Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.\\
We consider the following points:
$$\mathrm{A}(1; 3; 0), \quad \mathrm{B}(-1; 4; 5), \quad \mathrm{C}(0; 1; 0) \quad \text{and} \quad \mathrm{D}(-2; 2; 1).$$

\begin{enumerate}
  \item Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ determine a plane.
  \item Show that the triangle ABC is right-angled at A.
  \item Let $\Delta$ be the line passing through point D and with direction vector $\vec{u}\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$.\\
  a. Prove that the line $\Delta$ is orthogonal to the plane (ABC).\\
  b. Justify that the plane (ABC) admits the Cartesian equation:
  $$2x - y + z + 1 = 0$$
  c. Determine a parametric representation of the line $\Delta$.
  \item We call H the point with coordinates $\left(-\dfrac{2}{3}; \dfrac{4}{3}; \dfrac{5}{3}\right)$.\\
  Verify that $H$ is the orthogonal projection of point $D$ onto the plane (ABC).
  \item We recall that the volume of a tetrahedron is given by $V = \dfrac{1}{3} B \times h$, where $B$ is the area of a base of the tetrahedron and $h$ is its height relative to this base.\\
  a. Show that $\mathrm{DH} = \dfrac{2\sqrt{6}}{3}$.\\
  b. Deduce the volume of the tetrahedron ABCD.
  \item We consider the line $d$ with parametric representation:
  $$\left\{\begin{aligned} x &= 1 - 2k \\ y &= -3k \\ z &= 1 + k \end{aligned}\right. \text{ where } k \text{ describes } \mathbb{R}.$$
  Are the line $d$ and the plane (ABC) secant or parallel?
\end{enumerate}