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 \( \overrightarrow{ JD }=\dfrac{1}{4}\overrightarrow{ CD } \).

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240}, "size": {"width": 300, "height": 300}, "materials_options": {"surface": {"color": 16744448}, "solid_line": {"color": 16744448, "linewidth": 2}, "hidden_solid_line": {"color": 16744448, "linewidth": 2}, "dashed_line": {"color": 16744448, "linewidth": 2}}}, "data": {"Cuboid_(ABCDEFGH)": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 25, "ly": 15, "lz": 40}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": -13.0, "y": -8.0, "z": -20.0}, "text": "A", "position_offset": {"x": -2.4801773365735524, "y": -1.5262629763529554, "z": -3.8156574408823887}}, "Label_for_Point3D_B": {"type": "Label", "position": {"x": 12.0, "y": -8.0, "z": -20.0}, "text": "B", "position_offset": {"x": 2.33598846428298, "y": -1.5573256428553202, "z": -3.8933141071383}}, "Label_for_Point3D_C": {"type": "Label", "position": {"x": 12.0, "y": 7.0, "z": -20.0}, "text": "C", "position_offset": {"x": 2.3653484995771636, "y": 1.3797866247533455, "z": -3.9422474992952723}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": -13.0, "y": 7.0, "z": -20.0}, "text": "D", "position_offset": {"x": -2.5100961185324846, "y": 1.3515902176713377, "z": -3.8616863362038223}}, "Label_for_Point3D_E": {"type": "Label", "position": {"x": -13.0, "y": -8.0, "z": 20.0}, "text": "E", "position_offset": {"x": -2.4801773365735524, "y": -1.5262629763529554, "z": 3.8156574408823887}}, "Label_for_Point3D_F": {"type": "Label", "position": {"x": 12.0, "y": -8.0, "z": 20.0}, "text": "F", "position_offset": {"x": 2.33598846428298, "y": -1.5573256428553202, "z": 3.8933141071383}}, "Label_for_Point3D_G": {"type": "Label", "position": {"x": 12.0, "y": 7.0, "z": 20.0}, "text": "G", "position_offset": {"x": 2.3653484995771636, "y": 1.3797866247533455, "z": 3.9422474992952723}}, "Label_for_Point3D_H": {"type": "Label", "position": {"x": -13.0, "y": 7.0, "z": 20.0}, "text": "H", "position_offset": {"x": -2.5100961185324846, "y": 1.3515902176713377, "z": 3.8616863362038223}}, "Point3D_I": {"type": "Point", "position": {"x": 12.0, "y": 3.25, "z": -20.0}}, "Label_for_Point3D_I": {"type": "Label", "position": {"x": 12.0, "y": 3.25, "z": -20.0}, "text": "I", "position_offset": {"x": 2.445948079649261, "y": 0.6624442715716748, "z": -4.076580132748768}}, "Point3D_J": {"type": "Point", "position": {"x": -6.75, "y": 7.0, "z": -20.0}}, "Label_for_Point3D_J": {"type": "Label", "position": {"x": -6.75, "y": 7.0, "z": -20.0}, "text": "J", "position_offset": {"x": -1.456915684247789, "y": 1.5108755244051146, "z": -4.316787212586042}}}}

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 \( \overrightarrow{ WN } + \overrightarrow{ UW } + \overrightarrow{ DU } \) 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 \( GFBC \), \( I \) le point défini par \( \overrightarrow{ DI }=0,75\overrightarrow{ DC } \) et \( J \) le point défini par \( \overrightarrow{ AJ }=0,25\overrightarrow{ AD } \).

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240}, "size": {"width": 300, "height": 300}, "materials_options": {"surface": {"color": 16744448}, "solid_line": {"color": 16744448, "linewidth": 2}, "hidden_solid_line": {"color": 16744448, "linewidth": 2}, "dashed_line": {"color": 16744448, "linewidth": 2}}}, "data": {"Cuboid_(ABCDEFGH)": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 40, "ly": 20, "lz": 60}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": -20.0, "y": -10.0, "z": -30.0}, "text": "A", "position_offset": {"x": -2.5657079223592736, "y": -1.2828539611796368, "z": -3.8485618835389106}}, "Label_for_Point3D_B": {"type": "Label", "position": {"x": 20.0, "y": -10.0, "z": -30.0}, "text": "B", "position_offset": {"x": 2.5657079223592736, "y": -1.2828539611796368, "z": -3.8485618835389106}}, "Label_for_Point3D_C": {"type": "Label", "position": {"x": 20.0, "y": 10.0, "z": -30.0}, "text": "C", "position_offset": {"x": 2.5657079223592736, "y": 1.2828539611796368, "z": -3.8485618835389106}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": -20.0, "y": 10.0, "z": -30.0}, "text": "D", "position_offset": {"x": -2.5657079223592736, "y": 1.2828539611796368, "z": -3.8485618835389106}}, "Label_for_Point3D_E": {"type": "Label", "position": {"x": -20.0, "y": -10.0, "z": 30.0}, "text": "E", "position_offset": {"x": -2.5657079223592736, "y": -1.2828539611796368, "z": 3.8485618835389106}}, "Label_for_Point3D_F": {"type": "Label", "position": {"x": 20.0, "y": -10.0, "z": 30.0}, "text": "F", "position_offset": {"x": 2.5657079223592736, "y": -1.2828539611796368, "z": 3.8485618835389106}}, "Label_for_Point3D_G": {"type": "Label", "position": {"x": 20.0, "y": 10.0, "z": 30.0}, "text": "G", "position_offset": {"x": 2.5657079223592736, "y": 1.2828539611796368, "z": 3.8485618835389106}}, "Label_for_Point3D_H": {"type": "Label", "position": {"x": -20.0, "y": 10.0, "z": 30.0}, "text": "H", "position_offset": {"x": -2.5657079223592736, "y": 1.2828539611796368, "z": 3.8485618835389106}}, "Point3D_I": {"type": "Point", "position": {"x": 10.0, "y": 10.0, "z": -30.0}}, "Label_for_Point3D_I": {"type": "Label", "position": {"x": 10.0, "y": 10.0, "z": -30.0}, "text": "I", "position_offset": {"x": 1.447254453973265, "y": 1.447254453973265, "z": -4.341763361919796}}, "Point3D_J": {"type": "Point", "position": {"x": -20.0, "y": -5.0, "z": -30.0}}, "Label_for_Point3D_J": {"type": "Label", "position": {"x": -20.0, "y": -5.0, "z": -30.0}, "text": "J", "position_offset": {"x": -2.637322827814829, "y": -0.6593307069537072, "z": -3.955984241722243}}, "Point3D_O": {"type": "Point", "position": {"x": 20.0, "y": 0.0, "z": 0.0}}, "Label_for_Point3D_O": {"type": "Label", "position": {"x": 20.0, "y": 0.0, "z": 0.0}, "text": "O", "position_offset": {"x": 4.8, "y": 0.0, "z": 0.0}}, "Segment3D_[GB]": {"type": "Segment", "pt1": {"x": 20.0, "y": 10.0, "z": 30.0}, "pt2": {"x": 20.0, "y": -10.0, "z": -30.0}, "label_render": {"visible": false}, "line_render": {"visible": true, "dashed": true}}}}

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

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

Exercice 4 : Relation de Chasles à plus de deux membres

Donner le résultat de la somme \( \overrightarrow{ LC } + \overrightarrow{ CS } + \overrightarrow{ ST } + \overrightarrow{ TM } \) 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 \( \overrightarrow{ BI }=\dfrac{1}{2}\overrightarrow{ BC } \) et \( J \) le point défini par \( \overrightarrow{ CJ }=\dfrac{1}{4}\overrightarrow{ CD } \).

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240}, "size": {"width": 300, "height": 300}, "materials_options": {"surface": {"color": 16744448}, "solid_line": {"color": 16744448, "linewidth": 2}, "hidden_solid_line": {"color": 16744448, "linewidth": 2}, "dashed_line": {"color": 16744448, "linewidth": 2}}}, "data": {"Cuboid_(ABCDEFGH)": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 25, "ly": 15, "lz": 40}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": -13.0, "y": -8.0, "z": -20.0}, "text": "A", "position_offset": {"x": -2.4801773365735524, "y": -1.5262629763529554, "z": -3.8156574408823887}}, "Label_for_Point3D_B": {"type": "Label", "position": {"x": 12.0, "y": -8.0, "z": -20.0}, "text": "B", "position_offset": {"x": 2.33598846428298, "y": -1.5573256428553202, "z": -3.8933141071383}}, "Label_for_Point3D_C": {"type": "Label", "position": {"x": 12.0, "y": 7.0, "z": -20.0}, "text": "C", "position_offset": {"x": 2.3653484995771636, "y": 1.3797866247533455, "z": -3.9422474992952723}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": -13.0, "y": 7.0, "z": -20.0}, "text": "D", "position_offset": {"x": -2.5100961185324846, "y": 1.3515902176713377, "z": -3.8616863362038223}}, "Label_for_Point3D_E": {"type": "Label", "position": {"x": -13.0, "y": -8.0, "z": 20.0}, "text": "E", "position_offset": {"x": -2.4801773365735524, "y": -1.5262629763529554, "z": 3.8156574408823887}}, "Label_for_Point3D_F": {"type": "Label", "position": {"x": 12.0, "y": -8.0, "z": 20.0}, "text": "F", "position_offset": {"x": 2.33598846428298, "y": -1.5573256428553202, "z": 3.8933141071383}}, "Label_for_Point3D_G": {"type": "Label", "position": {"x": 12.0, "y": 7.0, "z": 20.0}, "text": "G", "position_offset": {"x": 2.3653484995771636, "y": 1.3797866247533455, "z": 3.9422474992952723}}, "Label_for_Point3D_H": {"type": "Label", "position": {"x": -13.0, "y": 7.0, "z": 20.0}, "text": "H", "position_offset": {"x": -2.5100961185324846, "y": 1.3515902176713377, "z": 3.8616863362038223}}, "Point3D_I": {"type": "Point", "position": {"x": 12.0, "y": -0.5, "z": -20.0}}, "Label_for_Point3D_I": {"type": "Label", "position": {"x": 12.0, "y": -0.5, "z": -20.0}, "text": "I", "position_offset": {"x": 2.4690123630089134, "y": -0.10287551512537138, "z": -4.1150206050148554}}, "Point3D_J": {"type": "Point", "position": {"x": 5.75, "y": 7.0, "z": -20.0}}, "Label_for_Point3D_J": {"type": "Label", "position": {"x": 5.75, "y": 7.0, "z": -20.0}, "text": "J", "position_offset": {"x": 1.2570640536606852, "y": 1.530338847934747, "z": -4.372396708384993}}}}

On pourra faire pivoter la figure en cliquant dessus.

Parmi les égalités suivantes, la ou laquelles sont vraies ?

A. \( -\overrightarrow{ HF }=\overrightarrow{ BA }-2\overrightarrow{ CI } \)
B. \( -\overrightarrow{ IC }=\dfrac{1}{2}\overrightarrow{ BD }+\dfrac{1}{2}\overrightarrow{ EF } \)
C. \( \overrightarrow{ CA }=\overrightarrow{ EH }-4\overrightarrow{ CJ } \)
D. \( -\overrightarrow{ CJ }=- \dfrac{1}{4}\overrightarrow{ DB }- \dfrac{1}{4}\overrightarrow{ HE } \)

Revenir au choix du niveau