Vecteurs, droites et plans de l’espace - Spécialité

Bases et repères de l’espace

Exercice 1 : Appliquer la relation de Chasles dans un parallélépipède (QCM)

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

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On pourra faire pivoter la figure en cliquant dessus.

Parmi les égalités suivantes, la ou laquelles sont vraies ?

A. \( -\overrightarrow{ HF }=-\overrightarrow{ HG }- \dfrac{4}{3}\overrightarrow{ IB } \)
B. \( -\overrightarrow{ IB }=\dfrac{3}{4}\overrightarrow{ AB }+\dfrac{3}{4}\overrightarrow{ BD } \)
C. \( -\overrightarrow{ HA }=-\overrightarrow{ DA }+4\overrightarrow{ JD } \)
D. \( -\overrightarrow{ JD }=\dfrac{1}{4}\overrightarrow{ AD }+\dfrac{1}{4}\overrightarrow{ BD } \)

Exercice 2 : Relation de Chasles à plus de deux membres

Donner le résultat de la somme \( \overrightarrow{ JY } + \overrightarrow{ YT } + \overrightarrow{ TS } + \overrightarrow{ SV } + \overrightarrow{ VN } \) sous forme d'un seul vecteur.

Exercice 3 : 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 \( \overrightarrow{ GI }=0,25\overrightarrow{ GF } \) et \( J \) le point défini par \( \overrightarrow{ FJ }=0,5\overrightarrow{ FB } \).

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

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

Exercice 4 : Relation de Chasles à plus de deux membres

Donner le résultat de la somme \( \overrightarrow{ UC } + \overrightarrow{ NU } + \overrightarrow{ SN } \) sous forme d'un seul vecteur.

Exercice 5 : Appliquer la relation de Chasles dans un parallélépipède (QCM)

\( ABCDEFGH \) est le parallélépipède rectangle représenté ci-dessous. \( I \) le point défini par \( \overrightarrow{ BI }=\dfrac{3}{4}\overrightarrow{ BF } \) et \( J \) le point défini par \( \overrightarrow{ JG }=\dfrac{1}{2}\overrightarrow{ FG } \).

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On pourra faire pivoter la figure en cliquant dessus.

Parmi les égalités suivantes, la ou laquelles sont vraies ?

A. \( -\overrightarrow{ HA }=\overrightarrow{ AD }+\dfrac{4}{3}\overrightarrow{ BI } \)
B. \( \overrightarrow{ IF }=- \dfrac{1}{4}\overrightarrow{ AD }- \dfrac{1}{4}\overrightarrow{ HA } \)
C. \( -\overrightarrow{ FH }=\overrightarrow{ DH }-2\overrightarrow{ JG } \)
D. \( -\overrightarrow{ JG }=- \dfrac{1}{2}\overrightarrow{ DH }- \dfrac{1}{2}\overrightarrow{ GB } \)

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