\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
In space, consider the points \(A(5, -5, 2)\), \(B(-1, 1, 0)\), \(C(0, 1, 2)\), and \(D(6, 6, -1)\).
  1. Determine the nature of triangle \(BCD\) and calculate its area.
    1. Show that the vector \(\Vect{n}\begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix}\) is normal to the plane \((BCD)\).
    2. Determine a Cartesian equation of the plane \((BCD)\).
  3. Determine a parametric representation of the line \(d\) orthogonal to the plane \((BCD)\) and passing through point \(A\).
  4. Determine the coordinates of point \(H\), the intersection of line \(d\) and the plane \((BCD)\).
  5. Determine the volume of the tetrahedron \(ABCD\).
  6. Determine an approximate value, rounded to the nearest \(0.1^{\circ}\), of the angle \(\Angle{BAC}\).

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