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\hfill + \begin{minipage}{0.4\linewidth} + \begin{tikzpicture} + \repereOIJ{-1}{5}{-1}{5} + \draw [->, very thick] (1, 1) -- node [midway, below] {$\vect{u}$} (3, 2); + \draw [->, very thick] (3, 2) -- node [midway, below] {$\vect{v}$} (4, 4); + \draw [->, very thick] (1, 1) -- node [midway, above left] {$\vect{u} + \vect{v}$} (4, 4); + \end{tikzpicture} + \end{minipage} +\end{propriete} + +\paragraph{Exemple}:~ +Soient 3 vecteurs $\vect{u} \vectCoord{2}{4}$, $\vect{v} \vectCoord{1}{-2}$ et $\vect{w} \vectCoord{-6}{5}$. Calculer les coordonnées des vecteurs suivants: + +\begin{itemize} + \item $\vect{u} + \vect{v} $ + \\[0.5cm] + \item $\vect{u} + \vect{v} - \vect{w} $ + \\[0.5cm] +\end{itemize} + +\afaire{compléter les exemples} + +\begin{definition}[Multiplication par un réel] + Soient $\vect{u}\vectCoord{x}{y}$ un vecteur et $k$ un nombre réel. Alors le vecteur $k\vect{u}$ est le vecteur de coordonnées + \[ + k\vect{v}\quad \vectCoord{kx}{ky} + \] + On dira alors que $\vect{u}$ et $k\vect{u}$ sont \textbf{colinéaires}. +\end{definition} + +\paragraph{Exemple}:~ +On reprend les vecteurs de l'exemple précédent. Calculer les coordonnées des vecteurs suivants +\begin{itemize} + \item $5\vect{u} $ + \\[0.5cm] + \item $\vect{u} + 2\vect{v}$ + \\[0.5cm] +\end{itemize} +\afaire{compléter les exemples} + +\paragraph{Remarque:} Si l'on a $\vect{AI} = \dfrac{1}{2} \vect{AB}$, alors \dotfill + +\afaire{compléter la phrase} \end{document} diff --git a/2nd/18_Vecteur_et_coordonnées/2E_operation.pdf b/2nd/18_Vecteur_et_coordonnées/2E_operation.pdf new file mode 100644 index 0000000000000000000000000000000000000000..270add6cd1c5b8d71e2bc61ab0754ff00592d8ba GIT binary patch literal 22676 zcmce-1CVW9vhQ2AtM)G2wr$(CZQHhO+qP}ncI|S#?>qOM*Bx>Do_KL1x>v-SYsHut zGb8_zbL1~`63YpRP}9>eLlI9c_b&~v%?*t}G2zkS*&0|tadF|%D!JPkf*y}nK)}`wPxG%+>G0_3{?}I;VCQe*xY>8QU2BzkB}r_xJs?{&nU5oE>ccHapn= zqju2$?ZJPKL+Jk&!GF#p^bG$#Oa8OkqW|}ai{Wo~|L1(d@V5m1J?dg${LfnQcbNP) zS;F$4wd8Lx|9dQ9{AZv1k0G9&k>wvj{vRQJT0=5+lLevYwssCzErfB9w2p4CsvFS3 zva=Phv}L@QohkME%cn?;n{2#cTRRV$%1C?=|1|T@TF1xE`ORXWiu~Q`euZHlaai)r z=^vY~;o4lBZy&s>C2VJulN%(bZ6R 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70bd905..d2df98d 100644 --- a/2nd/18_Vecteur_et_coordonnées/exercises.tex +++ b/2nd/18_Vecteur_et_coordonnées/exercises.tex @@ -23,7 +23,6 @@ \item $S$ image de $D$ par la translation de vecteur $2\vect{u}$ \end{enumerate} \end{enumerate} - \end{minipage} \begin{minipage}{0.4\linewidth} \begin{tikzpicture}[scale=0.7] @@ -51,23 +50,34 @@ \begin{enumerate} \item $\vect{AB}$ \item $\vect{AC}$ - \item $\vect{BC}$ - \item $\vect{DC}$ \item $\vect{DE}$ \item $\vect{ED}$ \item $\vect{AE}$ \item $\vect{BE}$ - \item $\vect{EC}$ + \item $\vect{EC}$ \item $\vect{EF}$ + \item $\vect{FA}$ \item $\vect{FG}$ \end{enumerate} \end{multicols} \end{exercise} +\begin{exercise}[subtitle={Égalité entre vecteurs}, step={1}, origin={Création}, topics={ Vecteur et coordonnées }, tags={ vecteurs }] + \begin{enumerate} + \item Dans les cas suivant, justifier si les vecteurs $\vect{AB}$ et $\vect{CD}$ sont égaux + \begin{enumerate} + \item $A(-2; -1)$, $B(1; 3)$, $C(1; 1)$ et $D(-2; -1)$ + \item $A(0; -1)$, $B(1; 0)$, $C(0; -2)$ et $D(1; -1)$ + \end{enumerate} + \item Écrire un algorithme pour déterminer deux vecteurs sont égaux en partant des coordonnées des points. + \item On donne 3 points $A(1; 2)$, $B(1; 4)$ et $C(x; 6)$. Quelle doit être la valeur de $x$ pour que les vecteurs $\vect{AB}$ et $\vect{BC}$ soient égaux? + \item On donne 4 points $A(x-1; 2)$, $B(-1; y-5)$, $C(0; -2)$ et $D(4; 3)$. Quelle doivent être les valeurs de $x$ et $y$ pour que les vecteurs $\vect{AB}$ et $\vect{CD}$ soient égaux? + \end{enumerate} +\end{exercise} \begin{exercise}[subtitle={Coordonnée de points et transformations}, step={1}, origin={Création}, topics={ Vecteur et coordonnées }, tags={ vecteurs }] Calculer les coordonnées des points suivants @@ -77,3 +87,48 @@ \item $F$ image du point $E(0; 3)$ par la translation de vecteur $\vect{v}\vectCoord{-3}{-2}$. \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 + \[ + \vect{u} \vectCoord{2}{5} \qquad + \vect{v} \vectCoord{0}{2} \qquad + \vect{w} \vectCoord{1}{-4} \qquad + \vect{x} \vectCoord{-3}{2} + \] + et les points suivants + \[ + A(2; 5) \qquad + B(4; 1) \qquad + C(2; -2) \qquad + D(-3; 1) + \] + Calculer les coordonnées des vecteurs suivants + \begin{multicols}{4} + \begin{enumerate} + \item $\vect{u} +\vect{x}$ + \item $\vect{w} +\vect{x}$ + + \item $\vect{w} - \vect{v}$ + \item $\vect{u} + \vect{x} + \vect{v} - 2\vect{w}$ + + \item $2\vect{w} +\vect{x} - 2\vect{x}$ + \item $\vect{AB} +\vect{x}$ + + \item $\vect{AC} + 2\vect{CD}$ + \item $\vect{AC} - 3\vect{AB}$ + \end{enumerate} + \end{multicols} +\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} + \begin{enumerate} + \item $\vect{AM} = \dfrac{1}{2}\vect{CB}$ + \item $2\vect{AB} + 3\vect{CM} = \vect{0}$ + \item $\vect{BM} = 3\vect{AB} - \vect{CB}$ + \item $3\vect{BM} = 2\vect{AM}$ + \end{enumerate} + \end{multicols} +\end{exercise} diff --git a/2nd/18_Vecteur_et_coordonnées/index.rst b/2nd/18_Vecteur_et_coordonnées/index.rst index 319dd72..3c545aa 100644 --- a/2nd/18_Vecteur_et_coordonnées/index.rst +++ b/2nd/18_Vecteur_et_coordonnées/index.rst @@ -50,6 +50,18 @@ Exercices Étape 2: Opérations avec des vecteurs ===================================== +.. image:: ./2B_operations.pdf + :height: 200px + :alt: Cours sur les opérations sur les coordonnées de vecteurs + +Exercices + +.. image:: ./2E_operation.pdf + :height: 200px + :alt: Calculs avec des vecteurs + + + Étape 3: Norme et distance ==========================