In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ of space, we consider the plane $(P)$ with equation:

$$(P) : \quad 2x + 2y - 3z + 1 = 0 .$$

We consider the three points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ with coordinates:

$$\mathrm{A}(1;0;1), \quad \mathrm{B}(2;-1;1) \quad \text{and} \quad \mathrm{C}(-4;-6;5).$$

The purpose of this exercise is to study the ratio of areas between a triangle and its orthogonal projection onto a plane.

\textbf{Part A}

\begin{enumerate}
  \item For each of the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$, verify whether it belongs to the plane $(P)$.
  \item Show that the point $\mathrm{C}^{\prime}(0;-2;-1)$ is the orthogonal projection of point $\mathrm{C}$ onto the plane $(P)$.
  \item Determine a parametric representation of the line (AB).
  \item We admit the existence of a unique point H satisfying the two conditions
$$\left\{ \begin{array}{l} \mathrm{H} \in (\mathrm{AB}) \\ (\mathrm{AB}) \text{ and } (\mathrm{HC}) \text{ are orthogonal.} \end{array} \right.$$
Determine the coordinates of point H.
\end{enumerate}

\textbf{Part B}

We admit that the coordinates of the vector $\overrightarrow{\mathrm{HC}}$ are: $\overrightarrow{\mathrm{HC}} \left( \begin{array}{c} -\frac{11}{2} \\ -\frac{11}{2} \\ 4 \end{array} \right)$.

\begin{enumerate}
  \item Calculate the exact value of $\| \overrightarrow{\mathrm{HC}} \|$.
  \item Let $S$ be the area of triangle ABC. Determine the exact value of $S$.
\end{enumerate}

\textbf{Part C}

We admit that $\mathrm{HC}^{\prime} = \sqrt{\frac{17}{2}}$.

\begin{enumerate}
  \item Let $\alpha = \widehat{\mathrm{CHC}^{\prime}}$. Determine the value of $\cos(\alpha)$.
  \item a. Show that the lines $(\mathrm{C}^{\prime}\mathrm{H})$ and (AB) are perpendicular.\\
b. Calculate $S^{\prime}$ the area of triangle $\mathrm{ABC}^{\prime}$, give the exact value.\\
c. Give a relationship between $S$, $S^{\prime}$ and $\cos(\alpha)$.
\end{enumerate}