Use of homogeneous coordinates to intersect lines

Question

Here we are in $P(\mathbb{R}^2)$. Let the two lines $x=1$ and $x=-1$ lives in the related affine space.

How to intersect these two lines in the projective space ?

I'm a bit confused about homogeneous coordinates and how to use them.

Answer 1

Ket $X,Y,T$ the homogeneous coordinates in $P(\mathbf R^2)$. The equations of the projective lines are $X=T$ and $X=-T$. They intersect at a point such that $\: X=T=-T$, which implies $T=X=0$, so that the intersection point is $$(X:Y:T)=(0:1:0)$$ i.e. the point at infinity of the $Y$-axis.