Space is referred to an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. We are given the points $\mathrm{A}(1;0;-1)$, $\mathrm{B}(1;2;3)$, $\mathrm{C}(-5;5;0)$ and $\mathrm{D}(11;1;-2)$. The points I and J are the midpoints of the segments $[\mathrm{AB}]$ and $[\mathrm{CD}]$ respectively. The point K is defined by $\overrightarrow{\mathrm{BK}} = \frac{1}{3}\overrightarrow{\mathrm{BC}}$.

\begin{enumerate}
  \item a. Determine the coordinates of points I, J and K.\\
b. Prove that the points I, J and K define a plane.\\
c. Show that the vector $\vec{n}$ with coordinates $(3;1;4)$ is a normal vector to the plane (IJK). Deduce a Cartesian equation of this plane.
  \item Let $\mathscr{P}$ be the plane with equation $3x + y + 4z - 8 = 0$.\\
a. Determine a parametric representation of the line (BD).\\
b. Prove that the plane $\mathscr{P}$ and the line $(\mathrm{BD})$ are secant and give the coordinates of L, the point of intersection of the plane $\mathscr{P}$ and the line (BD).\\
c. Is the point L the symmetric of point D with respect to point B?
\end{enumerate}