In space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider:
\begin{itemize}
  \item the plane $\mathscr{P}_1$ whose Cartesian equation is $2x + y - z + 2 = 0$,
  \item the plane $\mathscr{P}_2$ passing through point $\mathrm{B}(1; 1; 2)$ and whose normal vector is $\overrightarrow{n_2}\left(\begin{array}{c}1\\-1\\1\end{array}\right)$.
\end{itemize}

\begin{enumerate}
  \item a. Give the coordinates of a vector $\overrightarrow{n_1}$ normal to the plane $\mathscr{P}_1$.\\
b. We recall that two planes are perpendicular if a normal vector to one of the planes is orthogonal to a normal vector to the other plane. Show that the planes $\mathscr{P}_1$ and $\mathscr{P}_2$ are perpendicular.
  \item a. Determine a Cartesian equation of the plane $\mathscr{P}_2$.\\
b. We denote by $\Delta$ the line whose parametric representation is:
$$\left\{\begin{array}{rl} x &= 0 \\ y &= -2 + t \\ z &= t \end{array},\quad t \in \mathbb{R}\right.$$
Show that the line $\Delta$ is the intersection of the planes $\mathscr{P}_1$ and $\mathscr{P}_2$.
  \item We consider the point $\mathrm{A}(1; 1; 1)$ and we admit that point A belongs to neither $\mathscr{P}_1$ nor $\mathscr{P}_2$. We denote by H the orthogonal projection of point A onto the line $\Delta$.\\
We recall that, from question 2.b, the line $\Delta$ is the set of points $M_t$ with coordinates $(0; -2+t; t)$, where $t$ denotes any real number.\\
a. Show that, for every real $t$, $\mathrm{A}M_t = \sqrt{2t^2 - 8t + 11}$.\\
b. Deduce that $\mathrm{AH} = \sqrt{3}$.
  \item We denote by $\mathscr{D}_1$ the line perpendicular to the plane $\mathscr{P}_1$ passing through point A and $\mathrm{H}_1$ the orthogonal projection of point A onto the plane $\mathscr{P}_1$.\\
a. Determine a parametric representation of the line $\mathscr{D}_1$.\\
b. Deduce that the point $\mathrm{H}_1$ has coordinates $\left(-\frac{1}{3}; \frac{1}{3}; \frac{5}{3}\right)$.
  \item Let $\mathrm{H}_2$ be the orthogonal projection of A onto the plane $\mathscr{P}_2$.\\
We admit that $\mathrm{H}_2$ has coordinates $\left(\frac{4}{3}; \frac{2}{3}; \frac{4}{3}\right)$ and that H has coordinates $(0; 0; 2)$.\\
Show that $\mathrm{AH}_1\mathrm{HH}_2$ is a rectangle.
\end{enumerate}