In $\Bbb P^2$, how could I find the equation of the line joining, for example $P =(1:1:0)$ and $Q=(\alpha:0:\beta)$?
I was thinking about $\lambda(1:1:0) + \mu(\alpha:0:\beta)=(\lambda+\mu\alpha:\lambda:\mu\beta)$.
But then the follow up question came...
'Write down the point of intersection of the 2 lines $x+y+z=0$ and $\alpha x+\beta y=0$'. I thought lines in projective space where given by ratios and not really formulas like this?
This is the equation of a plane in $k^3$, i.e a line in $\mathbb P^2$. Notice that this equation makes perfectly sense, since it is homogenous : if $(a,b,c)$ verify $a + b + c = 0$ then any equivalent point $(\lambda a, \lambda b, \lambda c)$ will also verify $\lambda a + \lambda b + \lambda c = 0$. The value of $x + y + z$ is not defined, but if is is zero or not is, so this makes sense to talk about the line defined by $x + y + z = 0$.