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Номер 605, страница 155 - 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 6. Векторы в пространстве. Параграф 3. Компланарные векторы - номер 605, страница 155.
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" => 745020 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "155" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-605" "field_display_title" => "605" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #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" => "142" "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 {#1086 #items: array:1 [ 0 => App\Models\Book {#1049 #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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #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 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #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_class" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:2 [ 0 => App\Models\Term {#1053 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #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\Term {#1054 #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" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #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_publisher" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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_author" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:5 [ 0 => App\Models\Term {#1058 #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 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #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\Term {#1059 #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 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" …8 ] #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 {#1060 #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 {#1061 #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 {#1062 #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 {#1063 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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 {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #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 {#1068 #items: array:1 [ 0 => App\Models\Term {#1069 #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 {#1070 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1073 #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 {#1074 #items: array:1 [ 0 => App\Models\Term {#1075 #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 {#1076 #items: array:1 [ 0 => App\Models\Term {#1077 #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 {#1078 #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 => 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 {#1079 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1083 #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_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 {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055} "field_author" => Illuminate\Database\Eloquent\Collection {#1057} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_country" => Illuminate\Database\Eloquent\Collection {#1066} "field_city" => Illuminate\Database\Eloquent\Collection {#1068} "field_series" => Illuminate\Database\Eloquent\Collection {#1070} "field_umk" => Illuminate\Database\Eloquent\Collection {#1071} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078} "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 => 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 {#1079} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082} "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 } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "142" "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 {#1086} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084} ] #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 {#1121 #items: array:2 [ 0 => App\Models\Branch {#1129 #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" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "142" "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 {#1131 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "142" "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 {#1131} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #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 {#1133 #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" => 744402 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Компланарные векторы" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1134 #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" => "150" "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 {#1136 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744402 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Компланарные векторы" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "150" "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 {#1136} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135} ] #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 {#1038 #items: array:6 [ 0 => App\Models\Element {#1109 #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" => 939315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Edition {#1106 #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 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_content_source" => 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 } "task" => array:2 [ "refs" => "745020" "type" => "task" ] "text" => """ <p><strong>605.</strong> Точки <em>М</em> и <em>N</em> являются серединами рёбер <em>AB</em> и <em>А₁D₁</em> параллелепипеда <em>ABCDA₁B₁C₁D₁.</em> Разложите, если это возможно, по векторам <span class="arrtoptoleft">AB</span> и <span class="arrtoptoleft">AD</span> вектор:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/605.webp?ts=1745747723" alt="Разложить, если это возможно, по векторам" loading="lazy" width="917" height="191">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "605-1.jpg" "alt" => null "width" => "1051" "height" => 303 "path" => "/media/geometrija_10/atanasjan-u23/0-00/605-1.webp?ts=1739436772" ] ] ] #original: array:7 [ "id" => 939315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098} "task" => array:2 [ "refs" => "745020" "type" => "task" ] "text" => """ <p><strong>605.</strong> Точки <em>М</em> и <em>N</em> являются серединами рёбер <em>AB</em> и <em>А₁D₁</em> параллелепипеда <em>ABCDA₁B₁C₁D₁.</em> Разложите, если это возможно, по векторам <span class="arrtoptoleft">AB</span> и <span class="arrtoptoleft">AD</span> вектор:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/605.webp?ts=1745747723" alt="Разложить, если это возможно, по векторам" loading="lazy" width="917" height="191">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "605-1.jpg" "alt" => null "width" => "1051" "height" => 303 "path" => "/media/geometrija_10/atanasjan-u23/0-00/605-1.webp?ts=1739436772" ] ] ] #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 {#1093 #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" => 942294 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Edition {#1087 #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 [ "id" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_content_source" => 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 } "task" => array:2 [ "refs" => "745020" "type" => "task" ] "img" => array:7 [ 0 => array:5 [ "name" => "605-1.png" "alt" => null "width" => "694" "height" => 1448 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-1.webp?ts=1739441486" ] 1 => array:5 [ "name" => "605-2.png" "alt" => null "width" => "694" "height" => 1936 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-2.webp?ts=1739441486" ] 2 => array:5 [ "name" => "605-3.png" "alt" => null "width" => "694" "height" => 1936 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-3.webp?ts=1739441486" ] 3 => array:5 [ "name" => "605-4.png" "alt" => null "width" => "694" "height" => 1908 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-4.webp?ts=1739441486" ] 4 => array:5 [ "name" => "605-5.png" "alt" => null "width" => "694" "height" => 1561 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-5.webp?ts=1739441486" ] 5 => array:5 [ "name" => "605-6.png" "alt" => null "width" => "694" "height" => 1908 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-6.webp?ts=1739441486" ] 6 => array:5 [ "name" => "605-7.png" "alt" => null "width" => "694" "height" => 1742 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-7.webp?ts=1739441486" ] ] ] #original: array:6 [ "id" => 942294 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1146} "task" => array:2 [ "refs" => "745020" "type" => "task" ] "img" => array:7 [ 0 => array:5 [ "name" => "605-1.png" "alt" => null "width" => "694" "height" => 1448 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-1.webp?ts=1739441486" ] 1 => array:5 [ "name" => "605-2.png" "alt" => null "width" => "694" "height" => 1936 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-2.webp?ts=1739441486" ] 2 => array:5 [ "name" => "605-3.png" "alt" => null "width" => "694" "height" => 1936 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-3.webp?ts=1739441486" ] 3 => array:5 [ "name" => "605-4.png" "alt" => null "width" => "694" "height" => 1908 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-4.webp?ts=1739441486" ] 4 => array:5 [ "name" => "605-5.png" "alt" => null "width" => "694" "height" => 1561 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-5.webp?ts=1739441486" ] 5 => array:5 [ "name" => "605-6.png" "alt" => null "width" => "694" "height" => 1908 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-6.webp?ts=1739441486" ] 6 => array:5 [ "name" => "605-7.png" "alt" => null "width" => "694" "height" => 1742 "path" => "/media/geometrija_10/atanasjan-u23/2-00/605-7.webp?ts=1739441486" ] ] ] #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 {#1089 #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" => 942218 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Edition {#1122 #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 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_content_source" => 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 } "task" => array:2 [ "refs" => "745020" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "605-1.jpg" "alt" => null "width" => "1373" "height" => 1246 "path" => "/media/geometrija_10/atanasjan-u23/3-00/605-1.webp?ts=1739437049" ] 1 => array:5 [ "name" => "605-2.jpg" "alt" => null "width" => "1373" "height" => 1372 "path" => "/media/geometrija_10/atanasjan-u23/3-00/605-2.webp?ts=1739437049" ] ] ] #original: array:6 [ "id" => 942218 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124} "task" => array:2 [ "refs" => "745020" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "605-1.jpg" "alt" => null "width" => "1373" "height" => 1246 "path" => "/media/geometrija_10/atanasjan-u23/3-00/605-1.webp?ts=1739437049" ] 1 => array:5 [ "name" => "605-2.jpg" "alt" => null "width" => "1373" "height" => 1372 "path" => "/media/geometrija_10/atanasjan-u23/3-00/605-2.webp?ts=1739437049" ] ] ] #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 {#1100 #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" => 941591 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 [ "id" => 2319 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2319 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_content_source" => 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 } "task" => array:2 [ "refs" => "745020" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "605-1.jpg" "alt" => null "width" => "1127" "height" => 1026 "path" => "/media/geometrija_10/atanasjan-u23/4-00/605-1.webp?ts=1739436651" ] ] ] #original: array:6 [ "id" => 941591 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "745020" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "605-1.jpg" "alt" => null "width" => "1127" "height" => 1026 "path" => "/media/geometrija_10/atanasjan-u23/4-00/605-1.webp?ts=1739436651" ] ] ] #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 {#1094 #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" => 941706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Edition {#1034 #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 [ "id" => 2320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_source" => 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 } "task" => array:2 [ "refs" => "745020" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "605-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/605-1.webp?ts=1739436668" ] ] ] #original: array:6 [ "id" => 941706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1039} "task" => array:2 [ "refs" => "745020" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "605-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/605-1.webp?ts=1739436668" ] ] ] #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 {#1142 #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" => 1352333 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Edition {#1147 #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 [ "id" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_content_source" => 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 } "task" => array:2 [ "refs" => "745020" "type" => "task" ] "text" => "<p>Для решения задачи введем базисные векторы, соответствующие ребрам параллелепипеда, выходящим из вершины A: $\vec{AB}$, $\vec{AD}$ и $\vec{AA_1}$. Разложение по векторам $\vec{AB}$ и $\vec{AD}$ возможно только для тех векторов, которые компланарны плоскости основания ABCD, то есть для векторов, которые можно представить в виде $k\vec{AB} + l\vec{AD}$ (их составляющая по вектору $\vec{AA_1}$ равна нулю).</p><p>По условию задачи, точка M — середина ребра AB, следовательно, $\vec{AM} = \frac{1}{2}\vec{AB}$. Точка N — середина ребра A?D?, следовательно, $\vec{A_1N} = \frac{1}{2}\vec{A_1D_1}$. Так как в параллелепипеде $\vec{A_1D_1} = \vec{AD}$, то $\vec{A_1N} = \frac{1}{2}\vec{AD}$.</p><p>Рассмотрим каждый вектор по отдельности.</p><p><strong>а) $\vec{AC}$</strong></p><p>Вектор $\vec{AC}$ является диагональю основания ABCD. По правилу параллелограмма для сложения векторов: $\vec{AC} = \vec{AB} + \vec{BC}$. В параллелепипеде $\vec{BC} = \vec{AD}$. Таким образом, $\vec{AC} = \vec{AB} + \vec{AD}$. Данный вектор лежит в плоскости основания, поэтому разложение возможно.</p><p>Ответ: $\vec{AC} = \vec{AB} + \vec{AD}$.</p><p><strong>б) $\vec{CM}$</strong></p><p>Выразим вектор $\vec{CM}$ через векторы с общим началом в точке A: $\vec{CM} = \vec{AM} - \vec{AC}$. Из пункта (а) мы знаем, что $\vec{AC} = \vec{AB} + \vec{AD}$, а по условию $\vec{AM} = \frac{1}{2}\vec{AB}$. Подставим эти выражения: $\vec{CM} = \frac{1}{2}\vec{AB} - (\vec{AB} + \vec{AD}) = \frac{1}{2}\vec{AB} - \vec{AB} - \vec{AD} = -\frac{1}{2}\vec{AB} - \vec{AD}$. Точки C и M лежат в плоскости основания ABCD, значит и вектор $\vec{CM}$ лежит в этой плоскости. Разложение возможно.</p><p>Ответ: $\vec{CM} = -\frac{1}{2}\vec{AB} - \vec{AD}$.</p><p><strong>в) $\vec{C_1N}$</strong></p><p>Выразим вектор $\vec{C_1N}$ через векторы, выходящие из вершины $A_1$, которая находится в одной плоскости с точками $C_1$ и $N$. $\vec{C_1N} = \vec{A_1N} - \vec{A_1C_1}$. По условию, $N$ — середина $A_1D_1$, поэтому $\vec{A_1N} = \frac{1}{2}\vec{A_1D_1}$. Вектор $\vec{A_1C_1}$ является диагональю грани $A_1B_1C_1D_1$, поэтому $\vec{A_1C_1} = \vec{A_1B_1} + \vec{A_1D_1}$. Тогда $\vec{C_1N} = \frac{1}{2}\vec{A_1D_1} - (\vec{A_1B_1} + \vec{A_1D_1}) = -\vec{A_1B_1} - \frac{1}{2}\vec{A_1D_1}$. Поскольку в параллелепипеде $\vec{A_1B_1} = \vec{AB}$ и $\vec{A_1D_1} = \vec{AD}$, получаем: $\vec{C_1N} = -\vec{AB} - \frac{1}{2}\vec{AD}$. Вектор лежит в плоскости $A_1B_1C_1D_1$, параллельной плоскости $ABCD$, поэтому разложение возможно.</p><p>Ответ: $\vec{C_1N} = -\vec{AB} - \frac{1}{2}\vec{AD}$.</p><p><strong>г) $\vec{AC_1}$</strong></p><p>Вектор $\vec{AC_1}$ является пространственной диагональю параллелепипеда. По правилу параллелепипеда: $\vec{AC_1} = \vec{AB} + \vec{AD} + \vec{AA_1}$. Этот вектор не компланарен векторам $\vec{AB}$ и $\vec{AD}$, так как имеет ненулевую составляющую $\vec{AA_1}$, которая не лежит в плоскости основания ABCD. Следовательно, разложить вектор $\vec{AC_1}$ только по векторам $\vec{AB}$ и $\vec{AD}$ невозможно.</p><p>Ответ: Разложение невозможно.</p><p><strong>д) $\vec{A_1N}$</strong></p><p>Точка N — середина ребра $A_1D_1$. Следовательно, вектор $\vec{A_1N}$ можно выразить как $\vec{A_1N} = \frac{1}{2}\vec{A_1D_1}$. В параллелепипеде противоположные ребра параллельны и равны, поэтому $\vec{A_1D_1} = \vec{AD}$. Таким образом, $\vec{A_1N} = \frac{1}{2}\vec{AD}$. Это и есть искомое разложение (коэффициент при $\vec{AB}$ равен 0). Вектор $\vec{A_1N}$ параллелен вектору $\vec{AD}$ и, следовательно, его можно разложить по базису $\vec{AB}$ и $\vec{AD}$.</p><p>Ответ: $\vec{A_1N} = \frac{1}{2}\vec{AD}$.</p><p><strong>е) $\vec{AN}$</strong></p><p>Выразим вектор $\vec{AN}$, двигаясь из точки A в точку N через вершину $A_1$. $\vec{AN} = \vec{AA_1} + \vec{A_1N}$. Из предыдущего пункта мы знаем, что $\vec{A_1N} = \frac{1}{2}\vec{AD}$. Тогда $\vec{AN} = \vec{AA_1} + \frac{1}{2}\vec{AD}$. Этот вектор имеет ненулевую составляющую $\vec{AA_1}$, которая не компланарна векторам $\vec{AB}$ и $\vec{AD}$. Следовательно, разложить вектор $\vec{AN}$ по векторам $\vec{AB}$ и $\vec{AD}$ невозможно.</p><p>Ответ: Разложение невозможно.</p><p><strong>ж) $\vec{MD}$</strong></p><p>Выразим вектор $\vec{MD}$ через векторы с общим началом в точке A: $\vec{MD} = \vec{AD} - \vec{AM}$. По условию, M — середина ребра AB, поэтому $\vec{AM} = \frac{1}{2}\vec{AB}$. Подставляя, получаем: $\vec{MD} = \vec{AD} - \frac{1}{2}\vec{AB}$. Точки M и D лежат в плоскости основания ABCD, значит и вектор $\vec{MD}$ лежит в этой плоскости. Разложение возможно.</p><p>Ответ: $\vec{MD} = -\frac{1}{2}\vec{AB} + \vec{AD}$.</p>" ] #original: array:6 [ "id" => 1352333 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159} "task" => array:2 [ "refs" => "745020" "type" => "task" ] "text" => "<p>Для решения задачи введем базисные векторы, соответствующие ребрам параллелепипеда, выходящим из вершины A: $\vec{AB}$, $\vec{AD}$ и $\vec{AA_1}$. Разложение по векторам $\vec{AB}$ и $\vec{AD}$ возможно только для тех векторов, которые компланарны плоскости основания ABCD, то есть для векторов, которые можно представить в виде $k\vec{AB} + l\vec{AD}$ (их составляющая по вектору $\vec{AA_1}$ равна нулю).</p><p>По условию задачи, точка M — середина ребра AB, следовательно, $\vec{AM} = \frac{1}{2}\vec{AB}$. Точка N — середина ребра A?D?, следовательно, $\vec{A_1N} = \frac{1}{2}\vec{A_1D_1}$. Так как в параллелепипеде $\vec{A_1D_1} = \vec{AD}$, то $\vec{A_1N} = \frac{1}{2}\vec{AD}$.</p><p>Рассмотрим каждый вектор по отдельности.</p><p><strong>а) $\vec{AC}$</strong></p><p>Вектор $\vec{AC}$ является диагональю основания ABCD. По правилу параллелограмма для сложения векторов: $\vec{AC} = \vec{AB} + \vec{BC}$. В параллелепипеде $\vec{BC} = \vec{AD}$. Таким образом, $\vec{AC} = \vec{AB} + \vec{AD}$. Данный вектор лежит в плоскости основания, поэтому разложение возможно.</p><p>Ответ: $\vec{AC} = \vec{AB} + \vec{AD}$.</p><p><strong>б) $\vec{CM}$</strong></p><p>Выразим вектор $\vec{CM}$ через векторы с общим началом в точке A: $\vec{CM} = \vec{AM} - \vec{AC}$. Из пункта (а) мы знаем, что $\vec{AC} = \vec{AB} + \vec{AD}$, а по условию $\vec{AM} = \frac{1}{2}\vec{AB}$. Подставим эти выражения: $\vec{CM} = \frac{1}{2}\vec{AB} - (\vec{AB} + \vec{AD}) = \frac{1}{2}\vec{AB} - \vec{AB} - \vec{AD} = -\frac{1}{2}\vec{AB} - \vec{AD}$. Точки C и M лежат в плоскости основания ABCD, значит и вектор $\vec{CM}$ лежит в этой плоскости. Разложение возможно.</p><p>Ответ: $\vec{CM} = -\frac{1}{2}\vec{AB} - \vec{AD}$.</p><p><strong>в) $\vec{C_1N}$</strong></p><p>Выразим вектор $\vec{C_1N}$ через векторы, выходящие из вершины $A_1$, которая находится в одной плоскости с точками $C_1$ и $N$. $\vec{C_1N} = \vec{A_1N} - \vec{A_1C_1}$. По условию, $N$ — середина $A_1D_1$, поэтому $\vec{A_1N} = \frac{1}{2}\vec{A_1D_1}$. Вектор $\vec{A_1C_1}$ является диагональю грани $A_1B_1C_1D_1$, поэтому $\vec{A_1C_1} = \vec{A_1B_1} + \vec{A_1D_1}$. Тогда $\vec{C_1N} = \frac{1}{2}\vec{A_1D_1} - (\vec{A_1B_1} + \vec{A_1D_1}) = -\vec{A_1B_1} - \frac{1}{2}\vec{A_1D_1}$. Поскольку в параллелепипеде $\vec{A_1B_1} = \vec{AB}$ и $\vec{A_1D_1} = \vec{AD}$, получаем: $\vec{C_1N} = -\vec{AB} - \frac{1}{2}\vec{AD}$. Вектор лежит в плоскости $A_1B_1C_1D_1$, параллельной плоскости $ABCD$, поэтому разложение возможно.</p><p>Ответ: $\vec{C_1N} = -\vec{AB} - \frac{1}{2}\vec{AD}$.</p><p><strong>г) $\vec{AC_1}$</strong></p><p>Вектор $\vec{AC_1}$ является пространственной диагональю параллелепипеда. По правилу параллелепипеда: $\vec{AC_1} = \vec{AB} + \vec{AD} + \vec{AA_1}$. Этот вектор не компланарен векторам $\vec{AB}$ и $\vec{AD}$, так как имеет ненулевую составляющую $\vec{AA_1}$, которая не лежит в плоскости основания ABCD. Следовательно, разложить вектор $\vec{AC_1}$ только по векторам $\vec{AB}$ и $\vec{AD}$ невозможно.</p><p>Ответ: Разложение невозможно.</p><p><strong>д) $\vec{A_1N}$</strong></p><p>Точка N — середина ребра $A_1D_1$. Следовательно, вектор $\vec{A_1N}$ можно выразить как $\vec{A_1N} = \frac{1}{2}\vec{A_1D_1}$. В параллелепипеде противоположные ребра параллельны и равны, поэтому $\vec{A_1D_1} = \vec{AD}$. Таким образом, $\vec{A_1N} = \frac{1}{2}\vec{AD}$. Это и есть искомое разложение (коэффициент при $\vec{AB}$ равен 0). Вектор $\vec{A_1N}$ параллелен вектору $\vec{AD}$ и, следовательно, его можно разложить по базису $\vec{AB}$ и $\vec{AD}$.</p><p>Ответ: $\vec{A_1N} = \frac{1}{2}\vec{AD}$.</p><p><strong>е) $\vec{AN}$</strong></p><p>Выразим вектор $\vec{AN}$, двигаясь из точки A в точку N через вершину $A_1$. $\vec{AN} = \vec{AA_1} + \vec{A_1N}$. Из предыдущего пункта мы знаем, что $\vec{A_1N} = \frac{1}{2}\vec{AD}$. Тогда $\vec{AN} = \vec{AA_1} + \frac{1}{2}\vec{AD}$. Этот вектор имеет ненулевую составляющую $\vec{AA_1}$, которая не компланарна векторам $\vec{AB}$ и $\vec{AD}$. Следовательно, разложить вектор $\vec{AN}$ по векторам $\vec{AB}$ и $\vec{AD}$ невозможно.</p><p>Ответ: Разложение невозможно.</p><p><strong>ж) $\vec{MD}$</strong></p><p>Выразим вектор $\vec{MD}$ через векторы с общим началом в точке A: $\vec{MD} = \vec{AD} - \vec{AM}$. По условию, M — середина ребра AB, поэтому $\vec{AM} = \frac{1}{2}\vec{AB}$. Подставляя, получаем: $\vec{MD} = \vec{AD} - \frac{1}{2}\vec{AB}$. Точки M и D лежат в плоскости основания ABCD, значит и вектор $\vec{MD}$ лежит в этой плоскости. Разложение возможно.</p><p>Ответ: $\vec{MD} = -\frac{1}{2}\vec{AB} + \vec{AD}$.</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" => "745021" "type" => "task" ] "previous" => array:2 [ "refs" => "745019" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\BookPage {#1115 #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" => 933937 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "155" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-155" "field_display_title" => "155" "field_folder" => "1" "field_image_name" => "155" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\Book {#1049} ] #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 {#1117 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1166 #items: array:1 [ 0 => App\Models\Element {#1165 #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" => "933938" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933936" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1267 #items: array:7 [ 0 => App\Models\Task {#1281 #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 {#1288 #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 {#1292 #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 {#1307 #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 {#1331 #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 {#1338 #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 {#1332 #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" => 933937 "created_at" => 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№605 (с. 155)
Условие. №605 (с. 155)
скриншот условия
605. Точки М и N являются серединами рёбер AB и А₁D₁ параллелепипеда ABCDA₁B₁C₁D₁. Разложите, если это возможно, по векторам AB и AD вектор:
Решение 2. №605 (с. 155)
Решение 3. №605 (с. 155)
Решение 4. №605 (с. 155)
Решение 5. №605 (с. 155)
Решение 6. №605 (с. 155)
Для решения задачи введем базисные векторы, соответствующие ребрам параллелепипеда, выходящим из вершины A: $\vec{AB}$, $\vec{AD}$ и $\vec{AA_1}$. Разложение по векторам $\vec{AB}$ и $\vec{AD}$ возможно только для тех векторов, которые компланарны плоскости основания ABCD, то есть для векторов, которые можно представить в виде $k\vec{AB} + l\vec{AD}$ (их составляющая по вектору $\vec{AA_1}$ равна нулю).
По условию задачи, точка M — середина ребра AB, следовательно, $\vec{AM} = \frac{1}{2}\vec{AB}$. Точка N — середина ребра A?D?, следовательно, $\vec{A_1N} = \frac{1}{2}\vec{A_1D_1}$. Так как в параллелепипеде $\vec{A_1D_1} = \vec{AD}$, то $\vec{A_1N} = \frac{1}{2}\vec{AD}$.
Рассмотрим каждый вектор по отдельности.
а) $\vec{AC}$
Вектор $\vec{AC}$ является диагональю основания ABCD. По правилу параллелограмма для сложения векторов: $\vec{AC} = \vec{AB} + \vec{BC}$. В параллелепипеде $\vec{BC} = \vec{AD}$. Таким образом, $\vec{AC} = \vec{AB} + \vec{AD}$. Данный вектор лежит в плоскости основания, поэтому разложение возможно.
Ответ: $\vec{AC} = \vec{AB} + \vec{AD}$.
б) $\vec{CM}$
Выразим вектор $\vec{CM}$ через векторы с общим началом в точке A: $\vec{CM} = \vec{AM} - \vec{AC}$. Из пункта (а) мы знаем, что $\vec{AC} = \vec{AB} + \vec{AD}$, а по условию $\vec{AM} = \frac{1}{2}\vec{AB}$. Подставим эти выражения: $\vec{CM} = \frac{1}{2}\vec{AB} - (\vec{AB} + \vec{AD}) = \frac{1}{2}\vec{AB} - \vec{AB} - \vec{AD} = -\frac{1}{2}\vec{AB} - \vec{AD}$. Точки C и M лежат в плоскости основания ABCD, значит и вектор $\vec{CM}$ лежит в этой плоскости. Разложение возможно.
Ответ: $\vec{CM} = -\frac{1}{2}\vec{AB} - \vec{AD}$.
в) $\vec{C_1N}$
Выразим вектор $\vec{C_1N}$ через векторы, выходящие из вершины $A_1$, которая находится в одной плоскости с точками $C_1$ и $N$. $\vec{C_1N} = \vec{A_1N} - \vec{A_1C_1}$. По условию, $N$ — середина $A_1D_1$, поэтому $\vec{A_1N} = \frac{1}{2}\vec{A_1D_1}$. Вектор $\vec{A_1C_1}$ является диагональю грани $A_1B_1C_1D_1$, поэтому $\vec{A_1C_1} = \vec{A_1B_1} + \vec{A_1D_1}$. Тогда $\vec{C_1N} = \frac{1}{2}\vec{A_1D_1} - (\vec{A_1B_1} + \vec{A_1D_1}) = -\vec{A_1B_1} - \frac{1}{2}\vec{A_1D_1}$. Поскольку в параллелепипеде $\vec{A_1B_1} = \vec{AB}$ и $\vec{A_1D_1} = \vec{AD}$, получаем: $\vec{C_1N} = -\vec{AB} - \frac{1}{2}\vec{AD}$. Вектор лежит в плоскости $A_1B_1C_1D_1$, параллельной плоскости $ABCD$, поэтому разложение возможно.
Ответ: $\vec{C_1N} = -\vec{AB} - \frac{1}{2}\vec{AD}$.
г) $\vec{AC_1}$
Вектор $\vec{AC_1}$ является пространственной диагональю параллелепипеда. По правилу параллелепипеда: $\vec{AC_1} = \vec{AB} + \vec{AD} + \vec{AA_1}$. Этот вектор не компланарен векторам $\vec{AB}$ и $\vec{AD}$, так как имеет ненулевую составляющую $\vec{AA_1}$, которая не лежит в плоскости основания ABCD. Следовательно, разложить вектор $\vec{AC_1}$ только по векторам $\vec{AB}$ и $\vec{AD}$ невозможно.
Ответ: Разложение невозможно.
д) $\vec{A_1N}$
Точка N — середина ребра $A_1D_1$. Следовательно, вектор $\vec{A_1N}$ можно выразить как $\vec{A_1N} = \frac{1}{2}\vec{A_1D_1}$. В параллелепипеде противоположные ребра параллельны и равны, поэтому $\vec{A_1D_1} = \vec{AD}$. Таким образом, $\vec{A_1N} = \frac{1}{2}\vec{AD}$. Это и есть искомое разложение (коэффициент при $\vec{AB}$ равен 0). Вектор $\vec{A_1N}$ параллелен вектору $\vec{AD}$ и, следовательно, его можно разложить по базису $\vec{AB}$ и $\vec{AD}$.
Ответ: $\vec{A_1N} = \frac{1}{2}\vec{AD}$.
е) $\vec{AN}$
Выразим вектор $\vec{AN}$, двигаясь из точки A в точку N через вершину $A_1$. $\vec{AN} = \vec{AA_1} + \vec{A_1N}$. Из предыдущего пункта мы знаем, что $\vec{A_1N} = \frac{1}{2}\vec{AD}$. Тогда $\vec{AN} = \vec{AA_1} + \frac{1}{2}\vec{AD}$. Этот вектор имеет ненулевую составляющую $\vec{AA_1}$, которая не компланарна векторам $\vec{AB}$ и $\vec{AD}$. Следовательно, разложить вектор $\vec{AN}$ по векторам $\vec{AB}$ и $\vec{AD}$ невозможно.
Ответ: Разложение невозможно.
ж) $\vec{MD}$
Выразим вектор $\vec{MD}$ через векторы с общим началом в точке A: $\vec{MD} = \vec{AD} - \vec{AM}$. По условию, M — середина ребра AB, поэтому $\vec{AM} = \frac{1}{2}\vec{AB}$. Подставляя, получаем: $\vec{MD} = \vec{AD} - \frac{1}{2}\vec{AB}$. Точки M и D лежат в плоскости основания ABCD, значит и вектор $\vec{MD}$ лежит в этой плоскости. Разложение возможно.
Ответ: $\vec{MD} = -\frac{1}{2}\vec{AB} + \vec{AD}$.
Другие задания:
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