Space is referred to an orthonormal coordinate system in which we consider:
\begin{itemize}
  \item the points $\mathrm{A}(2;-1;0)$, $\mathrm{B}(1;0;-3)$, $\mathrm{C}(6;6;1)$ and $\mathrm{E}(1;2;4)$;
  \item The plane $\mathscr{P}$ with Cartesian equation $2x - y - z + 4 = 0$.
\end{itemize}

\begin{enumerate}
  \item a. Prove that triangle ABC is right-angled at A.\\
b. Calculate the dot product $\overrightarrow{\mathrm{BA}} \cdot \overrightarrow{\mathrm{BC}}$ then the lengths BA and BC.\\
c. Deduce the measure in degrees of the angle $\widehat{\mathrm{ABC}}$ rounded to the nearest degree.
  \item a. Prove that the plane $\mathscr{P}$ is parallel to the plane ABC.\\
b. Deduce a Cartesian equation of the plane ABC.\\
c. Determine a parametric representation of the line $\mathscr{D}$ orthogonal to the plane ABC and passing through point E.\\
d. Prove that the orthogonal projection H of point E onto the plane ABC has coordinates $\left(4; \frac{1}{2}; \frac{5}{2}\right)$.
  \item Recall that the volume of a pyramid is given by $V = \frac{1}{3}\mathscr{B}h$ where $\mathscr{B}$ denotes the area of a base and $h$ the height of the pyramid associated with this base.\\
Calculate the area of triangle ABC then prove that the volume of the pyramid ABCE is equal to $16.5$ cubic units.
\end{enumerate}