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a/2nd/18_Vecteur_et_coordonnées/3E_norme.tex b/2nd/18_Vecteur_et_coordonnées/3E_norme.tex new file mode 100644 index 0000000..3baebfb --- /dev/null +++ b/2nd/18_Vecteur_et_coordonnées/3E_norme.tex @@ -0,0 +1,18 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Vecteur et coordonnées - Exercices} +\date{avril 2022} + +\DeclareExerciseCollection[step=3]{banque} +\xsimsetup{collect} + + +\begin{document} +\setcounter{exercise}{7} +\input{exercises.tex} + +\printcollection{banque} + +\end{document} diff --git a/2nd/18_Vecteur_et_coordonnées/4B_determinant_colinearite.pdf b/2nd/18_Vecteur_et_coordonnées/4B_determinant_colinearite.pdf new file mode 100644 index 0000000000000000000000000000000000000000..05d5a7211e373084490457def3e00811f5c0114b GIT binary patch literal 29875 zcmce-1CTJywk_B;zqW1Lwr$(CZQHhOd~Msd-Tk#W|9{`yIp@VYabsd8&Z&s3s>rI! zT`TsAjNEJQB#{>urD33Dfg+h&?q3>Sn;RN|V#cS(w==Ya;^xMuQ}(bo!KYJ~Gqf}@ 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zlnP3DPJJC|Tsl@x*zLApwU5FbH$)nnwA-~MbU8P|hrlp5M1v|NHRP$=Sfs*&Q@Ug=67jWdb=$$$!X;!TlcxZ=hKK literal 0 HcmV?d00001 diff --git a/2nd/18_Vecteur_et_coordonnées/4B_determinant_colinearite.tex b/2nd/18_Vecteur_et_coordonnées/4B_determinant_colinearite.tex index aa34466..f70a0e7 100644 --- a/2nd/18_Vecteur_et_coordonnées/4B_determinant_colinearite.tex +++ b/2nd/18_Vecteur_et_coordonnées/4B_determinant_colinearite.tex @@ -10,5 +10,63 @@ \begin{document} \maketitle +\setcounter{section}{3} + +\section{Colinéarité et déterminant} + +\begin{definition}[Colinéarité] + + Soit $\vect{u}$ et $\vect{v}$ deux vecteurs non nuls. + + S'il existe un nombre $k$ tel que $\vect{u} = k \vect{v}$ on dira alors que $\vect{u}$ et $\vect{v}$ sont \textbf{colinéaires}. + + \begin{center} + \begin{tikzpicture} + \repereOIJ{-1}{5}{-1}{5} + \draw [->, very thick] (1, 2) -- node [midway, above] {$\vect{u}$} (3, 3); + \draw [->, very thick] (1, 1) -- node [midway, above] {$\vect{v}$} (5, 3); + \draw [->, very thick] (4, 5) -- node [midway, above] {$\vect{w}$} (2, 4); + \end{tikzpicture} + \end{center} + +\end{definition} + +\paragraph{Exemples} +\begin{itemize} + \item Dans l'illustration précédentes, $\vect{u}$, $\vect{v}$ et $\vect{w}$ sont colinéaires car + \\ + \item $\vect{u}\,\vectCoord{2}{5}$ et $\vect{v}\, \vectCoord{-10}{-25}$ sont colinéaires car + \\ + \item $\vect{u}\,\vectCoord{2}{5}$ et $\vect{v}\, \vectCoord{4}{15}$ ne sont pas colinéaires car + \\ +\end{itemize} + +\begin{definition}[ Déterminant ] + On appelle \textbf{déterminant} des vecteurs $\vect{u}\; \vectCoord{x_u}{y_u}$ et $vect{v}\; \vectCoord{x_v}{y_v}$ le nombre + \[ + det(\vect{u}, \vect{v}) = x_u\times y_v - x_v\times y_u + \] + + Deux vecteurs sont colinéaires si et seulement si $det(\vect{u}, \vect{v}) = 0$. +\end{definition} + +\begin{multicols}{2} + \begin{propriete}[ Parallélisme ] + Deux droites $(AB)$ et $(CD)$ sont parallèles si et seulement si $\vect{AB}$ et $\vect{CD}$ sont colinéaires. + \end{propriete} + + \paragraph{Exemple}: Soient $A(0; 0)$, $B(1; 1)$, $C(3; 5)$ et $D(5; 7)$. Démontrer que les droites $(AB)$ et $(AC)$ sont parallèles. + \\[1cm] + + \begin{propriete}[ Allignement ] + Trois points $A$, $B$ et $C$ sont alignés si et seulement si $\vect{AB}$ et $\vect{AC}$ sont colinéaires. + \end{propriete} + + \paragraph{Exemple}: Soient $A(4; 2)$, $B(10; -5)$ et $C(-8; 16)$. Démontrer que $A$, $B$ et $C$ sont alignés. + \\[1cm] + +\end{multicols} + +\afaire{compléter les explications} \end{document} diff --git a/2nd/18_Vecteur_et_coordonnées/exercises.tex b/2nd/18_Vecteur_et_coordonnées/exercises.tex index d2df98d..3a5fcf7 100644 --- a/2nd/18_Vecteur_et_coordonnées/exercises.tex +++ b/2nd/18_Vecteur_et_coordonnées/exercises.tex @@ -88,6 +88,9 @@ \end{enumerate} \end{exercise} +% ------- + + \begin{exercise}[subtitle={Calculs avec les coordonnées de vecteurs}, step={2}, origin={Création}, topics={ Vecteur et coordonnées }, tags={ vecteurs }] On définit les vecteurs suivants \[ @@ -121,6 +124,21 @@ \end{multicols} \end{exercise} +\begin{exercise}[subtitle={Équilibre des forces}, step={2}, origin={Création}, topics={ Vecteur et coordonnées }, tags={ vecteurs }] + \begin{enumerate} + \item Un objet est modélisé par un point $O$. On applique dessus 3 forces: $\vect{F_1} \; \vectCoord{0}{-5}$, $\vect{F_2} \; \vectCoord{-2}{2}$ et $\vect{F_3}\; \vectCoord{2}{3}$. + \begin{enumerate} + \item Additionner ces trois forces. + \item Expliquer pourquoi on peut dit que l'objet est en équilibre + \end{enumerate} + \item Un objet est modélisé par un point $O$. On applique dessus 3 forces: $\vect{F_1} \; \vectCoord{-1}{2}$, $\vect{F_2} \; \vectCoord{3}{1}$ et $\vect{F_3}\; \vectCoord{2}{2}$. + \begin{enumerate} + \item Montrer que l'objet n'est pas en équilibre. + \item Quelle doit être la quatrième force à appliquer pour que l'objet soit en équilibre. + \end{enumerate} + \end{enumerate} +\end{exercise} + \begin{exercise}[subtitle={Coordonnée manquante}, step={2}, origin={Création}, topics={ Vecteur et coordonnées }, tags={ vecteurs }] Soient $A(-3; 7)$, $B(0; -3)$ et $(-2; 3)$ trois points du plan et un point $M(x;y)$ dont il faudra déterminer les coordonnées dans chacun des cas suivants \begin{multicols}{4} @@ -132,3 +150,28 @@ \end{enumerate} \end{multicols} \end{exercise} + +% ------- + + +\begin{exercise}[subtitle={Norme d'un vecteur}, step={3}, origin={Création}, topics={ Vecteur et coordonnées }, tags={ vecteurs }] + On définit les vecteurs suivants + \[ + \vect{u} \vectCoord{2}{5} \qquad + \vect{v} \vectCoord{0}{2} \qquad + \vect{w} \vectCoord{1}{-4} \qquad + \vect{x} \vectCoord{-3}{2} + \] + et les points suivants + \[ + A(2; 5) \qquad + B(4; 1) \qquad + C(2; \dfrac{1}{5}) \qquad + D(\dfrac{2}{3}; 1) + \] + Calculer les coordonnées des vecteurs suivants + \begin{enumerate} + \item Calculer la norme des vecteurs: $\vect{u}$, $\vect{v}$, $\vect{w}$ et $\vect{x}$ + \item Calculer la norme des vecteurs: $\vect{AB}$ et $\vect{CD}$ + \end{enumerate} +\end{exercise} diff --git a/2nd/18_Vecteur_et_coordonnées/index.rst b/2nd/18_Vecteur_et_coordonnées/index.rst index ef1212e..b062647 100644 --- a/2nd/18_Vecteur_et_coordonnées/index.rst +++ b/2nd/18_Vecteur_et_coordonnées/index.rst @@ -33,6 +33,8 @@ Pas de plan de travail pour cette fois. Le chapitre est technique et c'est la fi Étape 1: Coordonnées de vecteurs dans un repère =============================================== +(environ 1h30) + Cours à lire en deux fois .. image:: ./1B_coordonnees.pdf @@ -50,6 +52,10 @@ Exercices Étape 2: Opérations avec des vecteurs ===================================== +(environ une bonne heure) + +Cours: + .. image:: ./2B_operations.pdf :height: 200px :alt: Cours sur les opérations sur les coordonnées de vecteurs @@ -63,12 +69,31 @@ Exercices Étape 3: Norme et distance ========================== +(environ une demi heure) + +Cours: + .. image:: ./3B_norme_distance.pdf :height: 200px :alt: Cours sur la norme d'un vecteur +Exercice +.. image:: ./3E_norme.pdf + :height: 200px + :alt: exercice sur la norme d'un vecteur Étape 4: Déterminant et colinéarité =================================== + +(environ une heure) + +Cours: + +.. image:: ./4B_determinant_colinearite.pdf + :height: 200px + :alt: cours sur le déterminant et la colinéarité + + +