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Номер 706, страница 178 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Глава 7. Метод координат в пространстве. Движения. Параграф 2. Скалярное произведение векторов - номер 706, страница 178.
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" => 745121 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "178" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-706" "field_display_title" => "706" "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" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#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" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#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" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #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_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 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #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_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 [ "id" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy i uglublennyy" ] #original: array:6 [ "id" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy i uglublennyy" ] #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_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 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => 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_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 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] 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"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 [ "id" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ …6] "field_translit" => "korichnevyy" ] #original: array:6 [ "id" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ …6] "field_translit" => "korichnevyy" ] #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 "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" => 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=> "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" => 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array:24 [ "id" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. 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Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#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" => 744405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Метод координат в пространстве. Движения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "160" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#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" => 744407 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Скалярное произведение векторов" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#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" => "171" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#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" => 744407 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Скалярное произведение векторов" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "171" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#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" => 939593 "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" => "745121" "type" => "task" ] "text" => """ <p><strong>706.</strong> Дана правильная треугольная призма <em>ABCA₁B₁C₁,</em> в которой <span class="long">AA₁ = <span class="squareroot">2</span>AB</span> (рис. 193, <em>а</em>). Найдите угол между прямыми АС₁ и А₁В.</p><p><strong>Решение</strong></p><p>Пусть <span class="long"><em>AB = а,</em></span> тогда <span class="long">AA₁ = a<span class="squareroot">2</span>.</span> Введём прямоугольную систему координат так, как показано на рисунке 193, б. Вершины А, В, A₁, C₁ имеют следующие координаты (объясните почему):</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/706.webp?ts=1745782172" alt="Найти координаты векторов" loading="lazy" width="1401" height="293">\n </figure>\n </figure><p>Отсюда находим координаты векторов <span class="arrtoptoleft">AC₁</span> и <span class="arrtoptoleft">BA₁:</span></p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/706_0.webp?ts=1745782172" alt="Найти координаты векторов" loading="lazy" width="928" height="200">\n </figure>\n </figure><p>Векторы <span class="arrtoptoleft">AC₁</span> и <span class="arrtoptoleft">BA₁</span> являются направляющими векторами прямых АС₁ и A₁B. Искомый угол φ между ними можно найти с помощью формулы (2):</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/706_1.webp?ts=1745782172" alt="Найти координаты векторов" loading="lazy" width="1264" height="300">\n </figure>\n </figure><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/706_2.webp?ts=1745782172" alt="Найти координаты векторов" loading="lazy" width="1190" height="630">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:2 [ 0 => array:5 [ "name" => "706-1.jpg" "alt" => null "width" => "1562" "height" => 1112 "path" => "/media/geometrija_10/atanasjan-u23/0-00/706-1.webp?ts=1739436846" ] 1 => array:5 [ "name" => "706-2.jpg" "alt" => null "width" => "1562" "height" => 669 "path" => "/media/geometrija_10/atanasjan-u23/0-00/706-2.webp?ts=1739436846" ] ] ] #original: array:7 [ "id" => 939593 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098} "task" => array:2 [ "refs" => "745121" "type" => "task" ] "text" => """ <p><strong>706.</strong> Дана правильная треугольная призма <em>ABCA₁B₁C₁,</em> в которой <span class="long">AA₁ = <span class="squareroot">2</span>AB</span> (рис. 193, <em>а</em>). Найдите угол между прямыми АС₁ и А₁В.</p><p><strong>Решение</strong></p><p>Пусть <span class="long"><em>AB = а,</em></span> тогда <span class="long">AA₁ = a<span class="squareroot">2</span>.</span> Введём прямоугольную систему координат так, как показано на рисунке 193, б. Вершины А, В, A₁, C₁ имеют следующие координаты (объясните почему):</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/706.webp?ts=1745782172" alt="Найти координаты векторов" loading="lazy" width="1401" height="293">\n </figure>\n </figure><p>Отсюда находим координаты векторов <span class="arrtoptoleft">AC₁</span> и <span class="arrtoptoleft">BA₁:</span></p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/706_0.webp?ts=1745782172" alt="Найти координаты векторов" loading="lazy" width="928" height="200">\n </figure>\n </figure><p>Векторы <span class="arrtoptoleft">AC₁</span> и <span class="arrtoptoleft">BA₁</span> являются направляющими векторами прямых АС₁ и A₁B. Искомый угол φ между ними можно найти с помощью формулы (2):</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/706_1.webp?ts=1745782172" alt="Найти координаты векторов" loading="lazy" width="1264" height="300">\n </figure>\n </figure><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/706_2.webp?ts=1745782172" alt="Найти координаты векторов" loading="lazy" width="1190" height="630">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:2 [ 0 => array:5 [ "name" => "706-1.jpg" "alt" => null "width" => "1562" "height" => 1112 "path" => "/media/geometrija_10/atanasjan-u23/0-00/706-1.webp?ts=1739436846" ] 1 => array:5 [ "name" => "706-2.jpg" "alt" => null "width" => "1562" "height" => 669 "path" => "/media/geometrija_10/atanasjan-u23/0-00/706-2.webp?ts=1739436846" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1122 #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" => 942568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1145 #items: array:1 [ 0 => App\Models\Edition {#1125 #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 {#1139 …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 {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …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 {#1139 …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 {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …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" => "745121" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "706-1.png" "alt" => null "width" => "694" "height" => 3277 "path" => "/media/geometrija_10/atanasjan-u23/2-00/706-1.webp?ts=1739441614" ] ] ] #original: array:6 [ "id" => 942568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1145} "task" => array:2 [ "refs" => "745121" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "706-1.png" "alt" => null "width" => "694" "height" => 3277 "path" => "/media/geometrija_10/atanasjan-u23/2-00/706-1.webp?ts=1739441614" ] ] ] #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 {#1043 #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" => 942390 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Edition {#1041 #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 {#1120 …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 {#1119 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1124 …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 {#1120 …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 {#1119 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1124 …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" => "745121" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "706-1.jpg" "alt" => null "width" => "1305" "height" => 110 "path" => "/media/geometrija_10/atanasjan-u23/3-00/706-1.webp?ts=1739437149" ] ] ] #original: array:6 [ "id" => 942390 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123} "task" => array:2 [ "refs" => "745121" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "706-1.jpg" "alt" => null "width" => "1305" "height" => 110 "path" => "/media/geometrija_10/atanasjan-u23/3-00/706-1.webp?ts=1739437149" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Element {#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" => 941954 "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" => "745121" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "706-1.jpg" "alt" => null "width" => "1423" "height" => 121 "path" => "/media/geometrija_10/atanasjan-u23/4-00/706-1.webp?ts=1739436713" ] ] ] #original: array:6 [ "id" => 941954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "745121" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "706-1.jpg" "alt" => null "width" => "1423" "height" => 121 "path" => "/media/geometrija_10/atanasjan-u23/4-00/706-1.webp?ts=1739436713" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Element {#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" => 941980 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034 #items: array:1 [ 0 => App\Models\Edition {#1093 #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 {#1090 …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 {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …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 {#1090 …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 {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …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" => "745121" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "706-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/706-1.webp?ts=1739436728" ] ] ] #original: array:6 [ "id" => 941980 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034} "task" => array:2 [ "refs" => "745121" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "706-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/706-1.webp?ts=1739436728" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1143 #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" => 1352434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Edition {#1144 #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 {#1147 …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 {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …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 {#1147 …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 {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …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" => "745121" "type" => "task" ] "text" => "<p><strong>Решение</strong></p><p>Пусть сторона основания правильной треугольной призмы $AB = a$. По условию задачи, высота призмы $AA_1 = \sqrt{2}AB$, следовательно, $AA_1 = a\sqrt{2}$.</p><p>Для нахождения угла между скрещивающимися прямыми $AC_1$ и $A_1B$ введем прямоугольную систему координат, как показано на рисунке 193, б. Объясним, почему вершины имеют указанные в задаче координаты.</p><ol><li>Поместим начало координат в вершину $C$. Таким образом, $C$ имеет координаты $(0; 0; 0)$.</li> <li>Направим ось $Oz$ вдоль бокового ребра $CC_1$. Так как высота призмы равна $a\sqrt{2}$, то вершина $C_1$ будет иметь координаты $(0; 0; a\sqrt{2})$.</li> <li>Расположим основание $ABC$ в плоскости $Oxy$. Так как $ABC$ – правильный треугольник со стороной $a$, то $|AB| = |BC| = |CA| = a$.</li> <li>Пусть ось $Oy$ проходит через сторону $CB$. Тогда вершина $B$ будет иметь координаты $(0; a; 0)$, поскольку $|CB|=a$.</li> <li>Найдем координаты вершины $A(x_A, y_A, 0)$. Расстояние от $A$ до $C$ равно $a$, и расстояние от $A$ до $B$ равно $a$.<p>Из $|CA|=a$ следует: $\sqrt{(x_A-0)^2 + (y_A-0)^2 + (0-0)^2} = a \implies x_A^2 + y_A^2 = a^2$.</p><p>Из $|AB|=a$ следует: $\sqrt{(x_A-0)^2 + (y_A-a)^2 + (0-0)^2} = a \implies x_A^2 + (y_A-a)^2 = a^2 \implies x_A^2 + y_A^2 - 2ay_A + a^2 = a^2$.</p><p>Подставляя $x_A^2 + y_A^2 = a^2$ во второе уравнение, получаем: $a^2 - 2ay_A + a^2 = a^2 \implies a^2 - 2ay_A = 0$. Отсюда $y_A = \frac{a}{2}$.</p><p>Теперь найдем $x_A$: $x_A^2 + (\frac{a}{2})^2 = a^2 \implies x_A^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4}$. Поскольку по рисунку вершина $A$ находится в полуплоскости с $x>0$, то $x_A = \frac{a\sqrt{3}}{2}$.</p><p>Таким образом, координаты вершины $A$ равны $(\frac{a\sqrt{3}}{2}; \frac{a}{2}; 0)$.</p></li> <li>Вершина $A_1$ получается сдвигом вершины $A$ на вектор $\vec{CC_1} = \{0; 0; a\sqrt{2}\}$. Следовательно, координаты $A_1$ равны $(\frac{a\sqrt{3}}{2}; \frac{a}{2}; a\sqrt{2})$.</li></ol><p>Теперь у нас есть все необходимые координаты для нахождения векторов:</p><p>$A(\frac{a\sqrt{3}}{2}; \frac{a}{2}; 0)$, $B(0; a; 0)$, $C_1(0; 0; a\sqrt{2})$, $A_1(\frac{a\sqrt{3}}{2}; \frac{a}{2}; a\sqrt{2})$.</p><p>Найдем координаты направляющих векторов для прямых $AC_1$ и $A_1B$. В качестве направляющего вектора для прямой $AC_1$ возьмем вектор $\vec{AC_1}$. В качестве направляющего вектора для прямой $A_1B$ возьмем вектор $\vec{BA_1}$.</p><p>$\vec{AC_1} = \{x_{C_1}-x_A; y_{C_1}-y_A; z_{C_1}-z_A\} = \{0 - \frac{a\sqrt{3}}{2}; 0 - \frac{a}{2}; a\sqrt{2} - 0\} = \{-\frac{a\sqrt{3}}{2}; -\frac{a}{2}; a\sqrt{2}\}$</p><p>$\vec{BA_1} = \{x_{A_1}-x_B; y_{A_1}-y_B; z_{A_1}-z_B\} = \{\frac{a\sqrt{3}}{2} - 0; \frac{a}{2} - a; a\sqrt{2} - 0\} = \{\frac{a\sqrt{3}}{2}; -\frac{a}{2}; a\sqrt{2}\}$</p><p>Угол $\phi$ между прямыми $AC_1$ и $A_1B$ равен углу между их направляющими векторами. Косинус этого угла можно найти по формуле:</p><p>$\cos \phi = \frac{|\vec{AC_1} \cdot \vec{BA_1}|}{|\vec{AC_1}| \cdot |\vec{BA_1}|}$</p><p>Найдем скалярное произведение векторов:</p><p>$\vec{AC_1} \cdot \vec{BA_1} = (-\frac{a\sqrt{3}}{2}) \cdot (\frac{a\sqrt{3}}{2}) + (-\frac{a}{2}) \cdot (-\frac{a}{2}) + (a\sqrt{2}) \cdot (a\sqrt{2}) = -\frac{3a^2}{4} + \frac{a^2}{4} + 2a^2 = -\frac{2a^2}{4} + 2a^2 = -\frac{a^2}{2} + 2a^2 = \frac{3a^2}{2}$.</p><p>Найдем длины (модули) векторов:</p><p>$|\vec{AC_1}| = \sqrt{(-\frac{a\sqrt{3}}{2})^2 + (-\frac{a}{2})^2 + (a\sqrt{2})^2} = \sqrt{\frac{3a^2}{4} + \frac{a^2}{4} + 2a^2} = \sqrt{a^2 + 2a^2} = \sqrt{3a^2} = a\sqrt{3}$.</p><p>$|\vec{BA_1}| = \sqrt{(\frac{a\sqrt{3}}{2})^2 + (-\frac{a}{2})^2 + (a\sqrt{2})^2} = \sqrt{\frac{3a^2}{4} + \frac{a^2}{4} + 2a^2} = \sqrt{a^2 + 2a^2} = \sqrt{3a^2} = a\sqrt{3}$.</p><p>Теперь подставим найденные значения в формулу для косинуса угла:</p><p>$\cos \phi = \frac{|\frac{3a^2}{2}|}{a\sqrt{3} \cdot a\sqrt{3}} = \frac{\frac{3a^2}{2}}{3a^2} = \frac{1}{2}$.</p><p>Отсюда находим угол $\phi$ (угол между прямыми берется в диапазоне от $0^\circ$ до $90^\circ$):</p><p>$\phi = \arccos(\frac{1}{2}) = 60^\circ$.</p><p>Ответ: $60^\circ$.</p>" ] #original: array:6 [ "id" => 1352434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1153} "task" => array:2 [ "refs" => "745121" "type" => "task" ] "text" => "<p><strong>Решение</strong></p><p>Пусть сторона основания правильной треугольной призмы $AB = a$. По условию задачи, высота призмы $AA_1 = \sqrt{2}AB$, следовательно, $AA_1 = a\sqrt{2}$.</p><p>Для нахождения угла между скрещивающимися прямыми $AC_1$ и $A_1B$ введем прямоугольную систему координат, как показано на рисунке 193, б. Объясним, почему вершины имеют указанные в задаче координаты.</p><ol><li>Поместим начало координат в вершину $C$. Таким образом, $C$ имеет координаты $(0; 0; 0)$.</li> <li>Направим ось $Oz$ вдоль бокового ребра $CC_1$. Так как высота призмы равна $a\sqrt{2}$, то вершина $C_1$ будет иметь координаты $(0; 0; a\sqrt{2})$.</li> <li>Расположим основание $ABC$ в плоскости $Oxy$. Так как $ABC$ – правильный треугольник со стороной $a$, то $|AB| = |BC| = |CA| = a$.</li> <li>Пусть ось $Oy$ проходит через сторону $CB$. Тогда вершина $B$ будет иметь координаты $(0; a; 0)$, поскольку $|CB|=a$.</li> <li>Найдем координаты вершины $A(x_A, y_A, 0)$. Расстояние от $A$ до $C$ равно $a$, и расстояние от $A$ до $B$ равно $a$.<p>Из $|CA|=a$ следует: $\sqrt{(x_A-0)^2 + (y_A-0)^2 + (0-0)^2} = a \implies x_A^2 + y_A^2 = a^2$.</p><p>Из $|AB|=a$ следует: $\sqrt{(x_A-0)^2 + (y_A-a)^2 + (0-0)^2} = a \implies x_A^2 + (y_A-a)^2 = a^2 \implies x_A^2 + y_A^2 - 2ay_A + a^2 = a^2$.</p><p>Подставляя $x_A^2 + y_A^2 = a^2$ во второе уравнение, получаем: $a^2 - 2ay_A + a^2 = a^2 \implies a^2 - 2ay_A = 0$. Отсюда $y_A = \frac{a}{2}$.</p><p>Теперь найдем $x_A$: $x_A^2 + (\frac{a}{2})^2 = a^2 \implies x_A^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4}$. Поскольку по рисунку вершина $A$ находится в полуплоскости с $x>0$, то $x_A = \frac{a\sqrt{3}}{2}$.</p><p>Таким образом, координаты вершины $A$ равны $(\frac{a\sqrt{3}}{2}; \frac{a}{2}; 0)$.</p></li> <li>Вершина $A_1$ получается сдвигом вершины $A$ на вектор $\vec{CC_1} = \{0; 0; a\sqrt{2}\}$. Следовательно, координаты $A_1$ равны $(\frac{a\sqrt{3}}{2}; \frac{a}{2}; a\sqrt{2})$.</li></ol><p>Теперь у нас есть все необходимые координаты для нахождения векторов:</p><p>$A(\frac{a\sqrt{3}}{2}; \frac{a}{2}; 0)$, $B(0; a; 0)$, $C_1(0; 0; a\sqrt{2})$, $A_1(\frac{a\sqrt{3}}{2}; \frac{a}{2}; a\sqrt{2})$.</p><p>Найдем координаты направляющих векторов для прямых $AC_1$ и $A_1B$. В качестве направляющего вектора для прямой $AC_1$ возьмем вектор $\vec{AC_1}$. В качестве направляющего вектора для прямой $A_1B$ возьмем вектор $\vec{BA_1}$.</p><p>$\vec{AC_1} = \{x_{C_1}-x_A; y_{C_1}-y_A; z_{C_1}-z_A\} = \{0 - \frac{a\sqrt{3}}{2}; 0 - \frac{a}{2}; a\sqrt{2} - 0\} = \{-\frac{a\sqrt{3}}{2}; -\frac{a}{2}; a\sqrt{2}\}$</p><p>$\vec{BA_1} = \{x_{A_1}-x_B; y_{A_1}-y_B; z_{A_1}-z_B\} = \{\frac{a\sqrt{3}}{2} - 0; \frac{a}{2} - a; a\sqrt{2} - 0\} = \{\frac{a\sqrt{3}}{2}; -\frac{a}{2}; a\sqrt{2}\}$</p><p>Угол $\phi$ между прямыми $AC_1$ и $A_1B$ равен углу между их направляющими векторами. Косинус этого угла можно найти по формуле:</p><p>$\cos \phi = \frac{|\vec{AC_1} \cdot \vec{BA_1}|}{|\vec{AC_1}| \cdot |\vec{BA_1}|}$</p><p>Найдем скалярное произведение векторов:</p><p>$\vec{AC_1} \cdot \vec{BA_1} = (-\frac{a\sqrt{3}}{2}) \cdot (\frac{a\sqrt{3}}{2}) + (-\frac{a}{2}) \cdot (-\frac{a}{2}) + (a\sqrt{2}) \cdot (a\sqrt{2}) = -\frac{3a^2}{4} + \frac{a^2}{4} + 2a^2 = -\frac{2a^2}{4} + 2a^2 = -\frac{a^2}{2} + 2a^2 = \frac{3a^2}{2}$.</p><p>Найдем длины (модули) векторов:</p><p>$|\vec{AC_1}| = \sqrt{(-\frac{a\sqrt{3}}{2})^2 + (-\frac{a}{2})^2 + (a\sqrt{2})^2} = \sqrt{\frac{3a^2}{4} + \frac{a^2}{4} + 2a^2} = \sqrt{a^2 + 2a^2} = \sqrt{3a^2} = a\sqrt{3}$.</p><p>$|\vec{BA_1}| = \sqrt{(\frac{a\sqrt{3}}{2})^2 + (-\frac{a}{2})^2 + (a\sqrt{2})^2} = \sqrt{\frac{3a^2}{4} + \frac{a^2}{4} + 2a^2} = \sqrt{a^2 + 2a^2} = \sqrt{3a^2} = a\sqrt{3}$.</p><p>Теперь подставим найденные значения в формулу для косинуса угла:</p><p>$\cos \phi = \frac{|\frac{3a^2}{2}|}{a\sqrt{3} \cdot a\sqrt{3}} = \frac{\frac{3a^2}{2}}{3a^2} = \frac{1}{2}$.</p><p>Отсюда находим угол $\phi$ (угол между прямыми берется в диапазоне от $0^\circ$ до $90^\circ$):</p><p>$\phi = \arccos(\frac{1}{2}) = 60^\circ$.</p><p>Ответ: $60^\circ$.</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" => "745122" "type" => "task" ] "previous" => array:2 [ "refs" => "745120" "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" 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"content" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Element {#1159 #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" => 1108224 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1161 …2} "book_page" => array:2 [ …2] …1 ] #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" => "933961" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933959" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1261 #items: array:3 [ 0 => App\Models\Task {#1271 #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 {#1278 #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 {#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] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 933960 "created_at" => "2026-04-10 13:58:26" "updated_at" => null 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№706 (с. 178)
Условие. №706 (с. 178)
скриншот условия
706. Дана правильная треугольная призма ABCA₁B₁C₁, в которой AA₁ = 2AB (рис. 193, а). Найдите угол между прямыми АС₁ и А₁В.
Решение
Пусть AB = а, тогда AA₁ = a2. Введём прямоугольную систему координат так, как показано на рисунке 193, б. Вершины А, В, A₁, C₁ имеют следующие координаты (объясните почему):
Отсюда находим координаты векторов AC₁ и BA₁:
Векторы AC₁ и BA₁ являются направляющими векторами прямых АС₁ и A₁B. Искомый угол φ между ними можно найти с помощью формулы (2):
Решение 2. №706 (с. 178)
Решение 3. №706 (с. 178)
Решение 4. №706 (с. 178)
Решение 5. №706 (с. 178)
Решение 6. №706 (с. 178)
Решение
Пусть сторона основания правильной треугольной призмы $AB = a$. По условию задачи, высота призмы $AA_1 = \sqrt{2}AB$, следовательно, $AA_1 = a\sqrt{2}$.
Для нахождения угла между скрещивающимися прямыми $AC_1$ и $A_1B$ введем прямоугольную систему координат, как показано на рисунке 193, б. Объясним, почему вершины имеют указанные в задаче координаты.
- Поместим начало координат в вершину $C$. Таким образом, $C$ имеет координаты $(0; 0; 0)$.
- Направим ось $Oz$ вдоль бокового ребра $CC_1$. Так как высота призмы равна $a\sqrt{2}$, то вершина $C_1$ будет иметь координаты $(0; 0; a\sqrt{2})$.
- Расположим основание $ABC$ в плоскости $Oxy$. Так как $ABC$ – правильный треугольник со стороной $a$, то $|AB| = |BC| = |CA| = a$.
- Пусть ось $Oy$ проходит через сторону $CB$. Тогда вершина $B$ будет иметь координаты $(0; a; 0)$, поскольку $|CB|=a$.
- Найдем координаты вершины $A(x_A, y_A, 0)$. Расстояние от $A$ до $C$ равно $a$, и расстояние от $A$ до $B$ равно $a$.
Из $|CA|=a$ следует: $\sqrt{(x_A-0)^2 + (y_A-0)^2 + (0-0)^2} = a \implies x_A^2 + y_A^2 = a^2$.
Из $|AB|=a$ следует: $\sqrt{(x_A-0)^2 + (y_A-a)^2 + (0-0)^2} = a \implies x_A^2 + (y_A-a)^2 = a^2 \implies x_A^2 + y_A^2 - 2ay_A + a^2 = a^2$.
Подставляя $x_A^2 + y_A^2 = a^2$ во второе уравнение, получаем: $a^2 - 2ay_A + a^2 = a^2 \implies a^2 - 2ay_A = 0$. Отсюда $y_A = \frac{a}{2}$.
Теперь найдем $x_A$: $x_A^2 + (\frac{a}{2})^2 = a^2 \implies x_A^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4}$. Поскольку по рисунку вершина $A$ находится в полуплоскости с $x>0$, то $x_A = \frac{a\sqrt{3}}{2}$.
Таким образом, координаты вершины $A$ равны $(\frac{a\sqrt{3}}{2}; \frac{a}{2}; 0)$.
- Вершина $A_1$ получается сдвигом вершины $A$ на вектор $\vec{CC_1} = \{0; 0; a\sqrt{2}\}$. Следовательно, координаты $A_1$ равны $(\frac{a\sqrt{3}}{2}; \frac{a}{2}; a\sqrt{2})$.
Теперь у нас есть все необходимые координаты для нахождения векторов:
$A(\frac{a\sqrt{3}}{2}; \frac{a}{2}; 0)$, $B(0; a; 0)$, $C_1(0; 0; a\sqrt{2})$, $A_1(\frac{a\sqrt{3}}{2}; \frac{a}{2}; a\sqrt{2})$.
Найдем координаты направляющих векторов для прямых $AC_1$ и $A_1B$. В качестве направляющего вектора для прямой $AC_1$ возьмем вектор $\vec{AC_1}$. В качестве направляющего вектора для прямой $A_1B$ возьмем вектор $\vec{BA_1}$.
$\vec{AC_1} = \{x_{C_1}-x_A; y_{C_1}-y_A; z_{C_1}-z_A\} = \{0 - \frac{a\sqrt{3}}{2}; 0 - \frac{a}{2}; a\sqrt{2} - 0\} = \{-\frac{a\sqrt{3}}{2}; -\frac{a}{2}; a\sqrt{2}\}$
$\vec{BA_1} = \{x_{A_1}-x_B; y_{A_1}-y_B; z_{A_1}-z_B\} = \{\frac{a\sqrt{3}}{2} - 0; \frac{a}{2} - a; a\sqrt{2} - 0\} = \{\frac{a\sqrt{3}}{2}; -\frac{a}{2}; a\sqrt{2}\}$
Угол $\phi$ между прямыми $AC_1$ и $A_1B$ равен углу между их направляющими векторами. Косинус этого угла можно найти по формуле:
$\cos \phi = \frac{|\vec{AC_1} \cdot \vec{BA_1}|}{|\vec{AC_1}| \cdot |\vec{BA_1}|}$
Найдем скалярное произведение векторов:
$\vec{AC_1} \cdot \vec{BA_1} = (-\frac{a\sqrt{3}}{2}) \cdot (\frac{a\sqrt{3}}{2}) + (-\frac{a}{2}) \cdot (-\frac{a}{2}) + (a\sqrt{2}) \cdot (a\sqrt{2}) = -\frac{3a^2}{4} + \frac{a^2}{4} + 2a^2 = -\frac{2a^2}{4} + 2a^2 = -\frac{a^2}{2} + 2a^2 = \frac{3a^2}{2}$.
Найдем длины (модули) векторов:
$|\vec{AC_1}| = \sqrt{(-\frac{a\sqrt{3}}{2})^2 + (-\frac{a}{2})^2 + (a\sqrt{2})^2} = \sqrt{\frac{3a^2}{4} + \frac{a^2}{4} + 2a^2} = \sqrt{a^2 + 2a^2} = \sqrt{3a^2} = a\sqrt{3}$.
$|\vec{BA_1}| = \sqrt{(\frac{a\sqrt{3}}{2})^2 + (-\frac{a}{2})^2 + (a\sqrt{2})^2} = \sqrt{\frac{3a^2}{4} + \frac{a^2}{4} + 2a^2} = \sqrt{a^2 + 2a^2} = \sqrt{3a^2} = a\sqrt{3}$.
Теперь подставим найденные значения в формулу для косинуса угла:
$\cos \phi = \frac{|\frac{3a^2}{2}|}{a\sqrt{3} \cdot a\sqrt{3}} = \frac{\frac{3a^2}{2}}{3a^2} = \frac{1}{2}$.
Отсюда находим угол $\phi$ (угол между прямыми берется в диапазоне от $0^\circ$ до $90^\circ$):
$\phi = \arccos(\frac{1}{2}) = 60^\circ$.
Ответ: $60^\circ$.
Другие задания:
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