We place ourselves in space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.

We consider the point $\mathrm{A}(1; 1; 0)$ and the vector $\vec{u}\begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}$.

We consider the plane $\mathscr{P}$ with equation: $x + 4y + 2z + 1 = 0$.

\begin{enumerate}
  \item We denote (d) the line passing through A and directed by the vector $\vec{u}$. Determine a parametric representation of (d).
  \item Justify that the line (d) and the plane $\mathscr{P}$ intersect at a point B whose coordinates are $(1; -1; 1)$.
  \item We consider the point $\mathrm{C}(1; -1; -1)$.

  a. Verify that the points $\mathrm{A}$, $\mathrm{B}$ and C do indeed define a plane.

  b. Show that the vector $\vec{n}\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ is a normal vector to the plane (ABC).

  c. Determine a Cartesian equation of the plane (ABC).

  \item a. Justify that the triangle ABC is isosceles at A.

  b. Let H be the midpoint of segment [BC]. Calculate the length AH then the area of triangle ABC.

  \item Let D be the point with coordinates $(0; -1; 1)$.

  a. Show that the line (BD) is a height of the pyramid ABCD.

  b. Deduce from the previous questions the volume of the pyramid ABCD.
\end{enumerate}

We recall that the volume $V$ of a pyramid is given by:
$$V = \frac{1}{3}\mathscr{B} \times h,$$
where $\mathscr{B}$ is the area of a base and $h$ the corresponding height.