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Номер 684, страница 176 - 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 684, страница 176.
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" => 745099 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "176" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-684" "field_display_title" => "684" "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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1062 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Term {#1069 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1073 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Term {#1075 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Term {#1077 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1083 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055} "field_author" => Illuminate\Database\Eloquent\Collection {#1057} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_country" => Illuminate\Database\Eloquent\Collection {#1066} "field_city" => Illuminate\Database\Eloquent\Collection {#1068} "field_series" => Illuminate\Database\Eloquent\Collection {#1070} "field_umk" => Illuminate\Database\Eloquent\Collection {#1071} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 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} "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} "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" => 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 #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" => 939534 "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" => "745099" "type" => "task" ] "text" => """ <p><strong>684.</strong> Ребро куба <em>ABCDA₁B₁C₁D₁</em> равно <em>а</em>, точка <em>O₁</em> — центр грани <em>A₁B₁C₁D₁</em>. Вычислите скалярное произведение векторов:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/684.webp?ts=1745779210" alt="Вычислить скалярное произведение векторов" loading="lazy" width="1403" height="265">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "684-1.jpg" "alt" => null "width" => "1562" "height" => 294 "path" => "/media/geometrija_10/atanasjan-u23/0-00/684-1.webp?ts=1739436828" ] ] ] #original: array:7 [ "id" => 939534 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098} "task" => array:2 [ "refs" => "745099" "type" => "task" ] "text" => """ <p><strong>684.</strong> Ребро куба <em>ABCDA₁B₁C₁D₁</em> равно <em>а</em>, точка <em>O₁</em> — центр грани <em>A₁B₁C₁D₁</em>. Вычислите скалярное произведение векторов:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/684.webp?ts=1745779210" alt="Вычислить скалярное произведение векторов" loading="lazy" width="1403" height="265">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "684-1.jpg" "alt" => null "width" => "1562" "height" => 294 "path" => "/media/geometrija_10/atanasjan-u23/0-00/684-1.webp?ts=1739436828" ] ] ] #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" => 942508 "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" => "745099" "type" => "task" ] "img" => array:7 [ 0 => array:5 [ "name" => "684-1.png" "alt" => null "width" => "694" "height" => 2215 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-1.webp?ts=1739441580" ] 1 => array:5 [ "name" => "684-2.png" "alt" => null "width" => "694" "height" => 2405 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-2.webp?ts=1739441580" ] 2 => array:5 [ "name" => "684-3.png" "alt" => null "width" => "694" "height" => 2297 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-3.webp?ts=1739441580" ] 3 => array:5 [ "name" => "684-4.png" "alt" => null "width" => "694" "height" => 2488 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-4.webp?ts=1739441580" ] 4 => array:5 [ "name" => "684-5.png" "alt" => null "width" => "694" "height" => 2468 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-5.webp?ts=1739441580" ] 5 => array:5 [ "name" => "684-6.png" "alt" => null "width" => "694" "height" => 2471 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-6.webp?ts=1739441580" ] 6 => array:5 [ "name" => "684-7.png" "alt" => null "width" => "694" "height" => 2911 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-7.webp?ts=1739441580" ] ] ] #original: array:6 [ "id" => 942508 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1146} "task" => array:2 [ "refs" => "745099" "type" => "task" ] "img" => array:7 [ 0 => array:5 [ "name" => "684-1.png" "alt" => null "width" => "694" "height" => 2215 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-1.webp?ts=1739441580" ] 1 => array:5 [ "name" => "684-2.png" "alt" => null "width" => "694" "height" => 2405 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-2.webp?ts=1739441580" ] 2 => array:5 [ "name" => "684-3.png" "alt" => null "width" => "694" "height" => 2297 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-3.webp?ts=1739441580" ] 3 => array:5 [ "name" => "684-4.png" "alt" => null "width" => "694" "height" => 2488 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-4.webp?ts=1739441580" ] 4 => array:5 [ "name" => "684-5.png" "alt" => null "width" => "694" "height" => 2468 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-5.webp?ts=1739441580" ] 5 => array:5 [ "name" => "684-6.png" "alt" => null "width" => "694" "height" => 2471 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-6.webp?ts=1739441580" ] 6 => array:5 [ "name" => "684-7.png" "alt" => null "width" => "694" "height" => 2911 "path" => "/media/geometrija_10/atanasjan-u23/2-00/684-7.webp?ts=1739441580" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#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" => 942355 "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" …8 ] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745099" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "684-1.jpg" "alt" => null "width" => "1320" "height" => 2032 "path" => "/media/geometrija_10/atanasjan-u23/3-00/684-1.webp?ts=1739437149" ] 1 => array:5 [ "name" => "684-2.jpg" "alt" => null "width" => "1320" "height" => 396 "path" => "/media/geometrija_10/atanasjan-u23/3-00/684-2.webp?ts=1739437149" ] ] ] #original: array:6 [ "id" => 942355 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124} "task" => array:2 [ "refs" => "745099" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "684-1.jpg" "alt" => null "width" => "1320" "height" => 2032 "path" => "/media/geometrija_10/atanasjan-u23/3-00/684-1.webp?ts=1739437149" ] 1 => array:5 [ "name" => "684-2.jpg" "alt" => null "width" => "1320" "height" => 396 "path" => "/media/geometrija_10/atanasjan-u23/3-00/684-2.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" => 941800 "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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745099" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "684-1.jpg" "alt" => null "width" => "1404" "height" => 664 "path" => "/media/geometrija_10/atanasjan-u23/4-00/684-1.webp?ts=1739436691" ] 1 => array:5 [ "name" => "684-2.jpg" "alt" => null "width" => "1404" "height" => 1863 "path" => "/media/geometrija_10/atanasjan-u23/4-00/684-2.webp?ts=1739436691" ] ] ] #original: array:6 [ "id" => 941800 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "745099" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "684-1.jpg" "alt" => null "width" => "1404" "height" => 664 "path" => "/media/geometrija_10/atanasjan-u23/4-00/684-1.webp?ts=1739436691" ] 1 => array:5 [ "name" => "684-2.jpg" "alt" => null "width" => "1404" "height" => 1863 "path" => "/media/geometrija_10/atanasjan-u23/4-00/684-2.webp?ts=1739436691" ] ] ] #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" => 941928 "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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745099" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "684-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/684-1.webp?ts=1739436728" ] ] ] #original: array:6 [ "id" => 941928 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1039} "task" => array:2 [ "refs" => "745099" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "684-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/684-1.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 {#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" => 1352412 "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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745099" "type" => "task" ] "text" => "<p>Для решения задачи введем прямоугольную систему координат. Пусть точка $A$ — начало координат, а оси $Ox$, $Oy$ и $Oz$ направлены вдоль ребер $AB$, $AD$ и $AA_1$ соответственно. Ребро куба равно $a$. Тогда вершины куба имеют следующие координаты: $A(0,0,0)$, $B(a,0,0)$, $C(a,a,0)$, $D(0,a,0)$, $A_1(0,0,a)$, $B_1(a,0,a)$, $C_1(a,a,a)$, $D_1(0,a,a)$.</p><p>Точка $O_1$ — центр грани $A_1B_1C_1D_1$, является серединой диагонали $A_1C_1$. Ее координаты: $O_1 = \left(\frac{0+a}{2}, \frac{0+a}{2}, \frac{a+a}{2}\right) = \left(\frac{a}{2}, \frac{a}{2}, a\right)$.</p><p>Скалярное произведение векторов $\vec{u}=(x_1, y_1, z_1)$ и $\vec{v}=(x_2, y_2, z_2)$ вычисляется по формуле $\vec{u} \cdot \vec{v} = x_1x_2 + y_1y_2 + z_1z_2$.</p><p><strong>а)</strong> Вычислим скалярное произведение векторов $\vec{AD}$ и $\vec{B_1C_1}$.</p><p>Найдем координаты векторов:</p><p>$\vec{AD} = D - A = (0, a, 0) - (0, 0, 0) = (0, a, 0)$.</p><p>$\vec{B_1C_1} = C_1 - B_1 = (a, a, a) - (a, 0, a) = (0, a, 0)$.</p><p>Вычислим скалярное произведение:</p><p>$\vec{AD} \cdot \vec{B_1C_1} = 0 \cdot 0 + a \cdot a + 0 \cdot 0 = a^2$.</p><p>Другой способ: векторы $\vec{AD}$ и $\vec{B_1C_1}$ равны, так как они получаются параллельным переносом. Следовательно, их скалярное произведение равно квадрату длины любого из них: $\vec{AD} \cdot \vec{B_1C_1} = \vec{AD} \cdot \vec{AD} = |\vec{AD}|^2 = a^2$.</p><p>Ответ: $a^2$.</p><p><strong>б)</strong> Вычислим скалярное произведение векторов $\vec{AC}$ и $\vec{C_1A_1}$.</p><p>Найдем координаты векторов:</p><p>$\vec{AC} = C - A = (a, a, 0) - (0, 0, 0) = (a, a, 0)$.</p><p>$\vec{C_1A_1} = A_1 - C_1 = (0, 0, a) - (a, a, a) = (-a, -a, 0)$.</p><p>Вычислим скалярное произведение:</p><p>$\vec{AC} \cdot \vec{C_1A_1} = a \cdot (-a) + a \cdot (-a) + 0 \cdot 0 = -a^2 - a^2 = -2a^2$.</p><p>Ответ: $-2a^2$.</p><p><strong>в)</strong> Вычислим скалярное произведение векторов $\vec{D_1B}$ и $\vec{AC}$.</p><p>Найдем координаты векторов:</p><p>$\vec{D_1B} = B - D_1 = (a, 0, 0) - (0, a, a) = (a, -a, -a)$.</p><p>Координаты вектора $\vec{AC}$ из пункта б): $\vec{AC} = (a, a, 0)$.</p><p>Вычислим скалярное произведение:</p><p>$\vec{D_1B} \cdot \vec{AC} = a \cdot a + (-a) \cdot a + (-a) \cdot 0 = a^2 - a^2 + 0 = 0$.</p><p>Ответ: $0$.</p><p><strong>г)</strong> Вычислим скалярное произведение векторов $\vec{BA_1}$ и $\vec{BC_1}$.</p><p>Векторы $\vec{BA_1}$ и $\vec{BC_1}$ выходят из одной точки $B$. Найдем их длины и угол между ними. $BA_1$ — диагональ грани $ABB_1A_1$, ее длина $|\vec{BA_1}| = \sqrt{AB^2 + AA_1^2} = \sqrt{a^2 + a^2} = a\sqrt{2}$. $BC_1$ — диагональ грани $BCC_1B_1$, ее длина $|\vec{BC_1}| = \sqrt{BC^2 + CC_1^2} = \sqrt{a^2 + a^2} = a\sqrt{2}$. Рассмотрим треугольник $A_1BC_1$. Его стороны: $BA_1$, $BC_1$ и $A_1C_1$. Длина $A_1C_1$ — это диагональ грани $A_1B_1C_1D_1$, ее длина также равна $a\sqrt{2}$. Поскольку все три стороны треугольника $A_1BC_1$ равны $a\sqrt{2}$, он является равносторонним. Угол между векторами $\vec{BA_1}$ и $\vec{BC_1}$ — это угол $\angle A_1BC_1$, который равен $60^\circ$.</p><p>Скалярное произведение равно:</p><p>$\vec{BA_1} \cdot \vec{BC_1} = |\vec{BA_1}| \cdot |\vec{BC_1}| \cdot \cos(\angle A_1BC_1) = (a\sqrt{2}) \cdot (a\sqrt{2}) \cdot \cos(60^\circ) = 2a^2 \cdot \frac{1}{2} = a^2$.</p><p>Ответ: $a^2$.</p><p><strong>д)</strong> Вычислим скалярное произведение векторов $\vec{A_1O_1}$ и $\vec{A_1C_1}$.</p><p>Точка $O_1$ является центром грани $A_1B_1C_1D_1$, а значит, и серединой ее диагонали $A_1C_1$. Следовательно, вектор $\vec{A_1O_1}$ сонаправлен с вектором $\vec{A_1C_1}$, и его длина равна половине длины вектора $\vec{A_1C_1}$: $\vec{A_1O_1} = \frac{1}{2}\vec{A_1C_1}$. Длина диагонали грани $|\vec{A_1C_1}| = \sqrt{A_1B_1^2 + B_1C_1^2} = \sqrt{a^2+a^2} = a\sqrt{2}$.</p><p>Скалярное произведение равно:</p><p>$\vec{A_1O_1} \cdot \vec{A_1C_1} = \left(\frac{1}{2}\vec{A_1C_1}\right) \cdot \vec{A_1C_1} = \frac{1}{2} (\vec{A_1C_1})^2 = \frac{1}{2} |\vec{A_1C_1}|^2 = \frac{1}{2} (a\sqrt{2})^2 = \frac{1}{2} \cdot 2a^2 = a^2$.</p><p>Ответ: $a^2$.</p><p><strong>е)</strong> Вычислим скалярное произведение векторов $\vec{D_1O_1}$ и $\vec{B_1O_1}$.</p><p>Точка $O_1$ — центр квадрата $A_1B_1C_1D_1$. Векторы $\vec{D_1O_1}$ и $\vec{B_1O_1}$ лежат на одной прямой — диагонали $B_1D_1$, но направлены в противоположные стороны, так как оба направлены к центру $O_1$ от вершин. Угол между ними равен $180^\circ$. Длина каждого из этих векторов равна половине длины диагонали грани: $|\vec{D_1O_1}| = |\vec{B_1O_1}| = \frac{1}{2} |B_1D_1| = \frac{a\sqrt{2}}{2}$.</p><p>Скалярное произведение равно:</p><p>$\vec{D_1O_1} \cdot \vec{B_1O_1} = |\vec{D_1O_1}| \cdot |\vec{B_1O_1}| \cdot \cos(180^\circ) = \left(\frac{a\sqrt{2}}{2}\right) \cdot \left(\frac{a\sqrt{2}}{2}\right) \cdot (-1) = \frac{2a^2}{4} \cdot (-1) = -\frac{a^2}{2}$.</p><p>Ответ: $-\frac{a^2}{2}$.</p><p><strong>ж)</strong> Вычислим скалярное произведение векторов $\vec{BO_1}$ и $\vec{C_1B}$.</p><p>Найдем координаты векторов:</p><p>$\vec{BO_1} = O_1 - B = \left(\frac{a}{2}, \frac{a}{2}, a\right) - (a, 0, 0) = \left(-\frac{a}{2}, \frac{a}{2}, a\right)$.</p><p>$\vec{C_1B} = B - C_1 = (a, 0, 0) - (a, a, a) = (0, -a, -a)$.</p><p>Вычислим скалярное произведение:</p><p>$\vec{BO_1} \cdot \vec{C_1B} = \left(-\frac{a}{2}\right) \cdot 0 + \left(\frac{a}{2}\right) \cdot (-a) + a \cdot (-a) = 0 - \frac{a^2}{2} - a^2 = -\frac{3a^2}{2}$.</p><p>Ответ: $-\frac{3a^2}{2}$.</p>" ] #original: array:6 [ "id" => 1352412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159} "task" => array:2 [ "refs" => "745099" "type" => "task" ] "text" => "<p>Для решения задачи введем прямоугольную систему координат. Пусть точка $A$ — начало координат, а оси $Ox$, $Oy$ и $Oz$ направлены вдоль ребер $AB$, $AD$ и $AA_1$ соответственно. Ребро куба равно $a$. Тогда вершины куба имеют следующие координаты: $A(0,0,0)$, $B(a,0,0)$, $C(a,a,0)$, $D(0,a,0)$, $A_1(0,0,a)$, $B_1(a,0,a)$, $C_1(a,a,a)$, $D_1(0,a,a)$.</p><p>Точка $O_1$ — центр грани $A_1B_1C_1D_1$, является серединой диагонали $A_1C_1$. Ее координаты: $O_1 = \left(\frac{0+a}{2}, \frac{0+a}{2}, \frac{a+a}{2}\right) = \left(\frac{a}{2}, \frac{a}{2}, a\right)$.</p><p>Скалярное произведение векторов $\vec{u}=(x_1, y_1, z_1)$ и $\vec{v}=(x_2, y_2, z_2)$ вычисляется по формуле $\vec{u} \cdot \vec{v} = x_1x_2 + y_1y_2 + z_1z_2$.</p><p><strong>а)</strong> Вычислим скалярное произведение векторов $\vec{AD}$ и $\vec{B_1C_1}$.</p><p>Найдем координаты векторов:</p><p>$\vec{AD} = D - A = (0, a, 0) - (0, 0, 0) = (0, a, 0)$.</p><p>$\vec{B_1C_1} = C_1 - B_1 = (a, a, a) - (a, 0, a) = (0, a, 0)$.</p><p>Вычислим скалярное произведение:</p><p>$\vec{AD} \cdot \vec{B_1C_1} = 0 \cdot 0 + a \cdot a + 0 \cdot 0 = a^2$.</p><p>Другой способ: векторы $\vec{AD}$ и $\vec{B_1C_1}$ равны, так как они получаются параллельным переносом. Следовательно, их скалярное произведение равно квадрату длины любого из них: $\vec{AD} \cdot \vec{B_1C_1} = \vec{AD} \cdot \vec{AD} = |\vec{AD}|^2 = a^2$.</p><p>Ответ: $a^2$.</p><p><strong>б)</strong> Вычислим скалярное произведение векторов $\vec{AC}$ и $\vec{C_1A_1}$.</p><p>Найдем координаты векторов:</p><p>$\vec{AC} = C - A = (a, a, 0) - (0, 0, 0) = (a, a, 0)$.</p><p>$\vec{C_1A_1} = A_1 - C_1 = (0, 0, a) - (a, a, a) = (-a, -a, 0)$.</p><p>Вычислим скалярное произведение:</p><p>$\vec{AC} \cdot \vec{C_1A_1} = a \cdot (-a) + a \cdot (-a) + 0 \cdot 0 = -a^2 - a^2 = -2a^2$.</p><p>Ответ: $-2a^2$.</p><p><strong>в)</strong> Вычислим скалярное произведение векторов $\vec{D_1B}$ и $\vec{AC}$.</p><p>Найдем координаты векторов:</p><p>$\vec{D_1B} = B - D_1 = (a, 0, 0) - (0, a, a) = (a, -a, -a)$.</p><p>Координаты вектора $\vec{AC}$ из пункта б): $\vec{AC} = (a, a, 0)$.</p><p>Вычислим скалярное произведение:</p><p>$\vec{D_1B} \cdot \vec{AC} = a \cdot a + (-a) \cdot a + (-a) \cdot 0 = a^2 - a^2 + 0 = 0$.</p><p>Ответ: $0$.</p><p><strong>г)</strong> Вычислим скалярное произведение векторов $\vec{BA_1}$ и $\vec{BC_1}$.</p><p>Векторы $\vec{BA_1}$ и $\vec{BC_1}$ выходят из одной точки $B$. Найдем их длины и угол между ними. $BA_1$ — диагональ грани $ABB_1A_1$, ее длина $|\vec{BA_1}| = \sqrt{AB^2 + AA_1^2} = \sqrt{a^2 + a^2} = a\sqrt{2}$. $BC_1$ — диагональ грани $BCC_1B_1$, ее длина $|\vec{BC_1}| = \sqrt{BC^2 + CC_1^2} = \sqrt{a^2 + a^2} = a\sqrt{2}$. Рассмотрим треугольник $A_1BC_1$. Его стороны: $BA_1$, $BC_1$ и $A_1C_1$. Длина $A_1C_1$ — это диагональ грани $A_1B_1C_1D_1$, ее длина также равна $a\sqrt{2}$. Поскольку все три стороны треугольника $A_1BC_1$ равны $a\sqrt{2}$, он является равносторонним. Угол между векторами $\vec{BA_1}$ и $\vec{BC_1}$ — это угол $\angle A_1BC_1$, который равен $60^\circ$.</p><p>Скалярное произведение равно:</p><p>$\vec{BA_1} \cdot \vec{BC_1} = |\vec{BA_1}| \cdot |\vec{BC_1}| \cdot \cos(\angle A_1BC_1) = (a\sqrt{2}) \cdot (a\sqrt{2}) \cdot \cos(60^\circ) = 2a^2 \cdot \frac{1}{2} = a^2$.</p><p>Ответ: $a^2$.</p><p><strong>д)</strong> Вычислим скалярное произведение векторов $\vec{A_1O_1}$ и $\vec{A_1C_1}$.</p><p>Точка $O_1$ является центром грани $A_1B_1C_1D_1$, а значит, и серединой ее диагонали $A_1C_1$. Следовательно, вектор $\vec{A_1O_1}$ сонаправлен с вектором $\vec{A_1C_1}$, и его длина равна половине длины вектора $\vec{A_1C_1}$: $\vec{A_1O_1} = \frac{1}{2}\vec{A_1C_1}$. Длина диагонали грани $|\vec{A_1C_1}| = \sqrt{A_1B_1^2 + B_1C_1^2} = \sqrt{a^2+a^2} = a\sqrt{2}$.</p><p>Скалярное произведение равно:</p><p>$\vec{A_1O_1} \cdot \vec{A_1C_1} = \left(\frac{1}{2}\vec{A_1C_1}\right) \cdot \vec{A_1C_1} = \frac{1}{2} (\vec{A_1C_1})^2 = \frac{1}{2} |\vec{A_1C_1}|^2 = \frac{1}{2} (a\sqrt{2})^2 = \frac{1}{2} \cdot 2a^2 = a^2$.</p><p>Ответ: $a^2$.</p><p><strong>е)</strong> Вычислим скалярное произведение векторов $\vec{D_1O_1}$ и $\vec{B_1O_1}$.</p><p>Точка $O_1$ — центр квадрата $A_1B_1C_1D_1$. Векторы $\vec{D_1O_1}$ и $\vec{B_1O_1}$ лежат на одной прямой — диагонали $B_1D_1$, но направлены в противоположные стороны, так как оба направлены к центру $O_1$ от вершин. Угол между ними равен $180^\circ$. Длина каждого из этих векторов равна половине длины диагонали грани: $|\vec{D_1O_1}| = |\vec{B_1O_1}| = \frac{1}{2} |B_1D_1| = \frac{a\sqrt{2}}{2}$.</p><p>Скалярное произведение равно:</p><p>$\vec{D_1O_1} \cdot \vec{B_1O_1} = |\vec{D_1O_1}| \cdot |\vec{B_1O_1}| \cdot \cos(180^\circ) = \left(\frac{a\sqrt{2}}{2}\right) \cdot \left(\frac{a\sqrt{2}}{2}\right) \cdot (-1) = \frac{2a^2}{4} \cdot (-1) = -\frac{a^2}{2}$.</p><p>Ответ: $-\frac{a^2}{2}$.</p><p><strong>ж)</strong> Вычислим скалярное произведение векторов $\vec{BO_1}$ и $\vec{C_1B}$.</p><p>Найдем координаты векторов:</p><p>$\vec{BO_1} = O_1 - B = \left(\frac{a}{2}, \frac{a}{2}, a\right) - (a, 0, 0) = \left(-\frac{a}{2}, \frac{a}{2}, a\right)$.</p><p>$\vec{C_1B} = B - C_1 = (a, 0, 0) - (a, a, a) = (0, -a, -a)$.</p><p>Вычислим скалярное произведение:</p><p>$\vec{BO_1} \cdot \vec{C_1B} = \left(-\frac{a}{2}\right) \cdot 0 + \left(\frac{a}{2}\right) \cdot (-a) + a \cdot (-a) = 0 - \frac{a^2}{2} - a^2 = -\frac{3a^2}{2}$.</p><p>Ответ: $-\frac{3a^2}{2}$.</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" => "745100" "type" => "task" ] "previous" => array:2 [ "refs" => "745098" "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" => 933958 "created_at" 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#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" => "933959" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933957" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1267 #items: array:15 [ 0 => App\Models\Task {#1289 #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: 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#attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1345 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1347 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 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App\Models\Task {#1387 #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] } 10 => App\Models\Task {#1407 #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: [] 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#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" => 933958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "176" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-176" "field_display_title" => "176" "field_folder" => "1" "field_image_name" => "176" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} "field_weight" => "0" "field_book_parent" => 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#dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№684 (с. 176)
Условие. №684 (с. 176)
скриншот условия
684. Ребро куба ABCDA₁B₁C₁D₁ равно а, точка O₁ — центр грани A₁B₁C₁D₁. Вычислите скалярное произведение векторов:
Решение 2. №684 (с. 176)
Решение 3. №684 (с. 176)
Решение 4. №684 (с. 176)
Решение 5. №684 (с. 176)
Решение 6. №684 (с. 176)
Для решения задачи введем прямоугольную систему координат. Пусть точка $A$ — начало координат, а оси $Ox$, $Oy$ и $Oz$ направлены вдоль ребер $AB$, $AD$ и $AA_1$ соответственно. Ребро куба равно $a$. Тогда вершины куба имеют следующие координаты: $A(0,0,0)$, $B(a,0,0)$, $C(a,a,0)$, $D(0,a,0)$, $A_1(0,0,a)$, $B_1(a,0,a)$, $C_1(a,a,a)$, $D_1(0,a,a)$.
Точка $O_1$ — центр грани $A_1B_1C_1D_1$, является серединой диагонали $A_1C_1$. Ее координаты: $O_1 = \left(\frac{0+a}{2}, \frac{0+a}{2}, \frac{a+a}{2}\right) = \left(\frac{a}{2}, \frac{a}{2}, a\right)$.
Скалярное произведение векторов $\vec{u}=(x_1, y_1, z_1)$ и $\vec{v}=(x_2, y_2, z_2)$ вычисляется по формуле $\vec{u} \cdot \vec{v} = x_1x_2 + y_1y_2 + z_1z_2$.
а) Вычислим скалярное произведение векторов $\vec{AD}$ и $\vec{B_1C_1}$.
Найдем координаты векторов:
$\vec{AD} = D - A = (0, a, 0) - (0, 0, 0) = (0, a, 0)$.
$\vec{B_1C_1} = C_1 - B_1 = (a, a, a) - (a, 0, a) = (0, a, 0)$.
Вычислим скалярное произведение:
$\vec{AD} \cdot \vec{B_1C_1} = 0 \cdot 0 + a \cdot a + 0 \cdot 0 = a^2$.
Другой способ: векторы $\vec{AD}$ и $\vec{B_1C_1}$ равны, так как они получаются параллельным переносом. Следовательно, их скалярное произведение равно квадрату длины любого из них: $\vec{AD} \cdot \vec{B_1C_1} = \vec{AD} \cdot \vec{AD} = |\vec{AD}|^2 = a^2$.
Ответ: $a^2$.
б) Вычислим скалярное произведение векторов $\vec{AC}$ и $\vec{C_1A_1}$.
Найдем координаты векторов:
$\vec{AC} = C - A = (a, a, 0) - (0, 0, 0) = (a, a, 0)$.
$\vec{C_1A_1} = A_1 - C_1 = (0, 0, a) - (a, a, a) = (-a, -a, 0)$.
Вычислим скалярное произведение:
$\vec{AC} \cdot \vec{C_1A_1} = a \cdot (-a) + a \cdot (-a) + 0 \cdot 0 = -a^2 - a^2 = -2a^2$.
Ответ: $-2a^2$.
в) Вычислим скалярное произведение векторов $\vec{D_1B}$ и $\vec{AC}$.
Найдем координаты векторов:
$\vec{D_1B} = B - D_1 = (a, 0, 0) - (0, a, a) = (a, -a, -a)$.
Координаты вектора $\vec{AC}$ из пункта б): $\vec{AC} = (a, a, 0)$.
Вычислим скалярное произведение:
$\vec{D_1B} \cdot \vec{AC} = a \cdot a + (-a) \cdot a + (-a) \cdot 0 = a^2 - a^2 + 0 = 0$.
Ответ: $0$.
г) Вычислим скалярное произведение векторов $\vec{BA_1}$ и $\vec{BC_1}$.
Векторы $\vec{BA_1}$ и $\vec{BC_1}$ выходят из одной точки $B$. Найдем их длины и угол между ними. $BA_1$ — диагональ грани $ABB_1A_1$, ее длина $|\vec{BA_1}| = \sqrt{AB^2 + AA_1^2} = \sqrt{a^2 + a^2} = a\sqrt{2}$. $BC_1$ — диагональ грани $BCC_1B_1$, ее длина $|\vec{BC_1}| = \sqrt{BC^2 + CC_1^2} = \sqrt{a^2 + a^2} = a\sqrt{2}$. Рассмотрим треугольник $A_1BC_1$. Его стороны: $BA_1$, $BC_1$ и $A_1C_1$. Длина $A_1C_1$ — это диагональ грани $A_1B_1C_1D_1$, ее длина также равна $a\sqrt{2}$. Поскольку все три стороны треугольника $A_1BC_1$ равны $a\sqrt{2}$, он является равносторонним. Угол между векторами $\vec{BA_1}$ и $\vec{BC_1}$ — это угол $\angle A_1BC_1$, который равен $60^\circ$.
Скалярное произведение равно:
$\vec{BA_1} \cdot \vec{BC_1} = |\vec{BA_1}| \cdot |\vec{BC_1}| \cdot \cos(\angle A_1BC_1) = (a\sqrt{2}) \cdot (a\sqrt{2}) \cdot \cos(60^\circ) = 2a^2 \cdot \frac{1}{2} = a^2$.
Ответ: $a^2$.
д) Вычислим скалярное произведение векторов $\vec{A_1O_1}$ и $\vec{A_1C_1}$.
Точка $O_1$ является центром грани $A_1B_1C_1D_1$, а значит, и серединой ее диагонали $A_1C_1$. Следовательно, вектор $\vec{A_1O_1}$ сонаправлен с вектором $\vec{A_1C_1}$, и его длина равна половине длины вектора $\vec{A_1C_1}$: $\vec{A_1O_1} = \frac{1}{2}\vec{A_1C_1}$. Длина диагонали грани $|\vec{A_1C_1}| = \sqrt{A_1B_1^2 + B_1C_1^2} = \sqrt{a^2+a^2} = a\sqrt{2}$.
Скалярное произведение равно:
$\vec{A_1O_1} \cdot \vec{A_1C_1} = \left(\frac{1}{2}\vec{A_1C_1}\right) \cdot \vec{A_1C_1} = \frac{1}{2} (\vec{A_1C_1})^2 = \frac{1}{2} |\vec{A_1C_1}|^2 = \frac{1}{2} (a\sqrt{2})^2 = \frac{1}{2} \cdot 2a^2 = a^2$.
Ответ: $a^2$.
е) Вычислим скалярное произведение векторов $\vec{D_1O_1}$ и $\vec{B_1O_1}$.
Точка $O_1$ — центр квадрата $A_1B_1C_1D_1$. Векторы $\vec{D_1O_1}$ и $\vec{B_1O_1}$ лежат на одной прямой — диагонали $B_1D_1$, но направлены в противоположные стороны, так как оба направлены к центру $O_1$ от вершин. Угол между ними равен $180^\circ$. Длина каждого из этих векторов равна половине длины диагонали грани: $|\vec{D_1O_1}| = |\vec{B_1O_1}| = \frac{1}{2} |B_1D_1| = \frac{a\sqrt{2}}{2}$.
Скалярное произведение равно:
$\vec{D_1O_1} \cdot \vec{B_1O_1} = |\vec{D_1O_1}| \cdot |\vec{B_1O_1}| \cdot \cos(180^\circ) = \left(\frac{a\sqrt{2}}{2}\right) \cdot \left(\frac{a\sqrt{2}}{2}\right) \cdot (-1) = \frac{2a^2}{4} \cdot (-1) = -\frac{a^2}{2}$.
Ответ: $-\frac{a^2}{2}$.
ж) Вычислим скалярное произведение векторов $\vec{BO_1}$ и $\vec{C_1B}$.
Найдем координаты векторов:
$\vec{BO_1} = O_1 - B = \left(\frac{a}{2}, \frac{a}{2}, a\right) - (a, 0, 0) = \left(-\frac{a}{2}, \frac{a}{2}, a\right)$.
$\vec{C_1B} = B - C_1 = (a, 0, 0) - (a, a, a) = (0, -a, -a)$.
Вычислим скалярное произведение:
$\vec{BO_1} \cdot \vec{C_1B} = \left(-\frac{a}{2}\right) \cdot 0 + \left(\frac{a}{2}\right) \cdot (-a) + a \cdot (-a) = 0 - \frac{a^2}{2} - a^2 = -\frac{3a^2}{2}$.
Ответ: $-\frac{3a^2}{2}$.
Другие задания:
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