We consider a cube ABCDEFGH with edge 8 cm and centre $\Omega$.

The points P, Q and R are defined by $\overrightarrow{AP} = \frac{3}{4}\overrightarrow{AB}$, $\overrightarrow{AQ} = \frac{3}{4}\overrightarrow{AE}$ and $\overrightarrow{FR} = \frac{1}{4}\overrightarrow{FG}$.\\
We use the orthonormal coordinate system $(A; \vec{\imath}, \vec{\jmath}, \vec{k})$ with: $\vec{\imath} = \frac{1}{8}\overrightarrow{AB}$, $\vec{\jmath} = \frac{1}{8}\overrightarrow{AD}$ and $\vec{k} = \frac{1}{8}\overrightarrow{AE}$.

\section*{Part I}
\begin{enumerate}
  \item In this coordinate system, we admit that the coordinates of point R are $(8; 2; 8)$.
Give the coordinates of points P and Q.
  \item Show that the vector $\vec{n}(1; -5; 1)$ is a normal vector to the plane (PQR).
  \item Justify that a Cartesian equation of the plane (PQR) is $x - 5y + z - 6 = 0$.
\end{enumerate}

\section*{Part II}
We denote L the orthogonal projection of point $\Omega$ onto the plane (PQR).

\begin{enumerate}
  \item Justify that the coordinates of point $\Omega$ are $(4; 4; 4)$.
  \item Give a parametric representation of the line $d$ perpendicular to the plane (PQR) and passing through $\Omega$.
  \item Show that the coordinates of point L are $\left(\frac{14}{3}; \frac{2}{3}; \frac{14}{3}\right)$.
  \item Calculate the distance from point $\Omega$ to the plane (PQR).
\end{enumerate}