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Номер 619, страница 158 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Глава 6. Векторы в пространстве. Параграф 3. Компланарные векторы, дополнительные задачи - номер 619, страница 158.
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" => 745034 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "158" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-619" "field_display_title" => "619" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "142" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Book {#1049 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:2 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:5 [ 0 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null …3 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1062 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Term {#1069 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1073 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Term {#1075 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Term {#1077 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1083 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055} "field_author" => Illuminate\Database\Eloquent\Collection {#1057} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_country" => Illuminate\Database\Eloquent\Collection {#1066} "field_city" => Illuminate\Database\Eloquent\Collection {#1068} "field_series" => Illuminate\Database\Eloquent\Collection {#1070} "field_umk" => Illuminate\Database\Eloquent\Collection {#1071} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "142" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1086} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:2 [ 0 => App\Models\Branch {#1129 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "142" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Векторы в пространстве" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "142" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1131} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1133 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Компланарные векторы, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1134 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "157" "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" => 744404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Компланарные векторы, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "157" "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 {#1118 #items: array:5 [ 0 => App\Models\Element {#1108 #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" => 939351 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Edition {#1109 #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 {#1107 …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 {#1105 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1102 …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 {#1107 …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 {#1105 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1102 …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" => "745034" "type" => "task" ] "text" => "<p> <b>619.</b> При каких значениях <i>k</i> в равенстве <span class="long"><span class="arrtoptoleft">a</span> = k<span class="arrtoptoleft">b,</span></span> где <span class="long"><span class="arrtoptoleft">b</span> ≠ <span class="arrtoptoleft">0,</span></span> векторы <span class="long"><span class="arrtoptoleft">a</span> и <span class="arrtoptoleft">b:</span></span> </p> <p> а) коллинеарны; </p> <p> б) сонаправлены; </p> <p> в) противоположно направлены; </p> <p> г) являются противоположными? </p>" "img" => array:1 [ 0 => array:5 [ "name" => "619-1.jpg" "alt" => null "width" => "1545" "height" => 231 "path" => "/media/geometrija_10/atanasjan-u23/0-00/619-1.webp?ts=1739436772" ] ] ] #original: array:7 [ "id" => 939351 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101} "task" => array:2 [ "refs" => "745034" "type" => "task" ] "text" => "<p> <b>619.</b> При каких значениях <i>k</i> в равенстве <span class="long"><span class="arrtoptoleft">a</span> = k<span class="arrtoptoleft">b,</span></span> где <span class="long"><span class="arrtoptoleft">b</span> ≠ <span class="arrtoptoleft">0,</span></span> векторы <span class="long"><span class="arrtoptoleft">a</span> и <span class="arrtoptoleft">b:</span></span> </p> <p> а) коллинеарны; </p> <p> б) сонаправлены; </p> <p> в) противоположно направлены; </p> <p> г) являются противоположными? </p>" "img" => array:1 [ 0 => array:5 [ "name" => "619-1.jpg" "alt" => null "width" => "1545" "height" => 231 "path" => "/media/geometrija_10/atanasjan-u23/0-00/619-1.webp?ts=1739436772" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1087 #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" => 942329 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Edition {#1043 #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 {#1039 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1039 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1128 …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" => "745034" "type" => "task" ] "img" => array:4 [ 0 => array:5 [ "name" => "619-1.png" "alt" => null "width" => "694" "height" => 4005 "path" => "/media/geometrija_10/atanasjan-u23/2-00/619-1.webp?ts=1739441486" ] 1 => array:5 [ "name" => "619-2.png" "alt" => null "width" => "694" "height" => 2411 "path" => "/media/geometrija_10/atanasjan-u23/2-00/619-2.webp?ts=1739441486" ] 2 => array:5 [ "name" => "619-3.png" "alt" => null "width" => "694" "height" => 2412 "path" => "/media/geometrija_10/atanasjan-u23/2-00/619-3.webp?ts=1739441486" ] 3 => array:5 [ "name" => "619-4.png" "alt" => null "width" => "694" "height" => 2542 "path" => "/media/geometrija_10/atanasjan-u23/2-00/619-4.webp?ts=1739441486" ] ] ] #original: array:6 [ "id" => 942329 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1125} "task" => array:2 [ "refs" => "745034" "type" => "task" ] "img" => array:4 [ 0 => array:5 [ "name" => "619-1.png" "alt" => null "width" => "694" "height" => 4005 "path" => "/media/geometrija_10/atanasjan-u23/2-00/619-1.webp?ts=1739441486" ] 1 => array:5 [ "name" => "619-2.png" "alt" => null "width" => "694" "height" => 2411 "path" => "/media/geometrija_10/atanasjan-u23/2-00/619-2.webp?ts=1739441486" ] 2 => array:5 [ "name" => "619-3.png" "alt" => null "width" => "694" "height" => 2412 "path" => "/media/geometrija_10/atanasjan-u23/2-00/619-3.webp?ts=1739441486" ] 3 => array:5 [ "name" => "619-4.png" "alt" => null "width" => "694" "height" => 2542 "path" => "/media/geometrija_10/atanasjan-u23/2-00/619-4.webp?ts=1739441486" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1095 #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" => 942245 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ 0 => App\Models\Edition {#1094 #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 {#1089 …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 {#1090 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 …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 {#1089 …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 {#1090 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 …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" => "745034" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "619-1.jpg" "alt" => null "width" => "1308" "height" => 433 "path" => "/media/geometrija_10/atanasjan-u23/3-00/619-1.webp?ts=1739437074" ] ] ] #original: array:6 [ "id" => 942245 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1041} "task" => array:2 [ "refs" => "745034" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "619-1.jpg" "alt" => null "width" => "1308" "height" => 433 "path" => "/media/geometrija_10/atanasjan-u23/3-00/619-1.webp?ts=1739437074" ] ] ] #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 {#1103 #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" => 941740 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1100 #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 {#1099 …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 {#1097 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 …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 {#1099 …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 {#1097 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 …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" => "745034" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "619-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/619-1.webp?ts=1739436668" ] ] ] #original: array:6 [ "id" => 941740 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "745034" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "619-1.jpg" "alt" => null "width" => "1329" "height" => 907 "path" => "/media/geometrija_10/atanasjan-u23/5-00/619-1.webp?ts=1739436668" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Element {#1124 #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" => 1352347 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144 #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" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1138 …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 {#1139 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1141 …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 {#1138 …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 {#1139 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745034" "type" => "task" ] "text" => "<p><strong>а) коллинеарны;</strong> Векторы $\vec{a}$ и $\vec{b}$ называются коллинеарными, если они лежат на одной прямой или на параллельных прямых. По определению, два ненулевых вектора коллинеарны тогда и только тогда, когда один из них можно выразить через другой путем умножения на некоторое число (скаляр). Равенство $\vec{a} = k\vec{b}$ как раз и является математической записью этого условия. Это означает, что для любого действительного числа $k$ вектор $\vec{a}$ будет коллинеарен вектору $\vec{b}$. Если $k=0$, то $\vec{a} = 0 \cdot \vec{b} = \vec{0}$. Нулевой вектор по определению считается коллинеарным любому вектору.<br>Ответ: $k$ — любое действительное число ($k \in \mathbb{R}$).</p><p><strong>б) сонаправлены;</strong> Сонаправленные векторы ($\vec{a} \uparrow\uparrow \vec{b}$) — это коллинеарные векторы, имеющие одинаковое направление. При умножении вектора на положительный скаляр его направление не меняется. Следовательно, для того чтобы вектор $\vec{a}$ был сонаправлен с вектором $\vec{b}$ в равенстве $\vec{a} = k\vec{b}$, коэффициент $k$ должен быть положительным.<br>Ответ: $k > 0$.</p><p><strong>в) противоположно направлены;</strong> Противоположно направленные векторы ($\vec{a} \uparrow\downarrow \vec{b}$) — это коллинеарные векторы, имеющие противоположные направления. При умножении вектора на отрицательный скаляр его направление меняется на противоположное. Следовательно, для того чтобы вектор $\vec{a}$ был противоположно направлен вектору $\vec{b}$ в равенстве $\vec{a} = k\vec{b}$, коэффициент $k$ должен быть отрицательным.<br>Ответ: $k < 0$.</p><p><strong>г) являются противоположными?</strong> Противоположные векторы — это векторы, которые имеют равные длины (модули) и противоположные направления. Таким образом, должны выполняться два условия: 1. Направления векторов противоположны. Как мы выяснили в пункте (в), это означает, что $k < 0$. 2. Длины векторов равны: $|\vec{a}| = |\vec{b}|$. Найдем длину вектора $\vec{a}$ из исходного равенства: $|\vec{a}| = |k\vec{b}| = |k| \cdot |\vec{b}|$. Подставим это в условие равенства длин: $|k| \cdot |\vec{b}| = |\vec{b}|$. Поскольку по условию $\vec{b} \neq \vec{0}$, то и $|\vec{b}| \neq 0$. Значит, мы можем разделить обе части равенства на $|\vec{b}|$, получив $|k| = 1$. Уравнение $|k|=1$ имеет два решения: $k=1$ и $k=-1$. Выбираем то решение, которое удовлетворяет первому условию ($k < 0$). Этим решением является $k=-1$. При $k=-1$ равенство принимает вид $\vec{a} = -\vec{b}$, что полностью соответствует определению противоположных векторов.<br>Ответ: $k = -1$.</p>" ] #original: array:6 [ "id" => 1352347 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "745034" "type" => "task" ] "text" => "<p><strong>а) коллинеарны;</strong> Векторы $\vec{a}$ и $\vec{b}$ называются коллинеарными, если они лежат на одной прямой или на параллельных прямых. По определению, два ненулевых вектора коллинеарны тогда и только тогда, когда один из них можно выразить через другой путем умножения на некоторое число (скаляр). Равенство $\vec{a} = k\vec{b}$ как раз и является математической записью этого условия. Это означает, что для любого действительного числа $k$ вектор $\vec{a}$ будет коллинеарен вектору $\vec{b}$. Если $k=0$, то $\vec{a} = 0 \cdot \vec{b} = \vec{0}$. Нулевой вектор по определению считается коллинеарным любому вектору.<br>Ответ: $k$ — любое действительное число ($k \in \mathbb{R}$).</p><p><strong>б) сонаправлены;</strong> Сонаправленные векторы ($\vec{a} \uparrow\uparrow \vec{b}$) — это коллинеарные векторы, имеющие одинаковое направление. При умножении вектора на положительный скаляр его направление не меняется. Следовательно, для того чтобы вектор $\vec{a}$ был сонаправлен с вектором $\vec{b}$ в равенстве $\vec{a} = k\vec{b}$, коэффициент $k$ должен быть положительным.<br>Ответ: $k > 0$.</p><p><strong>в) противоположно направлены;</strong> Противоположно направленные векторы ($\vec{a} \uparrow\downarrow \vec{b}$) — это коллинеарные векторы, имеющие противоположные направления. При умножении вектора на отрицательный скаляр его направление меняется на противоположное. Следовательно, для того чтобы вектор $\vec{a}$ был противоположно направлен вектору $\vec{b}$ в равенстве $\vec{a} = k\vec{b}$, коэффициент $k$ должен быть отрицательным.<br>Ответ: $k < 0$.</p><p><strong>г) являются противоположными?</strong> Противоположные векторы — это векторы, которые имеют равные длины (модули) и противоположные направления. Таким образом, должны выполняться два условия: 1. Направления векторов противоположны. Как мы выяснили в пункте (в), это означает, что $k < 0$. 2. Длины векторов равны: $|\vec{a}| = |\vec{b}|$. Найдем длину вектора $\vec{a}$ из исходного равенства: $|\vec{a}| = |k\vec{b}| = |k| \cdot |\vec{b}|$. Подставим это в условие равенства длин: $|k| \cdot |\vec{b}| = |\vec{b}|$. Поскольку по условию $\vec{b} \neq \vec{0}$, то и $|\vec{b}| \neq 0$. Значит, мы можем разделить обе части равенства на $|\vec{b}|$, получив $|k| = 1$. Уравнение $|k|=1$ имеет два решения: $k=1$ и $k=-1$. Выбираем то решение, которое удовлетворяет первому условию ($k < 0$). Этим решением является $k=-1$. При $k=-1$ равенство принимает вид $\vec{a} = -\vec{b}$, что полностью соответствует определению противоположных векторов.<br>Ответ: $k = -1$.</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" => "745035" "type" => "task" ] "previous" => array:2 [ "refs" => "745033" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1405 #items: array:1 [ 0 => App\Models\BookPage {#1032 #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" => 933940 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "158" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-158" "field_display_title" => "158" "field_folder" => "1" "field_image_name" => "158" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026 #items: array:1 [ 0 => App\Models\Book {#1038 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1168 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\Element {#1170 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "933941" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933939" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1178 #items: array:11 [ 0 => App\Models\Task {#1179 #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 {#1235 #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 {#1252 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1269 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1286 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1303 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1320 #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 {#1337 #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] } 8 => App\Models\Task {#1354 #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] } 9 => App\Models\Task {#1369 #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 {#1386 #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" => 933940 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "158" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-158" "field_display_title" => "158" "field_folder" => "1" "field_image_name" => "158" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1168} "content" => Illuminate\Database\Eloquent\Collection {#1169} "next" => array:2 [ "refs" => "933941" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933939" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1178} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 745034 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "158" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-619" "field_display_title" => "619" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1121} "content" => Illuminate\Database\Eloquent\Collection {#1118} "next" => array:2 [ "refs" => "745035" "type" => "task" ] "previous" => array:2 [ "refs" => "745033" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1110} "page" => array:2 [ "refs" => "933940" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№619 (с. 158)
Условие. №619 (с. 158)
скриншот условия
619. При каких значениях k в равенстве a = kb, где b ≠ 0, векторы a и b:
а) коллинеарны;
б) сонаправлены;
в) противоположно направлены;
г) являются противоположными?
Решение 2. №619 (с. 158)
Решение 3. №619 (с. 158)
Решение 5. №619 (с. 158)
Решение 6. №619 (с. 158)
а) коллинеарны; Векторы $\vec{a}$ и $\vec{b}$ называются коллинеарными, если они лежат на одной прямой или на параллельных прямых. По определению, два ненулевых вектора коллинеарны тогда и только тогда, когда один из них можно выразить через другой путем умножения на некоторое число (скаляр). Равенство $\vec{a} = k\vec{b}$ как раз и является математической записью этого условия. Это означает, что для любого действительного числа $k$ вектор $\vec{a}$ будет коллинеарен вектору $\vec{b}$. Если $k=0$, то $\vec{a} = 0 \cdot \vec{b} = \vec{0}$. Нулевой вектор по определению считается коллинеарным любому вектору.
Ответ: $k$ — любое действительное число ($k \in \mathbb{R}$).
б) сонаправлены; Сонаправленные векторы ($\vec{a} \uparrow\uparrow \vec{b}$) — это коллинеарные векторы, имеющие одинаковое направление. При умножении вектора на положительный скаляр его направление не меняется. Следовательно, для того чтобы вектор $\vec{a}$ был сонаправлен с вектором $\vec{b}$ в равенстве $\vec{a} = k\vec{b}$, коэффициент $k$ должен быть положительным.
Ответ: $k > 0$.
в) противоположно направлены; Противоположно направленные векторы ($\vec{a} \uparrow\downarrow \vec{b}$) — это коллинеарные векторы, имеющие противоположные направления. При умножении вектора на отрицательный скаляр его направление меняется на противоположное. Следовательно, для того чтобы вектор $\vec{a}$ был противоположно направлен вектору $\vec{b}$ в равенстве $\vec{a} = k\vec{b}$, коэффициент $k$ должен быть отрицательным.
Ответ: $k < 0$.
г) являются противоположными? Противоположные векторы — это векторы, которые имеют равные длины (модули) и противоположные направления. Таким образом, должны выполняться два условия: 1. Направления векторов противоположны. Как мы выяснили в пункте (в), это означает, что $k < 0$. 2. Длины векторов равны: $|\vec{a}| = |\vec{b}|$. Найдем длину вектора $\vec{a}$ из исходного равенства: $|\vec{a}| = |k\vec{b}| = |k| \cdot |\vec{b}|$. Подставим это в условие равенства длин: $|k| \cdot |\vec{b}| = |\vec{b}|$. Поскольку по условию $\vec{b} \neq \vec{0}$, то и $|\vec{b}| \neq 0$. Значит, мы можем разделить обе части равенства на $|\vec{b}|$, получив $|k| = 1$. Уравнение $|k|=1$ имеет два решения: $k=1$ и $k=-1$. Выбираем то решение, которое удовлетворяет первому условию ($k < 0$). Этим решением является $k=-1$. При $k=-1$ равенство принимает вид $\vec{a} = -\vec{b}$, что полностью соответствует определению противоположных векторов.
Ответ: $k = -1$.
Другие задания:
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