In an orthonormal coordinate system in space, we consider the points
$$\mathrm { A } ( 5 ; - 5 ; 2 ) , \mathrm { B } ( - 1 ; 1 ; 0 ) , \mathrm { C } ( 0 ; 1 ; 2 ) \text { and } \mathrm { D } ( 6 ; 6 ; - 1 ) .$$

\begin{enumerate}
  \item Determine the nature of triangle BCD and calculate its area.
  \item a. Show that the vector $\vec { n } \left( \begin{array} { c } - 2 \\ 3 \\ 1 \end{array} \right)$ is a normal vector to the plane (BCD).\\
b. Determine a Cartesian equation of the plane (BCD).
  \item Determine a parametric representation of the line $\mathfrak { D }$ perpendicular to the plane (BCD) and passing through point A.
  \item Determine the coordinates of point H, the intersection of line $\mathcal { D }$ and plane (BCD).
  \item Determine the volume of tetrahedron ABCD.
\end{enumerate}

Recall that the volume of a tetrahedron is given by the formula $\mathcal { V } = \frac { 1 } { 3 } \mathcal { B } \times h$, where $\mathcal { B }$ is the area of a base of the tetrahedron and h is the corresponding height.\\
6. We admit that $\mathrm { AB } = \sqrt { 76 }$ and $\mathrm { AC } = \sqrt { 61 }$.

Determine an approximate value to the nearest tenth of a degree of the angle $\widehat { \mathrm { BAC } }$.