Parametric equation for plane with $\langle 0,1,1\rangle + s.\langle 1,0,-1\rangle$ and $\langle 0,0,-3\rangle + t.\langle 2,1,2\rangle$

Question

In $3$-dimensional space, two lines $l_1$ and $l_2$ are given parametrically as follows: $$ X = \langle 0,1,1\rangle + s.\langle 1,0,-1\rangle \text{ and } Y=\langle 0,0,-3\rangle + t.\langle 2,1,2\rangle $$ How would you write the parametric equation that describes the plane $E$ spanned by $l_1$ and $l_2$?

I am not sure if this is correct, but I substituted $s$ and $t$ for $1$, and then calculated the respective values. I then set up a parametric form of $p_1 + s.p_1p_2 = \langle 1,1,0\rangle + s.\langle 1,0,1\rangle$. I think this may be wrong but I am not sure.

Answer 1

The point $(2,1,-1)$ lies on both lines as can be seen by using $s=2$ and $t=1$. This means that the lines indeed lie on the same plane. Now the plane is simply given parametrically by $(0,1,1)+u(1,0,-1)+v(2,1,2)$, since $(1,0,-1)$ and $(2,1,2)$ are non-parallel vectors each parallel to the plane.

Answer 2

The plane will contain the point $(0,1,1)$. You have the vector $\vec{OP} = (0,1,1)$. The plane will also "contain" the two vectors $\vec{v}_1 = (1,0,-1)$ and $\vec{v}_2 = (2,1,2)$. So parametric equations for a plane that contains a point $P$ and the two vectors $\vec{v}_1$ and $\vec{v}_2$ are $$ \pmatrix{x\\y\\z} = \vec{OP} + \vec{s}v_2 + t\vec{v}_2 $$ for $s,t\in \mathbb{R}$.