A homeowner is interested in the shadow cast on his future veranda by the roof of his house when the sun is at its zenith. This veranda is schematized in cavalier perspective in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The roof of the veranda consists of two triangular faces SEF and SFG.

\begin{itemize}
  \item The planes (SOA) and (SOC) are perpendicular.
  \item The planes (SOC) and (EAB) are parallel, as are the planes (SOA) and (GCB).
  \item The edges [UV) and [EF] of the roofs are parallel.
\end{itemize}

The point K belongs to the segment [SE], the plane (UVK) separates the veranda into two zones, one illuminated and the other shaded. The plane (UVK) cuts the veranda along the polygonal line KMNP which is the shadow-sun boundary.

\begin{enumerate}
  \item Without calculation, justify that:\\
a. the segment $[\mathrm{KM}]$ is parallel to the segment $[\mathrm{UV}]$;\\
b. the segment [NP] is parallel to the segment [UK].
  \item In the rest of the exercise, we place ourselves in the orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The coordinates of the different points are as follows: $\mathrm{A}(4 ; 0 ; 0)$, \ldots
\end{enumerate}