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Номер 702, страница 177 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Глава 7. Метод координат в пространстве. Движения. Параграф 2. Скалярное произведение векторов - номер 702, страница 177.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745117 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "177" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-702" "field_display_title" => "702" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:2 [ 0 => App\Models\Branch {#1077 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Term {#1078 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Book {#1082 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1080} "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1081} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1118 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744407 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Скалярное произведение векторов" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "171" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Book {#1122 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1143 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" …17 ] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Branch {#1158 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744407 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Скалярное произведение векторов" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119} "field_page_start" => "171" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1121} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:6 [ 0 => App\Models\Element {#1218 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 939584 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Edition {#1219 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745117" "type" => "task" ] "text" => "<p> <b>702.</b> Все рёбра тетраэдра <i>ABCD</i> равны друг другу. Точки <i>М</i> и <i>N</i> — середины рёбер <i>AD</i> и <i>ВС</i>. Докажите, что <span class="long"><span class="arrtoptoleft">MN</span> ⋅ <span class="arrtoptoleft">AD</span> = <span class="arrtoptoleft">MN</span> ⋅ <span class="arrtoptoleft">BC</span> = 0.</span> </p>" "img" => array:1 [ 0 => array:5 [ "name" => "702-1.jpg" "alt" => null "width" => "1562" "height" => 177 "path" => "/media/geometrija_10/atanasjan-u23/0-00/702-1.webp?ts=1739436846" ] ] ] #original: array:7 [ "id" => 939584 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227} "task" => array:2 [ "refs" => "745117" "type" => "task" ] "text" => "<p> <b>702.</b> Все рёбра тетраэдра <i>ABCD</i> равны друг другу. Точки <i>М</i> и <i>N</i> — середины рёбер <i>AD</i> и <i>ВС</i>. Докажите, что <span class="long"><span class="arrtoptoleft">MN</span> ⋅ <span class="arrtoptoleft">AD</span> = <span class="arrtoptoleft">MN</span> ⋅ <span class="arrtoptoleft">BC</span> = 0.</span> </p>" "img" => array:1 [ 0 => array:5 [ "name" => "702-1.jpg" "alt" => null "width" => "1562" "height" => 177 "path" => "/media/geometrija_10/atanasjan-u23/0-00/702-1.webp?ts=1739436846" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1249 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 942552 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1259 #items: array:1 [ 0 => App\Models\Edition {#1250 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745117" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "702-1.png" "alt" => null "width" => "694" "height" => 3890 "path" => "/media/geometrija_10/atanasjan-u23/2-00/702-1.webp?ts=1739441580" ] ] ] #original: array:6 [ "id" => 942552 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1259} "task" => array:2 [ "refs" => "745117" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "702-1.png" "alt" => null "width" => "694" "height" => 3890 "path" => "/media/geometrija_10/atanasjan-u23/2-00/702-1.webp?ts=1739441580" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1241 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 942386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Edition {#1242 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745117" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "702-1.jpg" "alt" => null "width" => "1320" "height" => 959 "path" => "/media/geometrija_10/atanasjan-u23/3-00/702-1.webp?ts=1739437149" ] ] ] #original: array:6 [ "id" => 942386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1251} "task" => array:2 [ "refs" => "745117" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "702-1.jpg" "alt" => null "width" => "1320" "height" => 959 "path" => "/media/geometrija_10/atanasjan-u23/3-00/702-1.webp?ts=1739437149" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Element {#1225 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 941846 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\Edition {#1226 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745117" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "702-1.jpg" "alt" => null "width" => "1423" "height" => 796 "path" => "/media/geometrija_10/atanasjan-u23/4-00/702-1.webp?ts=1739436713" ] 1 => array:5 [ "name" => "702-2.jpg" "alt" => null "width" => "1423" "height" => 357 "path" => "/media/geometrija_10/atanasjan-u23/4-00/702-2.webp?ts=1739436713" ] ] ] #original: array:6 [ "id" => 941846 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1235} "task" => array:2 [ "refs" => "745117" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "702-1.jpg" "alt" => null "width" => "1423" "height" => 796 "path" => "/media/geometrija_10/atanasjan-u23/4-00/702-1.webp?ts=1739436713" ] 1 => array:5 [ "name" => "702-2.jpg" "alt" => null "width" => "1423" "height" => 357 "path" => "/media/geometrija_10/atanasjan-u23/4-00/702-2.webp?ts=1739436713" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Element {#1233 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 941972 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1243 #items: array:1 [ 0 => App\Models\Edition {#1234 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745117" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "702-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/702-1.webp?ts=1739436728" ] ] ] #original: array:6 [ "id" => 941972 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1243} "task" => array:2 [ "refs" => "745117" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "702-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/702-1.webp?ts=1739436728" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1257 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1352430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1267 #items: array:1 [ 0 => App\Models\Edition {#1258 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745117" "type" => "task" ] "text" => "<p>По условию, все рёбра тетраэдра $ABCD$ равны друг другу. Такой тетраэдр является правильным, а все его грани — равносторонние треугольники. Пусть длина ребра тетраэдра равна $a$. Точки $M$ и $N$ являются серединами рёбер $AD$ и $BC$ соответственно. Требуется доказать, что скалярное произведение вектора $\vec{MN}$ на вектор $\vec{AD}$ равно нулю, и скалярное произведение вектора $\vec{MN}$ на вектор $\vec{BC}$ также равно нулю.</p><p>Для доказательства воспользуемся векторным методом. Выберем вершину $A$ в качестве начала отсчёта. Тогда векторы, выходящие из этой вершины, будут радиус-векторами других вершин: $\vec{AB}$, $\vec{AC}$, $\vec{AD}$.</p><p>Поскольку $M$ — середина ребра $AD$, радиус-вектор точки $M$ выражается как $\vec{AM} = \frac{1}{2}\vec{AD}$.<br>Поскольку $N$ — середина ребра $BC$, радиус-вектор точки $N$ можно найти как полусумму радиус-векторов точек $B$ и $C$: $\vec{AN} = \frac{1}{2}(\vec{AB} + \vec{AC})$.</p><p>Теперь выразим вектор $\vec{MN}$ через векторы, выходящие из точки $A$:<br>$\vec{MN} = \vec{AN} - \vec{AM} = \frac{1}{2}(\vec{AB} + \vec{AC}) - \frac{1}{2}\vec{AD} = \frac{1}{2}(\vec{AB} + \vec{AC} - \vec{AD})$.</p><p>Так как тетраэдр правильный, длины векторов, соответствующих его рёбрам, равны $a$:$|\vec{AB}| = |\vec{AC}| = |\vec{AD}| = a$.Углы между рёбрами, выходящими из одной вершины, равны $60^\circ$ (углы равностороннего треугольника). Это позволяет нам вычислить скалярные произведения этих векторов:$\vec{AB} \cdot \vec{AC} = |\vec{AB}| \cdot |\vec{AC}| \cos(60^\circ) = a \cdot a \cdot \frac{1}{2} = \frac{a^2}{2}$<br>$\vec{AB} \cdot \vec{AD} = |\vec{AB}| \cdot |\vec{AD}| \cos(60^\circ) = a \cdot a \cdot \frac{1}{2} = \frac{a^2}{2}$<br>$\vec{AC} \cdot \vec{AD} = |\vec{AC}| \cdot |\vec{AD}| \cos(60^\circ) = a \cdot a \cdot \frac{1}{2} = \frac{a^2}{2}$</p><p><strong>Доказательство, что $\vec{MN} \cdot \vec{AD} = 0$</strong></p><p>Вычислим скалярное произведение $\vec{MN}$ и $\vec{AD}$:$\vec{MN} \cdot \vec{AD} = \frac{1}{2}(\vec{AB} + \vec{AC} - \vec{AD}) \cdot \vec{AD}$<br>Используя дистрибутивное свойство скалярного произведения, раскроем скобки:$\vec{MN} \cdot \vec{AD} = \frac{1}{2}(\vec{AB} \cdot \vec{AD} + \vec{AC} \cdot \vec{AD} - \vec{AD} \cdot \vec{AD})$<br>Скалярный квадрат вектора равен квадрату его длины: $\vec{AD} \cdot \vec{AD} = |\vec{AD}|^2 = a^2$.<br>Подставим ранее найденные значения скалярных произведений:$\vec{MN} \cdot \vec{AD} = \frac{1}{2}\left(\frac{a^2}{2} + \frac{a^2}{2} - a^2\right) = \frac{1}{2}(a^2 - a^2) = 0$.Так как скалярное произведение равно нулю, векторы $\vec{MN}$ и $\vec{AD}$ перпендикулярны.<br>Ответ: Равенство $\vec{MN} \cdot \vec{AD} = 0$ доказано.</p><p><strong>Доказательство, что $\vec{MN} \cdot \vec{BC} = 0$</strong></p><p>Выразим вектор $\vec{BC}$ через векторы с началом в точке $A$:$\vec{BC} = \vec{AC} - \vec{AB}$.Теперь вычислим скалярное произведение $\vec{MN}$ и $\vec{BC}$:$\vec{MN} \cdot \vec{BC} = \frac{1}{2}(\vec{AB} + \vec{AC} - \vec{AD}) \cdot (\vec{AC} - \vec{AB})$<br>Раскроем скобки:$\vec{MN} \cdot \vec{BC} = \frac{1}{2}(\vec{AB} \cdot \vec{AC} - \vec{AB} \cdot \vec{AB} + \vec{AC} \cdot \vec{AC} - \vec{AC} \cdot \vec{AB} - \vec{AD} \cdot \vec{AC} + \vec{AD} \cdot \vec{AB})$<br>Учитывая, что $\vec{AB} \cdot \vec{AC} = \vec{AC} \cdot \vec{AB}$, эти слагаемые взаимно уничтожаются.$\vec{MN} \cdot \vec{BC} = \frac{1}{2}(-\vec{AB} \cdot \vec{AB} + \vec{AC} \cdot \vec{AC} - \vec{AD} \cdot \vec{AC} + \vec{AD} \cdot \vec{AB})$<br>Подставим значения скалярных квадратов и скалярных произведений:$-\vec{AB} \cdot \vec{AB} = -|\vec{AB}|^2 = -a^2$<br>$\vec{AC} \cdot \vec{AC} = |\vec{AC}|^2 = a^2$<br>$-\vec{AD} \cdot \vec{AC} = -\frac{a^2}{2}$<br>$\vec{AD} \cdot \vec{AB} = \frac{a^2}{2}$<br>Получаем:$\vec{MN} \cdot \vec{BC} = \frac{1}{2}\left(-a^2 + a^2 - \frac{a^2}{2} + \frac{a^2}{2}\right) = \frac{1}{2}(0) = 0$.Так как скалярное произведение равно нулю, векторы $\vec{MN}$ и $\vec{BC}$ перпендикулярны.<br>Ответ: Равенство $\vec{MN} \cdot \vec{BC} = 0$ доказано.</p>" ] #original: array:6 [ "id" => 1352430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1267} "task" => array:2 [ "refs" => "745117" "type" => "task" ] "text" => "<p>По условию, все рёбра тетраэдра $ABCD$ равны друг другу. Такой тетраэдр является правильным, а все его грани — равносторонние треугольники. Пусть длина ребра тетраэдра равна $a$. Точки $M$ и $N$ являются серединами рёбер $AD$ и $BC$ соответственно. Требуется доказать, что скалярное произведение вектора $\vec{MN}$ на вектор $\vec{AD}$ равно нулю, и скалярное произведение вектора $\vec{MN}$ на вектор $\vec{BC}$ также равно нулю.</p><p>Для доказательства воспользуемся векторным методом. Выберем вершину $A$ в качестве начала отсчёта. Тогда векторы, выходящие из этой вершины, будут радиус-векторами других вершин: $\vec{AB}$, $\vec{AC}$, $\vec{AD}$.</p><p>Поскольку $M$ — середина ребра $AD$, радиус-вектор точки $M$ выражается как $\vec{AM} = \frac{1}{2}\vec{AD}$.<br>Поскольку $N$ — середина ребра $BC$, радиус-вектор точки $N$ можно найти как полусумму радиус-векторов точек $B$ и $C$: $\vec{AN} = \frac{1}{2}(\vec{AB} + \vec{AC})$.</p><p>Теперь выразим вектор $\vec{MN}$ через векторы, выходящие из точки $A$:<br>$\vec{MN} = \vec{AN} - \vec{AM} = \frac{1}{2}(\vec{AB} + \vec{AC}) - \frac{1}{2}\vec{AD} = \frac{1}{2}(\vec{AB} + \vec{AC} - \vec{AD})$.</p><p>Так как тетраэдр правильный, длины векторов, соответствующих его рёбрам, равны $a$:$|\vec{AB}| = |\vec{AC}| = |\vec{AD}| = a$.Углы между рёбрами, выходящими из одной вершины, равны $60^\circ$ (углы равностороннего треугольника). Это позволяет нам вычислить скалярные произведения этих векторов:$\vec{AB} \cdot \vec{AC} = |\vec{AB}| \cdot |\vec{AC}| \cos(60^\circ) = a \cdot a \cdot \frac{1}{2} = \frac{a^2}{2}$<br>$\vec{AB} \cdot \vec{AD} = |\vec{AB}| \cdot |\vec{AD}| \cos(60^\circ) = a \cdot a \cdot \frac{1}{2} = \frac{a^2}{2}$<br>$\vec{AC} \cdot \vec{AD} = |\vec{AC}| \cdot |\vec{AD}| \cos(60^\circ) = a \cdot a \cdot \frac{1}{2} = \frac{a^2}{2}$</p><p><strong>Доказательство, что $\vec{MN} \cdot \vec{AD} = 0$</strong></p><p>Вычислим скалярное произведение $\vec{MN}$ и $\vec{AD}$:$\vec{MN} \cdot \vec{AD} = \frac{1}{2}(\vec{AB} + \vec{AC} - \vec{AD}) \cdot \vec{AD}$<br>Используя дистрибутивное свойство скалярного произведения, раскроем скобки:$\vec{MN} \cdot \vec{AD} = \frac{1}{2}(\vec{AB} \cdot \vec{AD} + \vec{AC} \cdot \vec{AD} - \vec{AD} \cdot \vec{AD})$<br>Скалярный квадрат вектора равен квадрату его длины: $\vec{AD} \cdot \vec{AD} = |\vec{AD}|^2 = a^2$.<br>Подставим ранее найденные значения скалярных произведений:$\vec{MN} \cdot \vec{AD} = \frac{1}{2}\left(\frac{a^2}{2} + \frac{a^2}{2} - a^2\right) = \frac{1}{2}(a^2 - a^2) = 0$.Так как скалярное произведение равно нулю, векторы $\vec{MN}$ и $\vec{AD}$ перпендикулярны.<br>Ответ: Равенство $\vec{MN} \cdot \vec{AD} = 0$ доказано.</p><p><strong>Доказательство, что $\vec{MN} \cdot \vec{BC} = 0$</strong></p><p>Выразим вектор $\vec{BC}$ через векторы с началом в точке $A$:$\vec{BC} = \vec{AC} - \vec{AB}$.Теперь вычислим скалярное произведение $\vec{MN}$ и $\vec{BC}$:$\vec{MN} \cdot \vec{BC} = \frac{1}{2}(\vec{AB} + \vec{AC} - \vec{AD}) \cdot (\vec{AC} - \vec{AB})$<br>Раскроем скобки:$\vec{MN} \cdot \vec{BC} = \frac{1}{2}(\vec{AB} \cdot \vec{AC} - \vec{AB} \cdot \vec{AB} + \vec{AC} \cdot \vec{AC} - \vec{AC} \cdot \vec{AB} - \vec{AD} \cdot \vec{AC} + \vec{AD} \cdot \vec{AB})$<br>Учитывая, что $\vec{AB} \cdot \vec{AC} = \vec{AC} \cdot \vec{AB}$, эти слагаемые взаимно уничтожаются.$\vec{MN} \cdot \vec{BC} = \frac{1}{2}(-\vec{AB} \cdot \vec{AB} + \vec{AC} \cdot \vec{AC} - \vec{AD} \cdot \vec{AC} + \vec{AD} \cdot \vec{AB})$<br>Подставим значения скалярных квадратов и скалярных произведений:$-\vec{AB} \cdot \vec{AB} = -|\vec{AB}|^2 = -a^2$<br>$\vec{AC} \cdot \vec{AC} = |\vec{AC}|^2 = a^2$<br>$-\vec{AD} \cdot \vec{AC} = -\frac{a^2}{2}$<br>$\vec{AD} \cdot \vec{AB} = \frac{a^2}{2}$<br>Получаем:$\vec{MN} \cdot \vec{BC} = \frac{1}{2}\left(-a^2 + a^2 - \frac{a^2}{2} + \frac{a^2}{2}\right) = \frac{1}{2}(0) = 0$.Так как скалярное произведение равно нулю, векторы $\vec{MN}$ и $\vec{BC}$ перпендикулярны.<br>Ответ: Равенство $\vec{MN} \cdot \vec{BC} = 0$ доказано.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "745118" "type" => "task" ] "previous" => array:2 [ "refs" => "745116" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Book {#1216 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:1 [ 0 => App\Models\Term {#1196 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:2 [ 0 => App\Models\Term {#1203 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1202 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:1 [ 0 => App\Models\Term {#1199 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:5 [ 0 => App\Models\Term {#1188 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1189 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1190 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1195 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1194 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1193 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Term {#1192 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1180 #items: array:1 [ 0 => App\Models\Term {#1181 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1182 #items: array:1 [ 0 => App\Models\Term {#1187 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Term {#1184 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Term {#1173 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 #items: array:1 [ 0 => App\Models\Term {#1179 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => 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#casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1197} "field_class" => Illuminate\Database\Eloquent\Collection {#1198} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1201} "field_author" => Illuminate\Database\Eloquent\Collection {#1200} "field_author_foreign" => 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общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1177} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1175} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1176} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1165} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:1 [ 0 => App\Models\BookPage {#1205 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 933959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "177" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-177" "field_display_title" => "177" "field_folder" => "1" "field_image_name" => "177" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1215 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ 0 => App\Models\Book {#1216} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:1 [ 0 => App\Models\Element {#1268 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "933960" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933958" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1482 #items: array:8 [ 0 => App\Models\Task {#1496 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Task {#1533 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Task {#1506 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1527 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1513 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1630 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1638 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1675 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 933959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "177" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-177" "field_display_title" => "177" "field_folder" => "1" "field_image_name" => "177" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1215} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1214} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1212} "content" => Illuminate\Database\Eloquent\Collection {#1269} "next" => array:2 [ "refs" => "933960" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933958" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1482} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 745117 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "177" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-702" "field_display_title" => "702" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1079} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1163} "content" => Illuminate\Database\Eloquent\Collection {#1206} "next" => array:2 [ "refs" => "745118" "type" => "task" ] "previous" => array:2 [ "refs" => "745116" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1170} "page" => array:2 [ "refs" => "933959" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№702 (с. 177)
Условие. №702 (с. 177)
скриншот условия
702. Все рёбра тетраэдра ABCD равны друг другу. Точки М и N — середины рёбер AD и ВС. Докажите, что MN ⋅ AD = MN ⋅ BC = 0.
Решение 2. №702 (с. 177)
Решение 3. №702 (с. 177)
Решение 4. №702 (с. 177)
Решение 5. №702 (с. 177)
Решение 6. №702 (с. 177)
По условию, все рёбра тетраэдра $ABCD$ равны друг другу. Такой тетраэдр является правильным, а все его грани — равносторонние треугольники. Пусть длина ребра тетраэдра равна $a$. Точки $M$ и $N$ являются серединами рёбер $AD$ и $BC$ соответственно. Требуется доказать, что скалярное произведение вектора $\vec{MN}$ на вектор $\vec{AD}$ равно нулю, и скалярное произведение вектора $\vec{MN}$ на вектор $\vec{BC}$ также равно нулю.
Для доказательства воспользуемся векторным методом. Выберем вершину $A$ в качестве начала отсчёта. Тогда векторы, выходящие из этой вершины, будут радиус-векторами других вершин: $\vec{AB}$, $\vec{AC}$, $\vec{AD}$.
Поскольку $M$ — середина ребра $AD$, радиус-вектор точки $M$ выражается как $\vec{AM} = \frac{1}{2}\vec{AD}$.
Поскольку $N$ — середина ребра $BC$, радиус-вектор точки $N$ можно найти как полусумму радиус-векторов точек $B$ и $C$: $\vec{AN} = \frac{1}{2}(\vec{AB} + \vec{AC})$.
Теперь выразим вектор $\vec{MN}$ через векторы, выходящие из точки $A$:
$\vec{MN} = \vec{AN} - \vec{AM} = \frac{1}{2}(\vec{AB} + \vec{AC}) - \frac{1}{2}\vec{AD} = \frac{1}{2}(\vec{AB} + \vec{AC} - \vec{AD})$.
Так как тетраэдр правильный, длины векторов, соответствующих его рёбрам, равны $a$:$|\vec{AB}| = |\vec{AC}| = |\vec{AD}| = a$.Углы между рёбрами, выходящими из одной вершины, равны $60^\circ$ (углы равностороннего треугольника). Это позволяет нам вычислить скалярные произведения этих векторов:$\vec{AB} \cdot \vec{AC} = |\vec{AB}| \cdot |\vec{AC}| \cos(60^\circ) = a \cdot a \cdot \frac{1}{2} = \frac{a^2}{2}$
$\vec{AB} \cdot \vec{AD} = |\vec{AB}| \cdot |\vec{AD}| \cos(60^\circ) = a \cdot a \cdot \frac{1}{2} = \frac{a^2}{2}$
$\vec{AC} \cdot \vec{AD} = |\vec{AC}| \cdot |\vec{AD}| \cos(60^\circ) = a \cdot a \cdot \frac{1}{2} = \frac{a^2}{2}$
Доказательство, что $\vec{MN} \cdot \vec{AD} = 0$
Вычислим скалярное произведение $\vec{MN}$ и $\vec{AD}$:$\vec{MN} \cdot \vec{AD} = \frac{1}{2}(\vec{AB} + \vec{AC} - \vec{AD}) \cdot \vec{AD}$
Используя дистрибутивное свойство скалярного произведения, раскроем скобки:$\vec{MN} \cdot \vec{AD} = \frac{1}{2}(\vec{AB} \cdot \vec{AD} + \vec{AC} \cdot \vec{AD} - \vec{AD} \cdot \vec{AD})$
Скалярный квадрат вектора равен квадрату его длины: $\vec{AD} \cdot \vec{AD} = |\vec{AD}|^2 = a^2$.
Подставим ранее найденные значения скалярных произведений:$\vec{MN} \cdot \vec{AD} = \frac{1}{2}\left(\frac{a^2}{2} + \frac{a^2}{2} - a^2\right) = \frac{1}{2}(a^2 - a^2) = 0$.Так как скалярное произведение равно нулю, векторы $\vec{MN}$ и $\vec{AD}$ перпендикулярны.
Ответ: Равенство $\vec{MN} \cdot \vec{AD} = 0$ доказано.
Доказательство, что $\vec{MN} \cdot \vec{BC} = 0$
Выразим вектор $\vec{BC}$ через векторы с началом в точке $A$:$\vec{BC} = \vec{AC} - \vec{AB}$.Теперь вычислим скалярное произведение $\vec{MN}$ и $\vec{BC}$:$\vec{MN} \cdot \vec{BC} = \frac{1}{2}(\vec{AB} + \vec{AC} - \vec{AD}) \cdot (\vec{AC} - \vec{AB})$
Раскроем скобки:$\vec{MN} \cdot \vec{BC} = \frac{1}{2}(\vec{AB} \cdot \vec{AC} - \vec{AB} \cdot \vec{AB} + \vec{AC} \cdot \vec{AC} - \vec{AC} \cdot \vec{AB} - \vec{AD} \cdot \vec{AC} + \vec{AD} \cdot \vec{AB})$
Учитывая, что $\vec{AB} \cdot \vec{AC} = \vec{AC} \cdot \vec{AB}$, эти слагаемые взаимно уничтожаются.$\vec{MN} \cdot \vec{BC} = \frac{1}{2}(-\vec{AB} \cdot \vec{AB} + \vec{AC} \cdot \vec{AC} - \vec{AD} \cdot \vec{AC} + \vec{AD} \cdot \vec{AB})$
Подставим значения скалярных квадратов и скалярных произведений:$-\vec{AB} \cdot \vec{AB} = -|\vec{AB}|^2 = -a^2$
$\vec{AC} \cdot \vec{AC} = |\vec{AC}|^2 = a^2$
$-\vec{AD} \cdot \vec{AC} = -\frac{a^2}{2}$
$\vec{AD} \cdot \vec{AB} = \frac{a^2}{2}$
Получаем:$\vec{MN} \cdot \vec{BC} = \frac{1}{2}\left(-a^2 + a^2 - \frac{a^2}{2} + \frac{a^2}{2}\right) = \frac{1}{2}(0) = 0$.Так как скалярное произведение равно нулю, векторы $\vec{MN}$ и $\vec{BC}$ перпендикулярны.
Ответ: Равенство $\vec{MN} \cdot \vec{BC} = 0$ доказано.
Другие задания:
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ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по геометрии за 10-11 класс, для упражнения номер 702 расположенного на странице 177 к учебнику 2019 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по геометрии к упражнению №702 (с. 177), авторов: Атанасян (Левон Сергеевич), Бутузов (Валентин Фёдорович), Кадомцев (Сергей Борисович), Позняк (Эдуард Генрихович), Киселёва (Людмила Сергеевна), ФГОС (старый) базовый и углублённый уровень обучения учебного пособия издательства Просвещение.