The figure below corresponds to the model of an architectural project. It is a house with a cubic shape (ABCDEFGH) attached to a garage with a cubic shape (BIJKLMNO) where L is the midpoint of segment [BF] and K is the midpoint of segment [BC]. The garage is topped with a roof with a pyramidal shape (LMNOP) with square base LMNO and apex P positioned on the facade of the house.

We equip space with the orthonormal coordinate system $(A; \vec{\imath}, \vec{\jmath}, \vec{k})$, with $\vec{\imath} = \frac{1}{2}\overrightarrow{AB}$, $\vec{\jmath} = \frac{1}{2}\overrightarrow{AD}$ and $\vec{k} = \frac{1}{2}\overrightarrow{AE}$.

\begin{enumerate}
  \item a. By reading the graph, give the coordinates of points H, M and N.\\
  b. Determine a parametric representation of the line (HM).
  \item The architect places point P at the intersection of line (HM) and plane (BCF). Show that the coordinates of P are $\left(2; \frac{2}{3}; \frac{4}{3}\right)$.
  \item a. Calculate the dot product $\overrightarrow{PM} \cdot \overrightarrow{PN}$.\\
  b. Calculate the distance PM.\\
  We admit that the distance PN is equal to $\frac{\sqrt{11}}{3}$.\\
  c. To satisfy technical constraints, the roof can only be built if the angle $\widehat{MPN}$ does not exceed $55°$. Can the roof be built?
  \item Justify that the lines (HM) and (EN) are secant. What is their point of intersection?
\end{enumerate}