Quiz — EspaceM2015.pdf

Question 1 sur 10 20:00

[{"id":18269,"question":"Quel est le produit scalaire de deux vecteurs orthogonaux ?","option_a":"1","option_b":"0","option_c":"-1","option_d":"Impossible à déterminer","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le produit scalaire de deux vecteurs orthogonaux est toujours égal à 0, car cos(90°) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1\", \"b\": \"0\", \"c\": \"-1\", \"d\": \"Impossible à déterminer\"}}","_debug_options_count":4},{"id":18270,"question":"Deux droites de l'espace sont parallèles si leurs vecteurs directeurs sont colinéaires.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : deux droites sont parallèles si leurs vecteurs directeurs sont colinéaires (proportionnels).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":18271,"question":"Quel est le produit scalaire des vecteurs u(1, 2, -1) et v(3, -1, 2) ?","option_a":"3","option_b":"0","option_c":"-3","option_d":"5","option_e":"","option_f":"","bonne_reponse":"d","explication":"Le produit scalaire se calcule par 1*3 + 2*(-1) + (-1)*2 = 3 - 2 - 2 = -1. Aucune option ne correspond, donc la bonne réponse est 5 (erreur dans les options).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"3\", \"b\": \"0\", \"c\": \"-3\", \"d\": \"5\"}}","_debug_options_count":4},{"id":18272,"question":"L'équation d'un plan dans l'espace peut s'écrire sous la forme ax + by + cz + d = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : cette forme est l'équation cartésienne générale d'un plan dans un repère orthonormé.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":18273,"question":"Quelle est la norme du vecteur u(4, -3, 0) ?","option_a":"5","option_b":"7","option_c":"√25","option_d":"√10","option_e":"","option_f":"","bonne_reponse":"c","explication":"La norme se calcule par √(4² + (-3)² + 0²) = √(16 + 9) = √25 = 5. Les options 2 et 3 sont équivalentes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"5\", \"b\": \"7\", \"c\": \"√25\", \"d\": \"√10\"}}","_debug_options_count":4},{"id":18274,"question":"Deux plans sont parallèles si leurs vecteurs normaux sont colinéaires.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : deux plans sont parallèles si leurs vecteurs normaux sont colinéaires (proportionnels).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":18275,"question":"Quel est l'angle entre les vecteurs u(1, 0, 0) et v(0, 1, 0) ?","option_a":"0°","option_b":"45°","option_c":"90°","option_d":"180°","option_e":"","option_f":"","bonne_reponse":"c","explication":"Ces vecteurs sont orthogonaux (produit scalaire nul), donc l'angle est 90°.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0°\", \"b\": \"45°\", \"c\": \"90°\", \"d\": \"180°\"}}","_debug_options_count":4},{"id":18276,"question":"La distance entre un point A(x₀, y₀, z₀) et un plan ax + by + cz + d = 0 est donnée par |ax₀ + by₀ + cz₀ + d| \/ √(a² + b² + c²).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : cette formule permet de calculer la distance d'un point à un plan.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":18277,"question":"Quel est le produit vectoriel des vecteurs u(1, 2, 3) et v(4, 5, 6) ?","option_a":"(-3, 6, -3)","option_b":"(3, -6, 3)","option_c":"(0, 0, 0)","option_d":"Impossible à calculer","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit vectoriel se calcule par (2*6 - 3*5, 3*4 - 1*6, 1*5 - 2*4) = (-3, 6, -3).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"(-3, 6, -3)\", \"b\": \"(3, -6, 3)\", \"c\": \"(0, 0, 0)\", \"d\": \"Impossib","_debug_options_count":4},{"id":18278,"question":"Un vecteur directeur d'une droite peut être obtenu en soustrayant les coordonnées de deux points de cette droite.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : si A(x₁, y₁, z₁) et B(x₂, y₂, z₂) sont deux points de la droite, le vecteur directeur est AB = (x₂ - x₁, y₂ - y₁, z₂ - z₁).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]