I'm trying to solve the following problem:
Embed $\mathbb{R^2}$ in the projective plane $\mathbb{RP^2}$ by the map $(x,y)\rightarrow [1,x,y]$. Find the point of intersection in $\mathbb{RP^2}$ of the projective lines corresponding to the parallel lines $y = mx$ and $y = mx+c$ in $\mathbb{R^2}$.
So in the projective plane the two lines correspond to $[1,x,mx]$ and $[1,x,mx+c]$I get that somehow the point of intersection is the point at infinity, but that point would have coordinates $[0,1,m]$ and that doesn't fall on our line. So I'm not sure I understand how this works
The two lines correspond to the equations $c+mx-y=0$ and $0+mx-y=0$.
The method I was taught for finding the intersection of these is to take the cross product $\begin{bmatrix}c&m& -1\end{bmatrix}\times \begin{bmatrix}0& m& -1\end{bmatrix}=\begin{bmatrix}0&c& cm\end{bmatrix}=\begin{bmatrix}0&1& m\end{bmatrix}$ (if $c\neq 0$).
This last point is indeed an ideal point on both lines, which you can confirm with the two equations above (remember that the ideal points are of the form $[0,x,y]$ instead of the 'real' points $[1,x,y]$, but you use their $x,y$ values in the equation just the same.