The problem consists of two parts:

1. Parametric Representation of the Line $g$: You are given four points:

   • $A(1, 1, 1)$
   • $B(3, 1, 1)$
   • $C(1, 0, 1)$
   • $D_\alpha(2\alpha, 3\alpha, \alpha)$, where $\alpha$ is a parameter in $\mathbb{R}$.

   The first task is to find the parametric equation of the line $g$, which passes through points $A$ and $B$, and the line $g_\alpha$, which passes through points $C$ and $D_\alpha$.

2. Orthogonality Condition: The second task is to find the values of $\alpha$ for which the lines $g$ and $g_\alpha$ are orthogonal, and whether there exists a plane that contains both lines.

Let's go step by step. Do you want me to start with part (a) to determine the parametric equations of the lines first?