Seconde - Calculer les coordonnées d'un vecteur

Calculer les coordonnées d'un vecteur

Episode 4

🔑 Pour suivre cet épisode tu dois :

avoir regardé l'épisode 3 ;
avoir cherché sur papier l'exercice de la fin de l'épisode.

📝 Résumé de l'épisode :

Nous donnons les solutions détaillées de l'exercice d'entraînement qui a terminé l'épisode 3.

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=off,id=e1}{4}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-4}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=off,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

Voici la formule qui permet de calculer les coordonnées de $\overrightarrow{\mathrm{MG}}$

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=off,id=e1}{4}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-4}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=off,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

Appliquons cette formule.

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=hl,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=hl,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=hl,id=e1}{4}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-4}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=off,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

Le nombre $x_\mathrm{G}$ est l'abscisse du point $\mathrm{G}$, c'est $4$.

Attention : Le point $\mathrm{G}$ est donné en premier dans l'énoncé.

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=hl,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{4}\htmlData{state=hl,id=e2}{-}\htmlData{state=off,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-4}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=off,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

moins ...

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=hl,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=hl,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{4}\htmlData{state=on,id=e2}{-}\htmlData{state=hl,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-4}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=off,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

Le nombre $x_\mathrm{M}$ est l'abscisse du point $\mathrm{M}$, c'est $3$.

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=hl,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=hl,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=hl,id=e4}{-4}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=off,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

Le nombre $y_\mathrm{G}$ est l'ordonnée du point $\mathrm{G}$, c'est $-4$.

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=hl,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-4}\htmlData{state=hl,id=e5}{-}\htmlData{state=off,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=off,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

moins ...

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=hl,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=hl,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-4}\htmlData{state=on,id=e5}{-}\htmlData{state=hl,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=off,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

Le nombre $y_\mathrm{M}$ est l'ordonnée du point $\mathrm{M}$, c'est $-3$.

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-4}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=off,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

On effectue les calculs.

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=hl,id=e9}{\htmlData{state=on,id=e1}{4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-4}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=hl,id=e9}{1}\\ \htmlData{state=off,id=e8}{-1} \end{pmatrix}$

$4-3=1$

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{3}}\\ \htmlData{state=hl,id=e8}{\htmlData{state=on,id=e4}{-4}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{1}\\ \htmlData{state=hl,id=e8}{-1} \end{pmatrix}$

$-4-\left(-3\right)=-4+3=-1$

1. $\overrightarrow{\mathrm{MG}}$ avec $\mathrm{G}\left(\htmlData{state=on,id=e1}{4} ; \htmlData{state=on,id=e4}{-4}\right)$ et $\mathrm{M}\left(\htmlData{state=on,id=e3}{3} ;\htmlData{state=on,id=e6}{-3}\right)$.

$\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{G}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{M}}\\ \htmlData{state=on,id=e4}{y_\mathrm{G}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{M}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{3}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-4}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{(-3)}} \end{pmatrix}$ $\overrightarrow{\mathrm{MG}}\begin{pmatrix} \htmlData{state=on,id=e9}{1}\\ \htmlData{state=on,id=e8}{-1} \end{pmatrix}$

2. $\overrightarrow{\mathrm{DE}}$ avec $\mathrm{D}\left(-2 ; -6\right)$ et $\mathrm{E}\left(-2 ; -6\right)$

$\overrightarrow{\mathrm{DE}}\begin{pmatrix} 0\\0 \end{pmatrix}$

Nous pouvons remarquer que les points $\mathrm{D}$ et $\mathrm{E}$ ont les mêmes coordonnées.

Sans faire aucun calcul nous savons que $\overrightarrow{\mathrm{DE}}$ est le vecteur nul.

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=off,id=e1}{-2}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=off,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

Voici la formule qui permet de calculer les coordonnées de $\overrightarrow{\mathrm{PA}}$

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=off,id=e1}{-2}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=off,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

Appliquons cette formule.

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=hl,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=hl,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=hl,id=e1}{-2}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=off,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

Le nombre $x_\mathrm{A}$ est l'abscisse du point $\mathrm{A}$, c'est $-2$.

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=hl,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-2}\htmlData{state=hl,id=e2}{-}\htmlData{state=off,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=off,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

moins ...

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=hl,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=hl,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-2}\htmlData{state=on,id=e2}{-}\htmlData{state=hl,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=off,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

Le nombre $x_\mathrm{P}$ est l'abscisse du point $\mathrm{P}$, c'est $4$.

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=hl,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=hl,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-2}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=hl,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=off,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

Le nombre $y_\mathrm{A}$ est l'ordonnée du point $\mathrm{A}$, c'est $3$.

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=hl,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-2}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=hl,id=e5}{-}\htmlData{state=off,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=off,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

moins ...

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=hl,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=hl,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-2}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=hl,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=off,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

Le nombre $y_\mathrm{P}$ est l'ordonnée du point $\mathrm{P}$, c'est $8$.

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-2}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=off,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

On effectue les calculs.

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=hl,id=e9}{\htmlData{state=on,id=e1}{-2}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=hl,id=e9}{-6}\\ \htmlData{state=off,id=e8}{-5} \end{pmatrix}$

$-2-4=-6$

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-2}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{4}}\\ \htmlData{state=hl,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{-6}\\ \htmlData{state=hl,id=e8}{-5} \end{pmatrix}$

$3-8=-5$

3. $\overrightarrow{\mathrm{PA}}$ avec $\mathrm{P}\left(\htmlData{state=on,id=e3}{4} ;\htmlData{state=on,id=e6}{8}\right)$ et $\mathrm{A}\left(\htmlData{state=on,id=e1}{-2} ; \htmlData{state=on,id=e4}{3}\right)$.

$\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{A}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{P}}\\ \htmlData{state=on,id=e4}{y_\mathrm{A}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{P}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-2}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{4}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{8}} \end{pmatrix}$ $\overrightarrow{\mathrm{PA}}\begin{pmatrix} \htmlData{state=on,id=e9}{-6}\\ \htmlData{state=on,id=e8}{-5} \end{pmatrix}$

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=off,id=e1}{9}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=off,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

Voici la formule qui permet de calculer les coordonnées de $\overrightarrow{\mathrm{ET}}$

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=off,id=e1}{9}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=off,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

Appliquons cette formule.

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=hl,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=hl,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=hl,id=e1}{9}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=off,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

Le nombre $x_\mathrm{T}$ est l'abscisse du point $\mathrm{T}$, c'est $9$.

Attention : Le point $\mathrm{T}$ est donné en premier dans l'énoncé.

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=hl,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{9}\htmlData{state=hl,id=e2}{-}\htmlData{state=off,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=off,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

moins ...

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=hl,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=hl,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{9}\htmlData{state=on,id=e2}{-}\htmlData{state=hl,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=off,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

Le nombre $x_\mathrm{E}$ est l'abscisse du point $\mathrm{E}$, c'est $-8$.

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=hl,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=hl,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{9}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=hl,id=e4}{3}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=off,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

Le nombre $y_\mathrm{T}$ est l'ordonnée du point $\mathrm{T}$, c'est $3$.

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=hl,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{9}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=hl,id=e5}{-}\htmlData{state=off,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=off,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

moins ...

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=hl,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=hl,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{9}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=hl,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=off,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

Le nombre $y_\mathrm{E}$ est l'ordonnée du point $\mathrm{E}$, c'est $6$.

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{9}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=off,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

On effectue les calculs.

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=hl,id=e9}{\htmlData{state=on,id=e1}{9}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=hl,id=e9}{17}\\ \htmlData{state=off,id=e8}{-3} \end{pmatrix}$

$9-\left(-8\right)=9+8=17$

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{9}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{(-8)}}\\ \htmlData{state=hl,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{17}\\ \htmlData{state=hl,id=e8}{-3} \end{pmatrix}$

$3-6=-3$

4. $\overrightarrow{\mathrm{ET}}$ avec $\mathrm{T}\left(\htmlData{state=on,id=e1}{9} ; \htmlData{state=on,id=e4}{3}\right)$ et $\mathrm{E}\left(\htmlData{state=on,id=e3}{-8} ;\htmlData{state=on,id=e6}{6}\right)$.

$\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{T}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{E}}\\ \htmlData{state=on,id=e4}{y_\mathrm{T}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{E}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{9}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{(-8)}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{3}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{6}} \end{pmatrix}$ $\overrightarrow{\mathrm{ET}}\begin{pmatrix} \htmlData{state=on,id=e9}{17}\\ \htmlData{state=on,id=e8}{-3} \end{pmatrix}$

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=off,id=e1}{-4}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-1}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=off,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

Voici la formule qui permet de calculer les coordonnées de $\overrightarrow{\mathrm{FK}}$

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=off,id=e1}{-4}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-1}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=off,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

Appliquons cette formule.

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=hl,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=hl,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=hl,id=e1}{-4}\htmlData{state=off,id=e2}{-}\htmlData{state=off,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-1}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=off,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

Le nombre $x_\mathrm{K}$ est l'abscisse du point $\mathrm{K}$, c'est $-4$.

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=hl,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-4}\htmlData{state=hl,id=e2}{-}\htmlData{state=off,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-1}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=off,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

moins ...

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=hl,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=hl,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-4}\htmlData{state=on,id=e2}{-}\htmlData{state=hl,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=off,id=e4}{-1}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=off,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

Le nombre $x_\mathrm{F}$ est l'abscisse du point $\mathrm{F}$, c'est $1$.

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=hl,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=hl,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=hl,id=e4}{-1}\htmlData{state=off,id=e5}{-}\htmlData{state=off,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=off,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

Le nombre $y_\mathrm{K}$ est l'ordonnée du point $\mathrm{K}$, c'est $-1$.

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=hl,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-1}\htmlData{state=hl,id=e5}{-}\htmlData{state=off,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=off,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

moins ...

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=hl,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=hl,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-1}\htmlData{state=on,id=e5}{-}\htmlData{state=hl,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=off,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

Le nombre $y_\mathrm{F}$ est l'ordonnée du point $\mathrm{F}$, c'est $-8$.

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-1}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=off,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

On effectue les calculs.

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=hl,id=e9}{\htmlData{state=on,id=e1}{-4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-1}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=hl,id=e9}{-5}\\ \htmlData{state=off,id=e8}{7} \end{pmatrix}$

$-4-1=-5$

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{1}}\\ \htmlData{state=hl,id=e8}{\htmlData{state=on,id=e4}{-1}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{-5}\\ \htmlData{state=hl,id=e8}{7} \end{pmatrix}$

$-1-\left(-8\right)=-1+8=7$

5. $\overrightarrow{\mathrm{FK}}$ avec $\mathrm{F}\left(\htmlData{state=on,id=e3}{1} ;\htmlData{state=on,id=e6}{-8}\right)$ et $\mathrm{K}\left(\htmlData{state=on,id=e1}{-4} ; \htmlData{state=on,id=e4}{-1}\right)$.

$\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e1}{x_\mathrm{K}}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{x_\mathrm{F}}\\ \htmlData{state=on,id=e4}{y_\mathrm{K}}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{y_\mathrm{F}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{\htmlData{state=on,id=e1}{-4}\htmlData{state=on,id=e2}{-}\htmlData{state=on,id=e3}{1}}\\ \htmlData{state=on,id=e8}{\htmlData{state=on,id=e4}{-1}\htmlData{state=on,id=e5}{-}\htmlData{state=on,id=e6}{(-8)}} \end{pmatrix}$ $\overrightarrow{\mathrm{FK}}\begin{pmatrix} \htmlData{state=on,id=e9}{-5}\\ \htmlData{state=on,id=e8}{7} \end{pmatrix}$

Seconde - Calculer les coordonnées d'un vecteur - Episode 4

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