\textbf{Exercise 2 — 5 points}\\
Theme: geometry in space\\
Space is equipped with an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$.\\
We consider:
\begin{itemize}
  \item the point $A(1; -1; -1)$;
  \item the plane $\mathscr{P}_{1}$, with equation: $5x + 2y + 4z = 17$;
  \item the plane $\mathscr{P}_{2}$ with equation: $10x + 14y + 3z = 19$;
  \item the line $\mathscr{D}$ with parametric representation:
$$\left\{ \begin{aligned}
x & = 1 + 2t \\
y & = -t \\
z & = 3 - 2t
\end{aligned} \text{ where } t \text{ ranges over } \mathbb{R} . \right.$$
\end{itemize}

\begin{enumerate}
  \item Justify that the planes $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$ are not parallel.
  \item Prove that $\mathscr{D}$ is the line of intersection of $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$.
  \item a. Verify that A does not belong to $\mathscr{P}_{1}$.\\
b. Justify that A does not belong to $\mathscr{D}$.
  \item For every real $t$, we denote $M$ the point of $\mathscr{D}$ with coordinates $(1 + 2t; -t; 3 - 2t)$.\\
We then consider the function $f$ which associates to every real $t$ the value $AM^{2}$, that is $f(t) = AM^{2}$.\\
a. Prove that for every real $t$, we have: $f(t) = 9t^{2} - 18t + 17$.\\
b. Prove that the distance AM is minimal when $M$ has coordinates $(3; -1; 1)$.
  \item We denote H the point with coordinates $(3; -1; 1)$.\\
Prove that the line (AH) is perpendicular to $\mathscr{D}$.
\end{enumerate}