\begin{exercice*} \label{TSG3_E_mental_ps_cube}
On considère un cube $ABCDEFGH$ de côté $a>0$. Soient $I$ le milieu de $[EH]$ et $J$ le centre de la face $CDHG$.
\begin{center}
 	\def \dx {-0.4}
	\begin{tikzpicture}[general, scale=2.5]
	%points
	\coordinate (A) at (0,0); \draw (A) node[above left] {$A$};
	\coordinate (B) at (\dx,-0.5); \draw (B) node[below left] {$B$};
	\coordinate (C) at (1+\dx,-0.5); \draw (C) node[below] {$C$};
	\coordinate (D) at (1,0); \draw (D) node[right] {$D$};
	\coordinate (E) at (0,1); \draw (E) node[above left] {$E$};
	\coordinate (F) at (\dx,0.5); \draw (F) node[left] {$F$};
	\coordinate (G) at (1+\dx,0.5); \draw (G) node[right] {$G$};
	\coordinate (H) at (1,1); \draw (H) node[above right] {$H$};
	\coordinate (I) at (0.5,1); \draw (I) node[above right] {$I$}; \draw (I) node {$\times$};
	\coordinate (J) at ({(2+\dx)/2},0.25); \draw (J) node[right] {$J$};
	%arêtes
	\draw (F)--(B)--(C)--(D)--(H)--(G)--(C); \draw (G)--(F)--(E)--(H);
	\draw[dashed] (A)--(B); \draw[dashed] (A)--(D); \draw[dashed] (A)--(E);
	%diagonales de CDGH
	\draw[thin] (C)--(H); \draw[thin] (G)--(D);
	\end{tikzpicture}
\end{center}
Exprimer en fonction de $a$ les produits scalaires :
\begin{colenumerate}{2}
 \item $\vv{AB} \cdot \vv{AC}$
 \item $\vv{AB} \cdot \vv{FH}$
 \item $\vv{EH} \cdot \vv{GC}$
 \item $\vv{EH} \cdot \vv{FC}$
 \item $\vv{BC} \cdot \vv{BG}$
 \item $\vv{HC} \cdot \vv{GD}$
\end{colenumerate}
\begin{corrige}
\begin{colenumerate}{2}
 \item $a^2$
 \item $-a^2$
 \item 0
 \item $a^2$
 \item $a^2$
 \item 0
\end{colenumerate}
\end{corrige}
\end{exercice*}

