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I have just started learning projective geometry. There they define a projective point to be a line passing through origin alternatively they added that it can be said to be $[x:y:z]$ satisfying $[x:y:z]=[kx:ky:kz]$. I didn't understand what they meant by this notation.

I know the $x,y,z$ axis systems that we are taught in high school. Do they mean this relation by that axis system? I mean $(x,y,z)$ means $x$ units along x axis,y units along y axis and $z$ units along $z$ axis. We can represent that point with a vector passing through the origin which is $xi+yj+zk$. And as we know any two points on that vector will follow $\frac{x_1}{x_2}=\frac{y_1}{y_2}=\frac{z_1}{z_2}$.Do they mean this $xyz$ axes system?Or am i wrong? If i am wrong please educate me giving some pictures so that i can understand what actually is going on.

green_blue
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  • Welcome to Math.SE! <> The idea here is to regard two points (neither the origin) lying on a line through the origin of three-space as "equivalent." If $(x,y,z) \neq (0,0,0)$, the "equivalence class," the set of points equivalent to $(x,y,z)$, is denoted $[x:y:z]$. To write $[x:y:z]=[kx:ky:kz]$ (for $k \neq 0$) is an algebraic way to express the equivalence relation. Just as you say, an equivalence class is a projective point, i.e., a point of the projective plane. A point no longer has a unique triple of coordinates, however: $$[2:3:5]=[1:3/2:5/2]=[2/3:1:5/3]=[2/5:3/5:1]$$ for example. – Andrew D. Hwang Jan 15 '22 at 14:07
  • Could you please eleborate on why two points on that line are made equivalent? And why do we care to write [x:y:z] in other words : indicating proportion of x,y,z?Is this because of the "And as we know any two points on that vector will follow $\frac{x_1}{x_2}=\frac{y_1}{y_2}=\frac{z_1}{z_2}$" from his post? – madness Jan 15 '22 at 14:39

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