cross ratio and homogeneous coordinates

Question

the wikipedia article on cross ratio

https://en.wikipedia.org/wiki/Cross-ratio#Definition_in_homogeneous_coordinates

says:

Definition in homogeneous coordinates

If four collinear points are represented in homogeneous coordinates by vectors $a, b, c, d$ such that $c = a + b$ and $d = ka + b$, then their cross-ratio is $k$.

I am wondering about this if the vectors $a$ and $b$ are not collinear then the points $a, b, c, d$ are not collinear to start.

and also I doubt the rest. can somebody check?

(also i thought we had a cross ratio tag , is it gone?)

Answer 1

A collection of distinct points in projective space $\mathbb{P}^n$ are colinear if the vectors representing each point in $\mathbb{R}^{n+1}$ lie in the same plane.

Answer 2

Collinearity of points in projective space corresponds to coplanarity of the vectors representing them (since the plane is what collapses to the line in projective space).

For example, it is always the case that two linearly independent vectors are coplanar: this corresponds to the fact that there is always a projective line through the two points they represent. If you meant that the vectors $a,b$ were linearly dependent at the start (instead of "collinear", however one interprets that for vectors), then you're right: the two vectors represent the same projective point, and you didn't have a full set of four points in projective space at the start.

All definitions I am aware of rely on the four projective points being distinct (i.e. no two of the representing vectors are linearly dependent), so this should no be an issue.