Given the equations $x = t(1,2,2) + (0,1,-1)$ and $y = t(1,1,3) + (0,2,-2)$, I'm trying to write a set $P$ that contains $x$ and $y$ (call $z$ the vector that represents this).
What I have done is let $$P:\, z = t(1,2,2) + s(1,1,3) + (1,3,1)$$
I chose $(1,3,1)$ as a point because I know $x$ and $y$ intersect there (I set $x=y$ and found this point).
But I'm not sure what the logic is behind the direction vectors, or even what this represents. Could anyone clarify or help out? (I am still very much a beginner with lin. alg.)
Your answer is perfectly correct.
You have chosen the point $X_0 =(1,3,1)$ on the intersection of the two lines and the direction vectors make the parametrization possible.
Your equation is $$P=X_0 +tV_1+sV_2$$ where you have the point and the direction vectors figured out correctly.