In which case three projetive points exist on the same projetive line $RP^2$.

Question

How to prove that three different points in $RP^2$, $[x_1;y_1;z_1]$ $[x_2;y_2;z_2]$ $[x_3;y_3;z_3]$ Are on the same projective line $RP^2$ if and only if the det of the row's matrix of the three points (above) equals 0. Can you help in this, and by the way suggest me a good book on the topic of Projective space and geometry.

Answer 1

Using homogeneous coordinates, the lines in $\Bbb R^3$ through the origin correspond to the points of $\Bbb RP^2$, and theplanes through the origin correspond to the lines of $\Bbb RP^2$.

So, the 3 given points are collinear in $\Bbb RP^2$ iff the 3 given vectors are in a common plane (iff they don't span the whole space $\Bbb R^3$ iff they are linearly dependent) iff their determinant is $0$.