Parametric equation with plane equation given

Question

Let $2x + y + z = 2$ be a plane in space. Find the parametric equation of a line of your choice lying in the plane. I find $n=<2,1,1>$ but I need a point to complete.

Answer 1

Hints:

Why not use $(1,0,0)$ as a point on the plane?

Also, the direction vector of the line is a vector perpendicular to the normal vector of the plane, for example $\begin{pmatrix}1\\-2\\0\end{pmatrix}$.

Can you make the vector equation for the line now?

Answer 2

The intersection of the given plane with for example the plane $x=0$ is the line

$$(2x+y+z=2\quad;\quad x=0)$$ so introducing the parameter $m$ such that $y=m$ we get the parametric equation of this line

$$\left\{\begin{matrix}x&=&0\\y&=&m\\z&=&2-m\end{matrix}\right.$$

Answer 3

both point $P_1(2;0;0)$ and $P_2(0;0;1)$ are lying in the plane thus an equation is given by $x=2+2t;$ $y=0;$ $z=-t$