We consider a cube ABCDEFGH. The point I is the midpoint of segment $[\mathrm{EF}]$, the point J is the midpoint of segment [BC] and the point K is the midpoint of segment [AE].

\begin{enumerate}
  \item Are the lines $(\mathrm{AI})$ and $(\mathrm{KH})$ parallel? Justify your answer.
\end{enumerate}

In the following, we place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.\\
2. a. Give the coordinates of points I and J.\\
b. Show that the vectors $\overrightarrow{\mathrm{IJ}}, \overrightarrow{\mathrm{AE}}$ and $\overrightarrow{\mathrm{AC}}$ are coplanar.

We consider the plane $\mathscr{P}$ with equation $x + 3y - 2z + 2 = 0$ as well as the lines $d_1$ and $d_2$ defined by the parametric representations below:

$$d_{1} : \left\{ \begin{array}{rl} x & = 3 + t \\ y & = 8 - 2t \\ z & = -2 + 3t \end{array} , t \in \mathbb{R} \text{ and } d_{2} : \left\{ \begin{array}{rl} x & = 4 + t \\ y & = 1 + t \\ z & = 8 + 2t \end{array} , t \in \mathbb{R}. \right. \right.$$

\begin{enumerate}
  \setcounter{enumi}{2}
  \item Are the lines $d_1$ and $d_2$ parallel? Justify your answer.
  \item Show that the line $d_2$ is parallel to the plane $\mathscr{P}$.
  \item Show that the point $\mathrm{L}(4 ; 0 ; 3)$ is the orthogonal projection of the point $\mathrm{M}(5 ; 3 ; 1)$ onto the plane $\mathscr{P}$.
\end{enumerate}