Line between points in projective space? I am trying to find the line through the points $(0 : 1 : 0)$ and $(1 : 1 : 1)$ in $\mathbb P^2$
and $(0 :1 : 0: 1)$ and $(1: 1: 1: 0)$ in $\mathbb P^3.$
Would the first line be the set of points $\{(b, a+b, b)\}$ for all $a,b \in \mathbb P^1$ 
and the second, $\{(d, c+d, d,c)\}$ for all $c, d \in \mathbb P^1$?
Thanks

Yes, you can treat the points on a line as the set of all linear combinations of the two defining points, the way you did.
In my nomenclature, I'd usually use different vectors to describe the line itself. In $\mathbb P^2$ I'd use the cross product to obtain a vector normal to all the points. In $\mathbb P^3$ I'd use Plücker coordinates. But there are applications where the set of all points is more useful, so it depends.
