\textbf{Exercise 3} — Main topics covered: geometry in space.

A house is modelled by a rectangular parallelepiped ABCDEFGH topped with a pyramid EFGHS.\\
We have $\mathrm{DC} = 6$, $\mathrm{DA} = \mathrm{DH} = 4$.\\
Let the points I, J and K be such that
$$\overrightarrow{\mathrm{DI}} = \frac{1}{6}\overrightarrow{\mathrm{DC}}, \quad \overrightarrow{\mathrm{DJ}} = \frac{1}{4}\overrightarrow{\mathrm{DA}}, \quad \overrightarrow{\mathrm{DK}} = \frac{1}{4}\overrightarrow{\mathrm{DH}}.$$
We denote $\vec{\imath} = \overrightarrow{\mathrm{DI}}$, $\vec{\jmath} = \overrightarrow{\mathrm{DJ}}$, $\vec{k} = \overrightarrow{\mathrm{DK}}$.\\
We use the orthonormal coordinate system $(\mathrm{D}; \vec{\imath}, \vec{\jmath}, \vec{k})$.\\
We admit that point S has coordinates $(3; 2; 6)$.

\begin{enumerate}
  \item Give, without justification, the coordinates of points $\mathrm{B}$, $\mathrm{E}$, $\mathrm{F}$ and G.

  \item Prove that the volume of the pyramid EFGHS represents one seventh of the total volume of the house.\\
  Recall that the volume $V$ of a tetrahedron is given by the formula:
  $$V = \frac{1}{3} \times (\text{area of the base}) \times \text{height}.$$

  \item a. Prove that the vector $\vec{n}$ with coordinates $\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$ is normal to the plane (EFS).\\
  b. Deduce that a Cartesian equation of the plane (EFS) is $y + z - 8 = 0$.

  \item An antenna is installed on the roof, represented by the segment $[\mathrm{PQ}]$. We have the following data:
  \begin{itemize}
    \item point P belongs to the plane (EFS);
    \item point Q has coordinates $(2; 3; 5{,}5)$;
    \item the line (PQ) is directed by the vector $\vec{k}$.
  \end{itemize}
  a. Justify that a parametric representation of the line (PQ) is:
  $$\left\{\begin{aligned} x &= 2 \\ y &= 3 \\ z &= 5{,}5 + t \end{aligned} \quad (t \in \mathbb{R})\right.$$
  b. Deduce the coordinates of point $P$.\\
  c. Deduce the length PQ of the antenna.

  \item A bird flies following a trajectory modelled by the line $\Delta$ whose parametric representation is:
  $$\left\{\begin{aligned} x &= -4 + 6s \\ y &= 7 - 4s \\ z &= 2 + 4s \end{aligned} \quad (s \in \mathbb{R})\right.$$
  Determine the relative position of the lines (PQ) and $\Delta$.\\
  Will the bird collide with the antenna represented by the segment $[\mathrm{PQ}]$?
\end{enumerate}