Vecteurs, droites et plans de l’espace - Spécialité

Bases et repères de l’espace

Exercice 1 : Décomposer un vecteur dans une base en 3D

\( ABCDEFGH \) est le parallélépipède rectangle représenté ci-dessous. On note O le centre de la face \( GHEF \), \( I \) le point défini par \( \overrightarrow{ EI }=0,25\overrightarrow{ EA } \) et \( J \) le point défini par \( \overrightarrow{ EJ }=0,5\overrightarrow{ EH } \).

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Décomposer le vecteur \( \overrightarrow{OI} \) et dans la base \( (\overrightarrow{ HD },\overrightarrow{ HE },\overrightarrow{ HG }) \).

Décomposer le vecteur \( \overrightarrow{ GJ } \) et dans la base \( (\overrightarrow{ HD },\overrightarrow{ HE },\overrightarrow{ HG }) \).

Exercice 2 : Relation de Chasles à plus de deux membres

Donner le résultat de la somme \( \overrightarrow{ NG } + \overrightarrow{ GH } + \overrightarrow{ HA } + \overrightarrow{ AZ } + \overrightarrow{ ZI } \) sous forme d'un seul vecteur.

Exercice 3 : Appliquer la relation de Chasles dans un parallélépipède (QCM)

\( ABCDEFGH \) est le parallélépipède rectangle représenté ci-dessous. \( I \) le point défini par \( \overrightarrow{ IC }=\dfrac{1}{4}\overrightarrow{ BC } \) et \( J \) le point défini par \( \overrightarrow{ CJ }=\dfrac{3}{4}\overrightarrow{ CD } \).

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On pourra faire pivoter la figure en cliquant dessus.

Parmi les égalités suivantes, la ou laquelles sont vraies ?

A. \( \overrightarrow{ FH }=\overrightarrow{ BA }- \dfrac{4}{3}\overrightarrow{ IB } \)
B. \( -\overrightarrow{ IB }=\dfrac{3}{4}\overrightarrow{ BD }+\dfrac{3}{4}\overrightarrow{ HG } \)
C. \( \overrightarrow{ FH }=\overrightarrow{ FG }+4\overrightarrow{ DJ } \)
D. \( -\overrightarrow{ DJ }=\dfrac{1}{4}\overrightarrow{ BD }+\dfrac{1}{4}\overrightarrow{ DA } \)

Exercice 4 : Relation de Chasles à plus de deux membres

Donner le résultat de la somme \( \overrightarrow{ TX } + \overrightarrow{ XU } + \overrightarrow{ UO } \) sous forme d'un seul vecteur.

Exercice 5 : Décomposer un vecteur dans une base en 3D

\( ABCDEFGH \) est le parallélépipède rectangle représenté ci-dessous. On note O le centre de la face \( ABFE \), \( I \) le point défini par \( \overrightarrow{ AI }=0,5\overrightarrow{ AD } \) et \( J \) le point défini par \( \overrightarrow{ CJ }=0,5\overrightarrow{ CB } \).

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Décomposer le vecteur \( \overrightarrow{OI} \) et dans la base \( (\overrightarrow{ AD },\overrightarrow{ AE },\overrightarrow{ AB }) \).

Décomposer le vecteur \( \overrightarrow{ DJ } \) et dans la base \( (\overrightarrow{ AD },\overrightarrow{ AE },\overrightarrow{ AB }) \).

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