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Номер 587, страница 150 - 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 2. Сложение и вычитание векторов. Умножение вектора на число - номер 587, страница 150.
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" => 745002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-587" "field_display_title" => "587" "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" "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 ] #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\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 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "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 => "*" ] } 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 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "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 => "*" ] } 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 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "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 => "*" ] } ] #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 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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_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 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" …2 ] #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" => 744401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Сложение и вычитание векторов. Умножение вектора на число" "field_branch_order" => "2" "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" => "145" "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" => 744401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Сложение и вычитание векторов. Умножение вектора на число" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "145" "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" => 939282 "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" => "745002" "type" => "task" ] "text" => "<p> <b>587.</b> Известно, что <span class="long"><span class="arrtoptoleft">p</span> = <span class="arrtoptoleft">a</span> + <span class="arrtoptoleft">b</span> + <span class="arrtoptoleft">c</span>,</span> причём векторы <span class="long"><span class="arrtoptoleft">a,</span> <span class="arrtoptoleft">b,</span> и <span class="arrtoptoleft">с</span></span> попарно не сонаправлены. Докажите, что <span class="long">| <span class="arrtoptoleft">p</span> | = | <span class="arrtoptoleft">a</span> | + | <span class="arrtoptoleft">b</span> | + | <span class="arrtoptoleft">c</span> |.</span> </p>" "img" => array:1 [ 0 => array:5 [ "name" => "587-1.jpg" "alt" => null "width" => "1545" "height" => 168 "path" => "/media/geometrija_10/atanasjan-u23/0-00/587-1.webp?ts=1739436755" ] ] ] #original: array:7 [ "id" => 939282 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098} "task" => array:2 [ "refs" => "745002" "type" => "task" ] "text" => "<p> <b>587.</b> Известно, что <span class="long"><span class="arrtoptoleft">p</span> = <span class="arrtoptoleft">a</span> + <span class="arrtoptoleft">b</span> + <span class="arrtoptoleft">c</span>,</span> причём векторы <span class="long"><span class="arrtoptoleft">a,</span> <span class="arrtoptoleft">b,</span> и <span class="arrtoptoleft">с</span></span> попарно не сонаправлены. Докажите, что <span class="long">| <span class="arrtoptoleft">p</span> | = | <span class="arrtoptoleft">a</span> | + | <span class="arrtoptoleft">b</span> | + | <span class="arrtoptoleft">c</span> |.</span> </p>" "img" => array:1 [ 0 => array:5 [ "name" => "587-1.jpg" "alt" => null "width" => "1545" "height" => 168 "path" => "/media/geometrija_10/atanasjan-u23/0-00/587-1.webp?ts=1739436755" ] ] ] #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" => 942257 "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" => "745002" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "587-1.png" "alt" => null "width" => "694" "height" => 3059 "path" => "/media/geometrija_10/atanasjan-u23/2-00/587-1.webp?ts=1739441455" ] ] ] #original: array:6 [ "id" => 942257 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1146} "task" => array:2 [ "refs" => "745002" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "587-1.png" "alt" => null "width" => "694" "height" => 3059 "path" => "/media/geometrija_10/atanasjan-u23/2-00/587-1.webp?ts=1739441455" ] ] ] #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" => 942194 "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" => "745002" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "587-1.jpg" "alt" => null "width" => "1305" "height" => 1128 "path" => "/media/geometrija_10/atanasjan-u23/3-00/587-1.webp?ts=1739437049" ] ] ] #original: array:6 [ "id" => 942194 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124} "task" => array:2 [ "refs" => "745002" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "587-1.jpg" "alt" => null "width" => "1305" "height" => 1128 "path" => "/media/geometrija_10/atanasjan-u23/3-00/587-1.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" => 941479 "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" => "745002" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "587-1.jpg" "alt" => null "width" => "1156" "height" => 537 "path" => "/media/geometrija_10/atanasjan-u23/4-00/587-1.webp?ts=1739436631" ] ] ] #original: array:6 [ "id" => 941479 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "745002" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "587-1.jpg" "alt" => null "width" => "1156" "height" => 537 "path" => "/media/geometrija_10/atanasjan-u23/4-00/587-1.webp?ts=1739436631" ] ] ] #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" => 941577 "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" => "745002" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "587-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/587-1.webp?ts=1739436649" ] ] ] #original: array:6 [ "id" => 941577 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1039} "task" => array:2 [ "refs" => "745002" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "587-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/587-1.webp?ts=1739436649" ] ] ] #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" => 1352315 "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" => "745002" "type" => "task" ] "text" => "<p>Данную задачу в предложенной формулировке доказать невозможно, так как утверждение, которое требуется доказать, находится в прямом противоречии с исходным условием. Проведем детальный анализ, чтобы продемонстрировать это противоречие.</p><p>Утверждение, которое необходимо доказать: $|\vec{p}| = |\vec{a}| + |\vec{b}| + |\vec{c}|$, при условии, что $\vec{p} = \vec{a} + \vec{b} + \vec{c}$.</p><p>Это равенство представляет собой крайний случай так называемого неравенства треугольника (или неравенства многоугольника), которое для трех векторов в общем виде выглядит как $|\vec{a} + \vec{b} + \vec{c}| \le |\vec{a}| + |\vec{b}| + |\vec{c}|$. Равенство в этом выражении достигается тогда и только тогда, когда векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ сонаправлены (т.е. направлены в одну и ту же сторону).</p><p>Докажем это строго. Для этого возведем обе части доказываемого равенства в квадрат. Модули векторов неотрицательны, поэтому такое преобразование является эквивалентным.</p><p>Квадрат левой части равенства:<br>$|\vec{p}|^2 = |\vec{a} + \vec{b} + \vec{c}|^2$<br>Используя свойство скалярного произведения, согласно которому квадрат модуля вектора равен скалярному квадрату этого вектора ($|\vec{x}|^2 = \vec{x} \cdot \vec{x}$), получаем:<br>$|\vec{a} + \vec{b} + \vec{c}|^2 = (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c})$<br>Раскрывая скобки, получаем:<br>$|\vec{p}|^2 = \vec{a}\cdot\vec{a} + \vec{b}\cdot\vec{b} + \vec{c}\cdot\vec{c} + 2\vec{a}\cdot\vec{b} + 2\vec{a}\cdot\vec{c} + 2\vec{b}\cdot\vec{c}$<br>Вспоминая, что скалярное произведение векторов $\vec{x}$ и $\vec{y}$ равно $|\vec{x}||\vec{y}|\cos(\theta)$, где $\theta$ — угол между ними, перепишем выражение:<br>$|\vec{p}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{a}||\vec{b}|\cos(\widehat{\vec{a},\vec{b}}) + 2|\vec{a}||\vec{c}|\cos(\widehat{\vec{a},\vec{c}}) + 2|\vec{b}||\vec{c}|\cos(\widehat{\vec{b},\vec{c}})$</p><p>Квадрат правой части равенства:<br>$(|\vec{a}| + |\vec{b}| + |\vec{c}|)^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{a}||\vec{b}| + 2|\vec{a}||\vec{c}| + 2|\vec{b}||\vec{c}|$</p><p>Теперь приравняем выражения для квадратов левой и правой частей:<br>$|\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(|\vec{a}||\vec{b}|\cos(\widehat{\vec{a},\vec{b}}) + \dots) = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(|\vec{a}||\vec{b}| + \dots)$<br>После вычитания одинаковых слагаемых из обеих частей, получим:<br>$|\vec{a}||\vec{b}|\cos(\widehat{\vec{a},\vec{b}}) + |\vec{a}||\vec{c}|\cos(\widehat{\vec{a},\vec{c}}) + |\vec{b}||\vec{c}|\cos(\widehat{\vec{b},\vec{c}}) = |\vec{a}||\vec{b}| + |\vec{a}||\vec{c}| + |\vec{b}||\vec{c}|$<br>Перенесем все члены в левую часть и сгруппируем:<br>$|\vec{a}||\vec{b}|(1 - \cos(\widehat{\vec{a},\vec{b}})) + |\vec{a}||\vec{c}|(1 - \cos(\widehat{\vec{a},\vec{c}})) + |\vec{b}||\vec{c}|(1 - \cos(\widehat{\vec{b},\vec{c}})) = 0$</p><p>Проанализируем полученное уравнение. Предположим, что векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ не являются нулевыми (в противном случае условие "не сонаправлены" становится некорректным). Тогда их модули $|\vec{a}|$, $|\vec{b}|$ и $|\vec{c}|$ — строго положительные числа.<br>Для любого угла $\theta$ значение его косинуса $\cos(\theta)$ находится в диапазоне $[-1, 1]$. Следовательно, выражение $1 - \cos(\theta)$ всегда неотрицательно ($1 - \cos(\theta) \ge 0$).<br>Таким образом, каждое из трех слагаемых в левой части уравнения является произведением неотрицательных чисел и, следовательно, само по себе неотрицательно.<br>Сумма нескольких неотрицательных слагаемых равна нулю тогда и только тогда, когда каждое из этих слагаемых равно нулю. Это приводит к системе из трех условий:<br>1. $|\vec{a}||\vec{b}|(1 - \cos(\widehat{\vec{a},\vec{b}})) = 0 \implies 1 - \cos(\widehat{\vec{a},\vec{b}}) = 0 \implies \cos(\widehat{\vec{a},\vec{b}}) = 1$. Это значит, что угол между векторами $\vec{a}$ и $\vec{b}$ равен 0, то есть они сонаправлены.<br>2. $|\vec{a}||\vec{c}|(1 - \cos(\widehat{\vec{a},\vec{c}})) = 0 \implies 1 - \cos(\widehat{\vec{a},\vec{c}}) = 0 \implies \cos(\widehat{\vec{a},\vec{c}}) = 1$. Это значит, что векторы $\vec{a}$ и $\vec{c}$ сонаправлены.<br>3. $|\vec{b}||\vec{c}|(1 - \cos(\widehat{\vec{b},\vec{c}})) = 0 \implies 1 - \cos(\widehat{\vec{b},\vec{c}}) = 0 \implies \cos(\widehat{\vec{b},\vec{c}}) = 1$. Это значит, что векторы $\vec{b}$ и $\vec{c}$ сонаправлены.</p><p>Из этого следует, что равенство $|\vec{p}| = |\vec{a}| + |\vec{b}| + |\vec{c}|$ может выполняться только в том случае, если все три вектора $\vec{a}$, $\vec{b}$ и $\vec{c}$ сонаправлены друг с другом.</p><p>Однако в условии задачи дано, что "векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ <strong>попарно не сонаправлены</strong>". Это условие прямо противоречит необходимому условию для выполнения доказываемого равенства.</p><p><strong>Вывод:</strong> Утверждение задачи неверно, так как оно требует, чтобы векторы были сонаправлены, в то время как условие задачи это запрещает. Вероятнее всего, в формулировке задачи содержится опечатка (например, частица "не" является лишней).</p><p><strong>Ответ:</strong> Доказать утверждение невозможно, так как оно неверно при заданных условиях. Равенство $|\vec{a} + \vec{b} + \vec{c}| = |\vec{a}| + |\vec{b}| + |\vec{c}|$ справедливо только тогда, когда векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ сонаправлены. Условие же задачи гласит, что они попарно не сонаправлены, что создает неразрешимое противоречие.</p>" ] #original: array:6 [ "id" => 1352315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159} "task" => array:2 [ "refs" => "745002" "type" => "task" ] "text" => "<p>Данную задачу в предложенной формулировке доказать невозможно, так как утверждение, которое требуется доказать, находится в прямом противоречии с исходным условием. Проведем детальный анализ, чтобы продемонстрировать это противоречие.</p><p>Утверждение, которое необходимо доказать: $|\vec{p}| = |\vec{a}| + |\vec{b}| + |\vec{c}|$, при условии, что $\vec{p} = \vec{a} + \vec{b} + \vec{c}$.</p><p>Это равенство представляет собой крайний случай так называемого неравенства треугольника (или неравенства многоугольника), которое для трех векторов в общем виде выглядит как $|\vec{a} + \vec{b} + \vec{c}| \le |\vec{a}| + |\vec{b}| + |\vec{c}|$. Равенство в этом выражении достигается тогда и только тогда, когда векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ сонаправлены (т.е. направлены в одну и ту же сторону).</p><p>Докажем это строго. Для этого возведем обе части доказываемого равенства в квадрат. Модули векторов неотрицательны, поэтому такое преобразование является эквивалентным.</p><p>Квадрат левой части равенства:<br>$|\vec{p}|^2 = |\vec{a} + \vec{b} + \vec{c}|^2$<br>Используя свойство скалярного произведения, согласно которому квадрат модуля вектора равен скалярному квадрату этого вектора ($|\vec{x}|^2 = \vec{x} \cdot \vec{x}$), получаем:<br>$|\vec{a} + \vec{b} + \vec{c}|^2 = (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c})$<br>Раскрывая скобки, получаем:<br>$|\vec{p}|^2 = \vec{a}\cdot\vec{a} + \vec{b}\cdot\vec{b} + \vec{c}\cdot\vec{c} + 2\vec{a}\cdot\vec{b} + 2\vec{a}\cdot\vec{c} + 2\vec{b}\cdot\vec{c}$<br>Вспоминая, что скалярное произведение векторов $\vec{x}$ и $\vec{y}$ равно $|\vec{x}||\vec{y}|\cos(\theta)$, где $\theta$ — угол между ними, перепишем выражение:<br>$|\vec{p}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{a}||\vec{b}|\cos(\widehat{\vec{a},\vec{b}}) + 2|\vec{a}||\vec{c}|\cos(\widehat{\vec{a},\vec{c}}) + 2|\vec{b}||\vec{c}|\cos(\widehat{\vec{b},\vec{c}})$</p><p>Квадрат правой части равенства:<br>$(|\vec{a}| + |\vec{b}| + |\vec{c}|)^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{a}||\vec{b}| + 2|\vec{a}||\vec{c}| + 2|\vec{b}||\vec{c}|$</p><p>Теперь приравняем выражения для квадратов левой и правой частей:<br>$|\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(|\vec{a}||\vec{b}|\cos(\widehat{\vec{a},\vec{b}}) + \dots) = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(|\vec{a}||\vec{b}| + \dots)$<br>После вычитания одинаковых слагаемых из обеих частей, получим:<br>$|\vec{a}||\vec{b}|\cos(\widehat{\vec{a},\vec{b}}) + |\vec{a}||\vec{c}|\cos(\widehat{\vec{a},\vec{c}}) + |\vec{b}||\vec{c}|\cos(\widehat{\vec{b},\vec{c}}) = |\vec{a}||\vec{b}| + |\vec{a}||\vec{c}| + |\vec{b}||\vec{c}|$<br>Перенесем все члены в левую часть и сгруппируем:<br>$|\vec{a}||\vec{b}|(1 - \cos(\widehat{\vec{a},\vec{b}})) + |\vec{a}||\vec{c}|(1 - \cos(\widehat{\vec{a},\vec{c}})) + |\vec{b}||\vec{c}|(1 - \cos(\widehat{\vec{b},\vec{c}})) = 0$</p><p>Проанализируем полученное уравнение. Предположим, что векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ не являются нулевыми (в противном случае условие "не сонаправлены" становится некорректным). Тогда их модули $|\vec{a}|$, $|\vec{b}|$ и $|\vec{c}|$ — строго положительные числа.<br>Для любого угла $\theta$ значение его косинуса $\cos(\theta)$ находится в диапазоне $[-1, 1]$. Следовательно, выражение $1 - \cos(\theta)$ всегда неотрицательно ($1 - \cos(\theta) \ge 0$).<br>Таким образом, каждое из трех слагаемых в левой части уравнения является произведением неотрицательных чисел и, следовательно, само по себе неотрицательно.<br>Сумма нескольких неотрицательных слагаемых равна нулю тогда и только тогда, когда каждое из этих слагаемых равно нулю. Это приводит к системе из трех условий:<br>1. $|\vec{a}||\vec{b}|(1 - \cos(\widehat{\vec{a},\vec{b}})) = 0 \implies 1 - \cos(\widehat{\vec{a},\vec{b}}) = 0 \implies \cos(\widehat{\vec{a},\vec{b}}) = 1$. Это значит, что угол между векторами $\vec{a}$ и $\vec{b}$ равен 0, то есть они сонаправлены.<br>2. $|\vec{a}||\vec{c}|(1 - \cos(\widehat{\vec{a},\vec{c}})) = 0 \implies 1 - \cos(\widehat{\vec{a},\vec{c}}) = 0 \implies \cos(\widehat{\vec{a},\vec{c}}) = 1$. Это значит, что векторы $\vec{a}$ и $\vec{c}$ сонаправлены.<br>3. $|\vec{b}||\vec{c}|(1 - \cos(\widehat{\vec{b},\vec{c}})) = 0 \implies 1 - \cos(\widehat{\vec{b},\vec{c}}) = 0 \implies \cos(\widehat{\vec{b},\vec{c}}) = 1$. Это значит, что векторы $\vec{b}$ и $\vec{c}$ сонаправлены.</p><p>Из этого следует, что равенство $|\vec{p}| = |\vec{a}| + |\vec{b}| + |\vec{c}|$ может выполняться только в том случае, если все три вектора $\vec{a}$, $\vec{b}$ и $\vec{c}$ сонаправлены друг с другом.</p><p>Однако в условии задачи дано, что "векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ <strong>попарно не сонаправлены</strong>". Это условие прямо противоречит необходимому условию для выполнения доказываемого равенства.</p><p><strong>Вывод:</strong> Утверждение задачи неверно, так как оно требует, чтобы векторы были сонаправлены, в то время как условие задачи это запрещает. Вероятнее всего, в формулировке задачи содержится опечатка (например, частица "не" является лишней).</p><p><strong>Ответ:</strong> Доказать утверждение невозможно, так как оно неверно при заданных условиях. Равенство $|\vec{a} + \vec{b} + \vec{c}| = |\vec{a}| + |\vec{b}| + |\vec{c}|$ справедливо только тогда, когда векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ сонаправлены. Условие же задачи гласит, что они попарно не сонаправлены, что создает неразрешимое противоречие.</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" => "745003" "type" => "task" ] "previous" => array:2 [ "refs" => "745001" "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" => 933932 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-150" "field_display_title" => "150" "field_folder" => "1" "field_image_name" => "150" "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" => "933933" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933931" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:6 [ 0 => App\Models\Task {#1280 #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 {#1287 #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 {#1291 #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 {#1310 #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 {#1313 #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 {#1312 #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" => 933932 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-150" "field_display_title" => "150" "field_folder" => "1" "field_image_name" => "150" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1042} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => 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№587 (с. 150)
Условие. №587 (с. 150)
скриншот условия
587. Известно, что p = a + b + c, причём векторы a, b, и с попарно не сонаправлены. Докажите, что | p | = | a | + | b | + | c |.
Решение 2. №587 (с. 150)
Решение 3. №587 (с. 150)
Решение 4. №587 (с. 150)
Решение 5. №587 (с. 150)
Решение 6. №587 (с. 150)
Данную задачу в предложенной формулировке доказать невозможно, так как утверждение, которое требуется доказать, находится в прямом противоречии с исходным условием. Проведем детальный анализ, чтобы продемонстрировать это противоречие.
Утверждение, которое необходимо доказать: $|\vec{p}| = |\vec{a}| + |\vec{b}| + |\vec{c}|$, при условии, что $\vec{p} = \vec{a} + \vec{b} + \vec{c}$.
Это равенство представляет собой крайний случай так называемого неравенства треугольника (или неравенства многоугольника), которое для трех векторов в общем виде выглядит как $|\vec{a} + \vec{b} + \vec{c}| \le |\vec{a}| + |\vec{b}| + |\vec{c}|$. Равенство в этом выражении достигается тогда и только тогда, когда векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ сонаправлены (т.е. направлены в одну и ту же сторону).
Докажем это строго. Для этого возведем обе части доказываемого равенства в квадрат. Модули векторов неотрицательны, поэтому такое преобразование является эквивалентным.
Квадрат левой части равенства:
$|\vec{p}|^2 = |\vec{a} + \vec{b} + \vec{c}|^2$
Используя свойство скалярного произведения, согласно которому квадрат модуля вектора равен скалярному квадрату этого вектора ($|\vec{x}|^2 = \vec{x} \cdot \vec{x}$), получаем:
$|\vec{a} + \vec{b} + \vec{c}|^2 = (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c})$
Раскрывая скобки, получаем:
$|\vec{p}|^2 = \vec{a}\cdot\vec{a} + \vec{b}\cdot\vec{b} + \vec{c}\cdot\vec{c} + 2\vec{a}\cdot\vec{b} + 2\vec{a}\cdot\vec{c} + 2\vec{b}\cdot\vec{c}$
Вспоминая, что скалярное произведение векторов $\vec{x}$ и $\vec{y}$ равно $|\vec{x}||\vec{y}|\cos(\theta)$, где $\theta$ — угол между ними, перепишем выражение:
$|\vec{p}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{a}||\vec{b}|\cos(\widehat{\vec{a},\vec{b}}) + 2|\vec{a}||\vec{c}|\cos(\widehat{\vec{a},\vec{c}}) + 2|\vec{b}||\vec{c}|\cos(\widehat{\vec{b},\vec{c}})$
Квадрат правой части равенства:
$(|\vec{a}| + |\vec{b}| + |\vec{c}|)^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{a}||\vec{b}| + 2|\vec{a}||\vec{c}| + 2|\vec{b}||\vec{c}|$
Теперь приравняем выражения для квадратов левой и правой частей:
$|\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(|\vec{a}||\vec{b}|\cos(\widehat{\vec{a},\vec{b}}) + \dots) = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(|\vec{a}||\vec{b}| + \dots)$
После вычитания одинаковых слагаемых из обеих частей, получим:
$|\vec{a}||\vec{b}|\cos(\widehat{\vec{a},\vec{b}}) + |\vec{a}||\vec{c}|\cos(\widehat{\vec{a},\vec{c}}) + |\vec{b}||\vec{c}|\cos(\widehat{\vec{b},\vec{c}}) = |\vec{a}||\vec{b}| + |\vec{a}||\vec{c}| + |\vec{b}||\vec{c}|$
Перенесем все члены в левую часть и сгруппируем:
$|\vec{a}||\vec{b}|(1 - \cos(\widehat{\vec{a},\vec{b}})) + |\vec{a}||\vec{c}|(1 - \cos(\widehat{\vec{a},\vec{c}})) + |\vec{b}||\vec{c}|(1 - \cos(\widehat{\vec{b},\vec{c}})) = 0$
Проанализируем полученное уравнение. Предположим, что векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ не являются нулевыми (в противном случае условие "не сонаправлены" становится некорректным). Тогда их модули $|\vec{a}|$, $|\vec{b}|$ и $|\vec{c}|$ — строго положительные числа.
Для любого угла $\theta$ значение его косинуса $\cos(\theta)$ находится в диапазоне $[-1, 1]$. Следовательно, выражение $1 - \cos(\theta)$ всегда неотрицательно ($1 - \cos(\theta) \ge 0$).
Таким образом, каждое из трех слагаемых в левой части уравнения является произведением неотрицательных чисел и, следовательно, само по себе неотрицательно.
Сумма нескольких неотрицательных слагаемых равна нулю тогда и только тогда, когда каждое из этих слагаемых равно нулю. Это приводит к системе из трех условий:
1. $|\vec{a}||\vec{b}|(1 - \cos(\widehat{\vec{a},\vec{b}})) = 0 \implies 1 - \cos(\widehat{\vec{a},\vec{b}}) = 0 \implies \cos(\widehat{\vec{a},\vec{b}}) = 1$. Это значит, что угол между векторами $\vec{a}$ и $\vec{b}$ равен 0, то есть они сонаправлены.
2. $|\vec{a}||\vec{c}|(1 - \cos(\widehat{\vec{a},\vec{c}})) = 0 \implies 1 - \cos(\widehat{\vec{a},\vec{c}}) = 0 \implies \cos(\widehat{\vec{a},\vec{c}}) = 1$. Это значит, что векторы $\vec{a}$ и $\vec{c}$ сонаправлены.
3. $|\vec{b}||\vec{c}|(1 - \cos(\widehat{\vec{b},\vec{c}})) = 0 \implies 1 - \cos(\widehat{\vec{b},\vec{c}}) = 0 \implies \cos(\widehat{\vec{b},\vec{c}}) = 1$. Это значит, что векторы $\vec{b}$ и $\vec{c}$ сонаправлены.
Из этого следует, что равенство $|\vec{p}| = |\vec{a}| + |\vec{b}| + |\vec{c}|$ может выполняться только в том случае, если все три вектора $\vec{a}$, $\vec{b}$ и $\vec{c}$ сонаправлены друг с другом.
Однако в условии задачи дано, что "векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ попарно не сонаправлены". Это условие прямо противоречит необходимому условию для выполнения доказываемого равенства.
Вывод: Утверждение задачи неверно, так как оно требует, чтобы векторы были сонаправлены, в то время как условие задачи это запрещает. Вероятнее всего, в формулировке задачи содержится опечатка (например, частица "не" является лишней).
Ответ: Доказать утверждение невозможно, так как оно неверно при заданных условиях. Равенство $|\vec{a} + \vec{b} + \vec{c}| = |\vec{a}| + |\vec{b}| + |\vec{c}|$ справедливо только тогда, когда векторы $\vec{a}$, $\vec{b}$ и $\vec{c}$ сонаправлены. Условие же задачи гласит, что они попарно не сонаправлены, что создает неразрешимое противоречие.
Другие задания:
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