Coordinates for line in projective geometry with given points

Question

1. The points $P = (1: -2: 3), Q = (2: 2: -1)$ and $R = (3: 0: 2)$ lie on one Line g in $\mathbb{P}^{2}\mathbb{R}$. Choose a coordinate for g such that ${(P, Q, R)}$ has coordinates in ${(0, 1, ∞)}$.
2. How many possibilities are there?

Answer 1

As you suspect, cross-ratios are a good way to go here. Let $S(t)$ be some convenient parameterization of the line, $O=(0:0:1)$ a point not on the line, and $\mu$ the coordinate. You then have $${[O,S(t),Q][P,S(t),R]\over[O,S,R][P,S,Q]} = {\begin{vmatrix}\mu & 1 \\ 1 & 1\end{vmatrix} \begin{vmatrix}0&1\\1&0\end{vmatrix} \over \begin{vmatrix}\mu & 1 \\ 1 & 0\end{vmatrix} \begin{vmatrix}0&1\\1&1\end{vmatrix}}.$$ Solve for $t$ in terms of $\mu$ and substitute into $S(t)$.