\textbf{Exercise 4 — Candidates who have not followed the speciality course}

We consider the regular pyramid $SABCD$ with apex $S$ consisting of the square base $ABCD$ and equilateral triangles.

The point O is the centre of the base ABCD with $\mathrm{OB} = 1$.\\
We recall that the segment $[\mathrm{SO}]$ is the height of the pyramid and that all edges have the same length.

\begin{enumerate}
  \item Justify that the coordinate system $(\mathrm{O}; \overrightarrow{\mathrm{OB}}, \overrightarrow{\mathrm{OC}}, \overrightarrow{\mathrm{OS}})$ is orthonormal.
  \item We define the point K by the relation $\overrightarrow{\mathrm{SK}} = \frac{1}{3}\overrightarrow{\mathrm{SD}}$ and we denote by I the midpoint of segment $[\mathrm{SO}]$.\\
a. Determine the coordinates of point K.\\
b. Deduce that the points B, I and K are collinear.\\
c. We denote by L the point of intersection of the edge $[\mathrm{SA}]$ with the plane (BCI). Justify that the lines (AD) and (KL) are parallel.\\
d. Determine the coordinates of point L.
  \item We consider the vector $\vec{n}\begin{pmatrix}1\\1\\2\end{pmatrix}$ in the coordinate system $(\mathrm{O}; \overrightarrow{\mathrm{OB}}, \overrightarrow{\mathrm{OC}}, \overrightarrow{\mathrm{OS}})$.\\
a. Show that $\vec{n}$ is a normal vector to the plane (BCI).\\
b. Show that the vectors $\vec{n}$, $\overrightarrow{\mathrm{AS}}$ and $\overrightarrow{\mathrm{DS}}$ are coplanar.\\
c. What is the relative position of the planes (BCI) and (SAD)?
\end{enumerate}