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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 13165499510445--
%%--Produit scalaire--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}%
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Les points $A,$ $B,$ $C,$ $H$ et $K$ sont dispos\'{e}s comme indiqu\'{e}s sur
la figure ci-contre. Calculer les produits scalaires suivants:%
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\begin{enumerate}
\item $\overrightarrow{BA}\cdot\overrightarrow{BC};$

\item $\overrightarrow{AB}\cdot\overrightarrow{AH};$

\item $\overrightarrow{AC}\cdot\overrightarrow{AK};$

\item $\overrightarrow{AB}\cdot\left(  \overrightarrow{CA}+\overrightarrow
{AH}\right)  $

\item $\overrightarrow{AC}\cdot\left(  \overrightarrow{BA}+\overrightarrow
{AK}\right)  ;$

\item $\overrightarrow{KB}\cdot\overrightarrow{HC}.$
\end{enumerate}

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\end{exercise}

\begin{exercise}
$ABC$ est un triangle isoc\`{e}le en$A,$ $O$ est le milieu de $\left[
BC\right]  $, $H$ le projet\'{e} orthogonal de $O$ sur $\left(  AC\right)  $
et $I $ est le milieu de $\left[  OH\right]  .$

\begin{enumerate}
\item
\begin{enumerate}
\item D\'{e}montrer que :%
\[
2\overrightarrow{AI}\cdot\overrightarrow{BH}=\overrightarrow{AO}%
\cdot\overrightarrow{CH}+\overrightarrow{AH}\cdot\overrightarrow
{BC}+\overrightarrow{AH}\cdot\overrightarrow{CH};
\]


\item D\'{e}montrer que:%
\[
\overrightarrow{AO}\cdot\overrightarrow{CH}=\overrightarrow{AH}\cdot
\overrightarrow{CH}\quad\text{et}\quad\overrightarrow{AH}\cdot\overrightarrow
{BC}=2\overrightarrow{AH}\cdot\overrightarrow{HC}.
\]

\end{enumerate}

\item En d\'{e}duire que les droites $\left(  BH\right)  $ et $\left(
AI\right)  $ sont perpendiculaires.
\end{enumerate}
\end{exercise}

\begin{exercise}
\'{E}tant donn\'{e}s deux r\'{e}els non nuls $\alpha$ et $\beta,$ on
consid\`{e}re les points $A$ et $B$ de coordonn\'{e}es respectives $\left(
\alpha;0\right)  $ et $\left(  0;\beta\right)  .$ On note $H$ le projet\'{e}
orthogonal de $O$ sur $\left(  AB\right)  .$

\begin{enumerate}
\item D\'{e}montrer que:%
\[
\overrightarrow{AH}=\left(  \overrightarrow{AO}\cdot\overrightarrow
{AB}\right)  \frac{\overrightarrow{AB}}{\left\Vert \overrightarrow
{AB}\right\Vert ^{2}};
\]


\item En d\'{e}duire, en fonction de $\alpha$ et $\beta,$ les coordonn\'{e}es
de $H.$
\end{enumerate}
\end{exercise}


\end{document}