\begin{center}
\includegraphics{225_6_fig_25}
\end{center}
\begin{questions}
        \item  On a :
        \begin{align*}
        \gvec{AB} \cdot \gvec{AD} & = AB \times AD \times \cos \left(
\frac{2\pi}{3}\right) \\
        &= 6 \times 4 \times \left( -\frac{1}{2} \right) \\
        &= -12
        \end{align*}

        \item
        \begin{questions}
                \item  On a :
                \[
                \gvec{AC} = \gvec{AB}+\gvec{AD} \qquad \text{et}
\qquad \gvec{BD} = \gvec{AD} - \gvec{AB}
                \]
                \item  On a :
                \begin{align*}
                \gvec{AB} \cdot \gvec{AC} &= \gvec{AB} \cdot
(\gvec{AB} + \gvec{AD}) \\
                &= \gvec{AB}^2 + \gvec{AB} \cdot \gvec{AD} \\
                &= 36 - 12 = 24 \\
                \gvec{AB} \cdot \gvec{BD} &= \gvec{AB} \cdot
(\gvec{AD} - \gvec{AB}) \\
                &= \gvec{AB} \cdot \gvec{AD} - \gvec{AB}^2 \\
                &= -12-36 = -48
                \end{align*}

                \item  On a :
               \[
               \gvec{AB} \cdot \gvec{AC} = AB \times AC' = 24 \quad
               \text{et} \quad AB = 6\;,
                \]
                donc :
                \[
                AC' = 4
                \]
                \[
                \gvec{AB} \cdot \gvec{AD} = -AB \times AD' = -12
                \quad \text{et} \quad
                AB = 6.
                \]
                Ainsi :
                \[
                AD' = 2.
                \]
        \end{questions}
\end{questions}
