what it means for a point $(a, b, c)$ to lie on a line $[x, y, z]$ in the projective plane. I'm guessing that there is a prior post, and they are just reminding the reader of the definition, hence "notice that ...". – Calvin Lin Apr 10 '23 at 16:21
  • This the parent article https://plus.maths.org/content/projective-geometry-projective-plane-geometry#homo but I can't see any reference to It as a definition – riccardoventrella Apr 10 '23 at 17:08
  • You say "a point defines a line in 3d Euclidean space". I am not sure what you mean by this, but there is in projective geometry a point of view that in some sense a point of the projective plane "is" a line in ordinary 3d space, in fact one going through the origin (so rather than just 3d space it is this with an origin chossen in it). More precisely, what they call the point (a,b,c) is kind of identified with the line with parametric equations x=at, y=bt, z=ct. Here t is the parameter (which must take all real values, each t giving a point of the line). (Continued) – Ulysse Keller May 14 '23 at 18:46
  • Cont. of last comment: The line mentioned is indeed perpendicular to the plane with equation ax+by+cz=0 going through the origin (where the line and the plane meet). But this is not the point here. There is a conflict in the uses of the letters x,y,z: on one side (as often) they are the unknowns in an equation defining e.g. a plane: this is the set of all points in 3d space whose coordinates x,y,z represent a solution of the equation in other words make the equation true. (Here more than an origin is needed: a coord. system where the origin has coordinates 0,0,0 (continued) – Ulysse Keller May 14 '23 at 18:54
  • (Cont.:) On another side, they talk of a line [x,y,z] where x,y,z are supposed to be given i.e. fixed numbers. Using x,y,z in two ways in ONE context creates confusion that must be solved by changing the letters for one of the 2 uses. If you want to keep the first use, you best replace x,y,z in the sentence you refer to (beginning with "Notice"). Choose any 3 letters different from a,b,c,x,y,z. You might e. g. take p,q,r. (But OK if you prefer others.) You replace ALL occurrences of x,y,z (not only in the "name" of the line, but in the equation too). OK for you? If yes we'll continue – Ulysse Keller May 14 '23 at 19:15
  • @riccardoventrella Is this OK for you? If yes we'll continue when you confirm (or give another new comment) – Ulysse Keller May 14 '23 at 19:20

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