The problem involves finding the intersection of a plane and a line in 3D space and calculating the distance between specific points. Here's the step-by-step solution:

Given Information:

1. Plane equation: $E: x_1 + 2x_2 + 3x_3 = 6$
2. Line parametric equation:
   $G: \vec{x} = \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} + r \cdot \begin{pmatrix} -1 \\ -1 \\ 1 \end{pmatrix}, r \in \mathbb{R}$

Part (a): Find the intersection point $S$ of the plane $E$ and the line $G$.

The parametric equation of the line is:
$\vec{x}(r) = \begin{pmatrix} 1 - r \\ -2 - r \\ 1 + r \end{pmatrix}.$

Substitute the line's parametric coordinates into the plane equation $x_1 + 2x_2 + 3x_3 = 6$:

$(1 - r) + 2(-2 - r) + 3(1 + r) = 6.$

Simplify:

$1 - r - 4 - 2r + 3 + 3r = 6.$

Combine like terms:

$0 + 0r = 6.$

Since the left-hand side simplifies to 0, the system contradicts and intersection does not occur