Exercise 2 (5 points)

The space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:
\begin{itemize}
  \item the points $\mathrm{A}(-1; 2; 1)$, $\mathrm{B}(1; -1; 2)$ and $\mathrm{C}(1; 1; 1)$;
  \item the line $d$ whose parametric representation is given by:
$$d : \left\{ \begin{aligned} x &= \frac{3}{2} + 2t \\ y &= 2 + t \\ z &= 3 - t \end{aligned} \quad \text{with } t \in \mathbb{R}; \right.$$
  \item the line $d'$ whose parametric representation is given by:
$$d' : \left\{ \begin{aligned} x &= s \\ y &= \frac{3}{2} + s \\ z &= 3 - 2s \end{aligned} \quad \text{with } s \in \mathbb{R}. \right.$$
\end{itemize}

\textbf{Part A}
\begin{enumerate}
  \item Show that the lines $d$ and $d'$ intersect at the point $\mathrm{S}\left(-\frac{1}{2}; 1; 4\right)$.
  \item \begin{enumerate}
    \item[a.] Show that the vector $\vec{n}\begin{pmatrix}1\\2\\4\end{pmatrix}$ is a normal vector to the plane (ABC).
    \item[b.] Deduce that a Cartesian equation of the plane (ABC) is:
$$x + 2y + 4z - 7 = 0$$
  \end{enumerate}
  \item Prove that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and S are not coplanar.
  \item \begin{enumerate}
    \item[a.] Prove that the point $\mathrm{H}(-1; 0; 2)$ is the orthogonal projection of S onto the plane (ABC).
    \item[b.] Deduce that there is no point $M$ in the plane (ABC) such that $\mathrm{S}M < \frac{\sqrt{21}}{2}$.
  \end{enumerate}
\end{enumerate}

\textbf{Part B}

We consider a point $M$ belonging to the segment [CS]. We thus have $\overrightarrow{\mathrm{CM}} = k\overrightarrow{\mathrm{CS}}$ with $k$ a real number in the interval $[0; 1]$.
\begin{enumerate}
  \item Determine the coordinates of point $M$ as a function of $k$.
  \item Does there exist a point $M$ on the segment [CS] such that the triangle $(MAB)$ is right-angled at $M$?
\end{enumerate}