Correction DM 1 - ,Calcul vectoriel, Exercice 2
📅 October 30, 2024 | 👁️ Views: 348
\documentclass[12pt,a4paper]{article}
\usepackage{tabularx}
\usepackage{booktabs}
\usepackage{ragged2e}
\usepackage[left=1.00cm, right=1.00cm, top=0.50cm, bottom=2cm]{geometry}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{fontspec}
\usepackage{mathrsfs}
\usepackage{setspace}
\usepackage{multirow,multicol}
\usepackage{xcolor}
\usepackage[ddmmyyyy]{datetime}
\usepackage{tikz,tkz-tab}
\usetikzlibrary{shapes,decorations.text,calc}
\usepackage{hyperref}
\setstretch{1.7}
\setlength{\columnsep}{0.4cm} % Adjust space between columns
\setlength{\columnseprule}{0.4pt} % Width of the vertical line
\hypersetup{
colorlinks=true,
linkcolor=blue
}
\newcommand{\mylink}{\href{https://mosaid.xyz/cc}{www.mosaid.xyz}}
\newcommand{\stamp}[2]{
\begin{tikzpicture}[remember picture, overlay]
\coordinate (A) at (#1,#2);
\draw[red!50] (A) circle (1.9cm);
% Draw the inner circle
\draw[red!50] (A) circle (1.4cm);
% Draw the curved line
\draw[red!50, decorate, decoration={text along path,
text={|\fontspec{DejaVu Sans}\color{red!75}\bfseries|★MOSAID RADOUAN★},
text align={align=center}, raise=-3pt}] (A) ++ (180:1.6cm) arc (180:0:1.6cm);
\draw[decorate, decoration={text along path,
text={|\fontspec{DejaVu Sans}\color{red!75}\bfseries|∞★~mosaid.xyz~★∞ },
text align={align=center}, raise=-6.5pt}] (A) ++ (180:1.53cm) arc (-180:0:1.53cm);
\node[red!75,font=\fontsize{48}{48}\fontspec{DejaVu Sans}\bfseries\selectfont] at (A) {✷};
\end{tikzpicture}
}
\everymath{\displaystyle}
\begin{document}
\noindent
\begin{center}
\begin{tabular}{@{}p{0.22\textwidth}p{0.57\textwidth}p{0.17\textwidth}}
%\toprule
\multirow{2}{*}{\parbox{\linewidth}{Prof MOSAID \newline \mylink }}
& \Centering {Correction Devoir à Domicile 1/Exercice 2} & \hfill TCS \\
\bottomrule
\end{tabular}
\end{center}
\noindent
\textbf{\underline{Exercice 2:}}\\
1. La figure:\\
$\overrightarrow{BD}=\frac{2}{3}\overrightarrow{BC}$\\
$\overrightarrow{AE}=-2\overrightarrow{AD}$\\
$\overrightarrow{BF}=\frac{3}{5}\overrightarrow{BE}$\\[7cm]
\hspace*{4cm}
\begin{tikzpicture}[remember picture, overlay]
\coordinate (A) at (5,5);
\coordinate (B) at (3,2);
\coordinate (C) at (6,2);
\coordinate (D) at ($(B)!0.666!(C)$);
\coordinate (E) at ($(D)!3!(A)$);
\coordinate (F) at ($(B)!0.6!(E)$);
\draw (A) -- (B) -- (C) -- cycle;
\draw (A) -- (D) ;
\draw (B) -- (E) ;
\draw[->,thick,blue] (B) -- (D);
\draw[->,thick,red] (A) -- (E);
\draw[->,thick,red!75] (B) -- (F);
\draw[thick,blue!75] ($(A)!1.4!(C)$) -- ($(A)!1.4!(F)$);
\node[above right] at (A) {A};
\node[below left] at (B) {B};
\node[below right] at (C) {C};
\node[above left] at (D) {D};
\node[above right] at (E) {E};
\node[above right] at (F) {F};
\node[thick, font=\tiny\bfseries, rotate=-30] at ($(B)!0.333!(C)$) {|};
\node[thick, font=\tiny\bfseries, rotate=-30] at ($(A)!0.5!(E)$) {|};
\node[thick, font=\tiny\bfseries, rotate=30] at ($(B)!0.2!(E)$) {|};
\node[thick, font=\tiny\bfseries, rotate=30] at ($(B)!0.4!(E)$) {|};
\node[thick, font=\tiny\bfseries, rotate=30] at ($(B)!0.6!(E)$) {|};
\node[thick, font=\tiny\bfseries, rotate=30] at ($(B)!0.8!(E)$) {|};
\end{tikzpicture}
\stamp{9}{6}\\
2. Montrer que $\overrightarrow{EA}=2\overrightarrow{AB}+\dfrac{4}{3}\overrightarrow{BC}$ puis
$\overrightarrow{FB}=\dfrac{9}{5}\overrightarrow{AB}+\dfrac{4}{5}\overrightarrow{BC}$\\
\begin{minipage}{0.3\textwidth}
\noindent On a:\\
$\overrightarrow{AE}=-2\overrightarrow{AD}$\\
$\overrightarrow{EA}=2\overrightarrow{AD}$\\
$\overrightarrow{EA}=2(\overrightarrow{A\textcolor{red}{B}}+\overrightarrow{\textcolor{red}{B}D})$\\
$\overrightarrow{EA}=2\overrightarrow{AB}+2\overrightarrow{BD}$\\
$\overrightarrow{EA}=2\overrightarrow{AB}+2 \times \textcolor{red}{\frac{2}{3}\overrightarrow{BC}}$\\
$\overrightarrow{EA}=2\overrightarrow{AB}+\frac{4}{3}\overrightarrow{BC}$\\
\end{minipage}
\hspace*{0.2cm}\vline\hspace*{0.2cm}
\begin{minipage}{0.3\textwidth}
\noindent On a:\\
$\overrightarrow{BF}=\frac{3}{5}\overrightarrow{BE}$\\
$\overrightarrow{FB}=\frac{3}{5}\overrightarrow{EB}$\\
$\overrightarrow{FB}=\frac{3}{5}(\overrightarrow{E\textcolor{red}{A}}+\overrightarrow{\textcolor{red}{A}B})$\\
$\overrightarrow{FB}=\frac{3}{5}\overrightarrow{EA}+\frac{3}{5}\overrightarrow{AB}$\\
$\overrightarrow{FB}=\frac{3}{5}(\textcolor{red}{2\overrightarrow{AB}+\frac{4}{3}\overrightarrow{BC}})+\frac{3}{5}\overrightarrow{AB}$\\
$\overrightarrow{FB}=\frac{6}{5}\overrightarrow{AB}+\frac{3}{5}\times \frac{4}{3}\overrightarrow{BC}+\frac{3}{5}\overrightarrow{AB}$\\
$\overrightarrow{FB}=\frac{9}{5}\overrightarrow{AB}+\frac{4}{5}\overrightarrow{BC}$\\
\end{minipage}
\hspace*{0.2cm}\vline\hspace*{0.2cm}
\begin{minipage}{0.3\textwidth}
Pour montrer que les points A, F et C sont alignés, il suffit de montrer que
les vecteurs $\overrightarrow{AC}$ et $\overrightarrow{AF}$ sont colinèaires\\
On va utiliser les deux relations qu'on vient de démontrer:\\
\stamp{3}{-2}
\end{minipage}
\noindent On a:~
$\overrightarrow{FB}=\frac{9}{5}\overrightarrow{AB}+\frac{4}{5}\overrightarrow{BC}$~~donc~~
$\overrightarrow{FA}+\overrightarrow{AB}=\frac{9}{5}\overrightarrow{AB}+\frac{4}{5}(\overrightarrow{BA}
+\overrightarrow{AC})$\\
Donc~~
$\overrightarrow{FA}=-\overrightarrow{AB}+\frac{9}{5}\overrightarrow{AB}-\frac{4}{5}\overrightarrow{AB}
+\frac{4}{5}\overrightarrow{AC}$\\
Donc~~
$\overrightarrow{FA}=-\frac{5}{5}\overrightarrow{AB}+\frac{9}{5}\overrightarrow{AB}-\frac{4}{5}\overrightarrow{AB}
+\frac{4}{5}\overrightarrow{AC}$
~~donc~~
$\textcolor{red}{\mathbf{\overrightarrow{FA}=\frac{4}{5}\overrightarrow{AC}}}$\\
Alors les vecteurs $\overrightarrow{AC}$ et $\overrightarrow{FA}$ sont colinèaires
et les points A, F et C sont alignés. \\
\textcolor{white}{.}\hfill \underline{MOSAID le \today}\\
\vspace*{-1cm}
\textcolor{white}{.}\hfill \mylink
\end{document}
Related Courses, Exams, and Exercises
Solution PDF:
📥 Download Correction DM 1 - ,Calcul vectoriel, Exercice 2 (PDF)
if you find this content helpful, Please consider supporting me with a small donation
إن وجدت هذا المحتوى مفيدا، من فضلك إدعمني بمبلغ بسيط كتبرع
Buy me a coffee — إشتر لي قهوة
PayPal.me • عبر بايبالOr bank transfer • أو حوالة بنكية
Titulaire : RADOUAN MOSAID RIB : 230 090 6501953211022000 65 IBAN : MA64 2300 9065 0195 3211 0220 0065 BIC / SWIFT : CIHMMAMC
Recent Courses
Controle 02 S01, Généralités sur les fonctions et Ensembles
29 Nov 2025
Exam • Maths • 1 Bac Science Maths
Devoir Libre 02 - S01, Etude d’une fonction, Limite d’une suite numérique
28 Nov 2025
Exam • Maths • 2 Bac Science
Most Viewed Courses
Tous les sujets des suites numériques des examens nationaux 2008 - 2022
Views: 1.70K
Exam • Maths • 2 Bac Science
DM 1 - Arithmetiques, Calcul vectoriel et projection
Views: 1.57K
Exercise • Maths • Tronc Commun Sciences
MATHEMATIQUES Examens nationaux 2003-2021 2 Bac.Sciences expérimentales
Views: 1.53K
Exam • Maths • 2 Bac Science
Control 01 S01 En arithmétiques et Calcul vectoriel - A 2025-2026
Views: 1.51K
Exam • Maths • Tronc Commun Sciences
Recent Articles
Boosting LaTeX Editing with Custom Vim Mappings
23 Nov 2025
Quran Search: A Feature rich and modern design
13 Feb 2025
Most Viewed Articles
