5 POINTS

In space equipped with an orthonormal reference frame $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$ with unit 1 cm, we consider the points: $A(3; -1; 1)$; $B(4; -1; 0)$; $C(0; 3; 2)$; $D(4; 3; -2)$ and $S(2; 1; 4)$.

In this exercise we wish to show that SABDC is a pyramid with trapezoidal base ABDC and apex $S$, in order to calculate its volume.

\begin{enumerate}
  \item Show that the points $A$, $B$ and $C$ are not collinear.
  \item a. Show that the points $A$, $B$, $C$ and $D$ are coplanar.\\
b. Show that the quadrilateral ABDC is a trapezoid with bases $[AB]$ and $[CD]$.
\end{enumerate}

Recall that a trapezoid is a quadrilateral having two opposite parallel sides called bases.

3. a. Prove that the vector $\vec{n}(2; 1; 2)$ is a normal vector to the plane (ABC).\\
b. Deduce a Cartesian equation of the plane (ABC).\\
c. Determine a parametric representation of the line $\Delta$ passing through point $S$ and orthogonal to the plane (ABC).\\
d. Let I be the point of intersection of the line $\Delta$ and the plane (ABC). Show that point I has coordinates $\left(\frac{2}{3}; \frac{1}{3}; \frac{8}{3}\right)$, then show that $SI = 2$ cm.

4. a. Verify that the orthogonal projection H of point B onto the line (CD) has coordinates $H(3; 3; -1)$ and show that $HB = 3\sqrt{2}$ cm.\\
b. Calculate the exact value of the area of trapezoid ABDC.

Recall that the area of a trapezoid is given by the formula
$$\mathscr{A} = \frac{b + B}{2} \times h$$
where $b$ and $B$ are the lengths of the bases of the trapezoid and $h$ is its height.

5. Determine the volume of pyramid SABDC.

Recall that the volume $V$ of a pyramid is given by the formula
$$V = \frac{1}{3} \times \text{area of the base} \times \text{height}$$