We need to find the parametric equations for the line that satisfies the following conditions:

• It passes through the point $(0, 1, 2)$.
• It is parallel to the plane $x + y + z = 5$.
• It is perpendicular to the line $x = 1 + t, y = 1 - t, z = 2t$.

Step 1: Find the direction vector of the line

We know that the line we are looking for is parallel to the plane $x + y + z = 5$. The normal vector to this plane is $\vec{n} = \langle 1, 1, 1 \rangle$, since the plane's equation is $x + y + z = 5$, and the coefficients of $x, y, z$ represent the components of the normal vector.

For a line to be parallel to the plane, its direction vector $\vec{v}$ must be perpendicular to the normal vector of the plane. Hence, we need the direction vector $\vec{v} = \langle v_x, v_y, v_z \rangle$ to satisfy the condition:

[ \vec{v} \cdot \vec