In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ we consider the points:
$$\mathrm{A}(1;1;-4), \quad \mathrm{B}(2;-1;-3), \quad \mathrm{C}(0;-1;-1) \text{ and } \Omega(1;1;2).$$

\begin{enumerate}
  \item Prove that the points $\mathrm{A}$, $\mathrm{B}$ and C define a plane.
  \item a. Prove that the vector $\vec{n}$ with coordinates $\left(\begin{array}{l}1\\1\\1\end{array}\right)$ is normal to the plane (ABC).\\
b. Justify that a Cartesian equation of the plane (ABC) is $x + y + z + 2 = 0$.
  \item a. Justify that the point $\Omega$ does not belong to the plane (ABC).\\
b. Determine the coordinates of the point H, the orthogonal projection of the point $\Omega$ onto the plane (ABC).
\end{enumerate}

We admit that $\Omega\mathrm{H} = 2\sqrt{3}$.\\
We define the sphere $S$ with centre $\Omega$ and radius $2\sqrt{3}$ as the set of all points M in space such that $\Omega\mathrm{M} = 2\sqrt{3}$.\\
4. Justify, without calculation, that any point N of the plane (ABC), distinct from H, does not belong to the sphere $S$.\\
We say that a plane $\mathscr{P}$ is tangent to the sphere $S$ at a point K when the following two conditions are satisfied:
\begin{itemize}
  \item $\mathrm{K} \in \mathscr{P} \cap S$
  \item $(\Omega\mathrm{K}) \perp \mathscr{P}$
\end{itemize}

\begin{enumerate}
  \setcounter{enumi}{4}
  \item Let the plane $\mathscr{P}$ with Cartesian equation $x + y - z - 6 = 0$ and the point K with coordinates $\mathrm{K}(3;3;0)$.\\
Prove that the plane $\mathscr{P}$ is tangent to the sphere $S$ at point K.
  \item We admit that the planes (ABC) and $\mathscr{P}$ intersect along a line ($\Delta$).\\
Determine a parametric equation of the line ($\Delta$).
\end{enumerate}