I'm having some trouble with giving the formula through 2 (or more) points in projective geometry.
For example I have P=(1:0:0), and Q=(1:1:1).
How do I find a formula of a line through these points?
I thought I had to use $ax_0+bx_1+cx_2=0$, but then you get a line
L= {$(x_0,x_1,x_2)|x_1=x_2=x_3=0$} ?
Or for two points with for example $P=(1:0:0)$ and $Q=(m_1,m_2,m_3)$? Would that be $(1,x,x)$?
$$\begin{matrix}\\\scriptstyle\text{projective coordinates of }\;P\to \\\scriptstyle\text{projective coordinates of }\;Q\to\end{matrix}\begin{vmatrix}x_0&x_1&x_2\\1&0&1\\1&1&1\end{vmatrix}=0$$