The solid ABCDEFGH is a cube. We place ourselves in the orthonormal coordinate system (A ; $\vec { \imath } , \vec { \jmath } , \vec { k }$) of space in which the coordinates of points B, D and E are:
$$\mathrm { B } ( 3 ; 0 ; 0 ) , \quad \mathrm { D } ( 0 ; 3 ; 0 ) \quad \text { and } \quad \mathrm { E } ( 0 ; 0 ; 3 ) .$$
We consider the points $\mathrm { P } ( 0 ; 0 ; 1 ) , \quad \mathrm { Q } ( 0 ; 2 ; 3 )$ and $\mathrm { R } ( 1 ; 0 ; 3 )$.

\begin{enumerate}
  \item Place the points P, Q and R on the figure in the APPENDIX which must be returned with your work.
  \item Show that the triangle PQR is isosceles at R.
  \item Justify that the points P, Q and R define a plane.
  \item We are now interested in the distance between point E and the plane (PQR).\\
a. Show that the vector $\vec { u } ( 2 ; 1 ; - 1 )$ is normal to the plane (PQR).\\
b. Deduce a Cartesian equation of the plane (PQR).\\
c. Determine a parametric representation of the line (d) passing through point E and orthogonal to the plane (PQR).\\
d. Show that the point $\mathrm { L } \left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 8 } { 3 } \right)$ is the orthogonal projection of point E onto the plane (PQR).\\
e. Determine the distance between point E and the plane (PQR).
  \item By choosing the triangle EQR as the base, show that the volume of the tetrahedron EPQR is $\frac { 2 } { 3 }$.\\
We recall that the volume $V$ of a tetrahedron is given by the formula:
$$V = \frac { 1 } { 3 } \times \text { area of a base } \times \text { corresponding height. }$$
  \item Find, using the two previous questions, the area of triangle PQR.
\end{enumerate}