gdz.top
Номер 646, страница 167 - 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 1. Координаты точки и координаты вектора - номер 646, страница 167.
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" => 745061 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "167" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-646" "field_display_title" => "646" "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 {#1079 #items: array:1 [ 0 => App\Models\Branch {#1034 #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 {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #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 {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1076 #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 {#1039} "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 {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #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 {#1120 #items: array:2 [ 0 => App\Models\Branch {#1128 #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 {#1132 #items: array:1 [ 0 => App\Models\Term {#1129 #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 {#1168 #items: array:1 [ 0 => App\Models\Book {#1131 #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 {#1133 #items: array:1 [ 0 => App\Models\Term {#1130 #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 {#1134 #items: array:2 [ 0 => App\Models\Term {#1135 #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 {#1136 #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 {#1137 #items: array:1 [ 0 => App\Models\Term {#1138 #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 {#1139 #items: array:5 [ 0 => App\Models\Term {#1140 #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 {#1141 #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 {#1142 #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 {#1143 #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 {#1144 #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 {#1145 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1147 #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 {#1148 #items: array:1 [ 0 => App\Models\Term {#1149 #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 {#1150 #items: array:1 [ 0 => App\Models\Term {#1151 #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 {#1152 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1153 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Term {#1155 #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 {#1156 #items: array:1 [ 0 => App\Models\Term {#1157 #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 {#1158 #items: array:1 [ 0 => App\Models\Term {#1159 #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 {#1160 #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 {#1161 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1163 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Term {#1165 #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 {#1133} "field_class" => Illuminate\Database\Eloquent\Collection {#1134} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137} "field_author" => Illuminate\Database\Eloquent\Collection {#1139} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1145} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_country" => Illuminate\Database\Eloquent\Collection {#1148} "field_city" => Illuminate\Database\Eloquent\Collection {#1150} "field_series" => Illuminate\Database\Eloquent\Collection {#1152} "field_umk" => Illuminate\Database\Eloquent\Collection {#1153} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1156} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1160} "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 {#1161} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1162} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1163} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1164} "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 {#1166 #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 {#1132} "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 {#1168} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166} ] #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 {#1167 #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" => 744406 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Координаты точки и координаты вектора" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Term {#1169 #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" => "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 {#1171 #items: array:1 [ 0 => App\Models\Book {#1131} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744406 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Координаты точки и координаты вектора" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1172} "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 {#1171} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170} ] #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 {#1077 #items: array:6 [ 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" => 939424 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1105 #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 {#1104 …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 {#1103 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1101 …2} …3 ] #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" => "745061" "type" => "task" ] "text" => """ <p><strong>646.</strong> Даны векторы</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/646.webp?ts=1745751927" alt="Упражнение 646 найти координаты векторов" loading="lazy" width="1076" height="179">\n </figure>\n <figcaption> </figcaption></figure><p>Найдите координаты векторов:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/646_0.webp?ts=1745751927" alt="Упражнение 646 найти координаты векторов" loading="lazy" width="1402" height="229">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "646-1.jpg" "alt" => null "width" => "1557" "height" => 312 "path" => "/media/geometrija_10/atanasjan-u23/0-00/646-1.webp?ts=1739436793" ] ] ] #original: array:7 [ "id" => 939424 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "745061" "type" => "task" ] "text" => """ <p><strong>646.</strong> Даны векторы</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/646.webp?ts=1745751927" alt="Упражнение 646 найти координаты векторов" loading="lazy" width="1076" height="179">\n </figure>\n <figcaption> </figcaption></figure><p>Найдите координаты векторов:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/646_0.webp?ts=1745751927" alt="Упражнение 646 найти координаты векторов" loading="lazy" width="1402" height="229">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "646-1.jpg" "alt" => null "width" => "1557" "height" => 312 "path" => "/media/geometrija_10/atanasjan-u23/0-00/646-1.webp?ts=1739436793" ] ] ] #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 {#1121 #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" => 942395 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1180 #items: array:1 [ 0 => App\Models\Edition {#1124 #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" => "745061" "type" => "task" ] "img" => array:12 [ 0 => array:5 [ "name" => "646-1.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-1.webp?ts=1739441519" ] 1 => array:5 [ "name" => "646-2.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-2.webp?ts=1739441519" ] 2 => array:5 [ "name" => "646-3.png" "alt" => null "width" => "694" "height" => 1608 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-3.webp?ts=1739441519" ] 3 => array:5 [ "name" => "646-4.png" "alt" => null "width" => "694" "height" => 1648 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-4.webp?ts=1739441519" ] 4 => array:5 [ "name" => "646-5.png" "alt" => null "width" => "694" "height" => 1648 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-5.webp?ts=1739441519" ] 5 => array:5 [ "name" => "646-6.png" "alt" => null "width" => "694" "height" => 1738 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-6.webp?ts=1739441519" ] 6 => array:5 [ "name" => "646-7.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-7.webp?ts=1739441519" ] 7 => array:5 [ "name" => "646-8.png" "alt" => null "width" => "694" "height" => 1608 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-8.webp?ts=1739441519" ] 8 => array:5 [ "name" => "646-9.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-9.webp?ts=1739441519" ] 9 => array:5 [ "name" => "646-10.png" "alt" => null "width" => "694" "height" => 1608 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-10.webp?ts=1739441519" ] 10 => array:5 [ "name" => "646-11.png" "alt" => null "width" => "694" "height" => 1662 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-11.webp?ts=1739441519" ] 11 => array:5 [ "name" => "646-12.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-12.webp?ts=1739441519" ] ] ] #original: array:6 [ "id" => 942395 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1180} "task" => array:2 [ "refs" => "745061" "type" => "task" ] "img" => array:12 [ 0 => array:5 [ "name" => "646-1.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-1.webp?ts=1739441519" ] 1 => array:5 [ "name" => "646-2.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-2.webp?ts=1739441519" ] 2 => array:5 [ "name" => "646-3.png" "alt" => null "width" => "694" "height" => 1608 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-3.webp?ts=1739441519" ] 3 => array:5 [ "name" => "646-4.png" "alt" => null "width" => "694" "height" => 1648 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-4.webp?ts=1739441519" ] 4 => array:5 [ "name" => "646-5.png" "alt" => null "width" => "694" "height" => 1648 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-5.webp?ts=1739441519" ] 5 => array:5 [ "name" => "646-6.png" "alt" => null "width" => "694" "height" => 1738 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-6.webp?ts=1739441519" ] 6 => array:5 [ "name" => "646-7.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-7.webp?ts=1739441519" ] 7 => array:5 [ "name" => "646-8.png" "alt" => null "width" => "694" "height" => 1608 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-8.webp?ts=1739441519" ] 8 => array:5 [ "name" => "646-9.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-9.webp?ts=1739441519" ] 9 => array:5 [ "name" => "646-10.png" "alt" => null "width" => "694" "height" => 1608 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-10.webp?ts=1739441519" ] 10 => array:5 [ "name" => "646-11.png" "alt" => null "width" => "694" "height" => 1662 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-11.webp?ts=1739441519" ] 11 => array:5 [ "name" => "646-12.png" "alt" => null "width" => "694" "height" => 1620 "path" => "/media/geometrija_10/atanasjan-u23/2-00/646-12.webp?ts=1739441519" ] ] ] #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 {#1083 #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" => 942287 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1084 #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" => "745061" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "646-1.jpg" "alt" => null "width" => "1320" "height" => 356 "path" => "/media/geometrija_10/atanasjan-u23/3-00/646-1.webp?ts=1739437100" ] 1 => array:5 [ "name" => "646-2.jpg" "alt" => null "width" => "1320" "height" => 1650 "path" => "/media/geometrija_10/atanasjan-u23/3-00/646-2.webp?ts=1739437100" ] ] ] #original: array:6 [ "id" => 942287 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "745061" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "646-1.jpg" "alt" => null "width" => "1320" "height" => 356 "path" => "/media/geometrija_10/atanasjan-u23/3-00/646-1.webp?ts=1739437100" ] 1 => array:5 [ "name" => "646-2.jpg" "alt" => null "width" => "1320" "height" => 1650 "path" => "/media/geometrija_10/atanasjan-u23/3-00/646-2.webp?ts=1739437100" ] ] ] #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 {#1099 #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" => 941655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #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 [ …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" => "745061" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "646-1.jpg" "alt" => null "width" => "1386" "height" => 954 "path" => "/media/geometrija_10/atanasjan-u23/4-00/646-1.webp?ts=1739436673" ] 1 => array:5 [ "name" => "646-2.jpg" "alt" => null "width" => "1386" "height" => 245 "path" => "/media/geometrija_10/atanasjan-u23/4-00/646-2.webp?ts=1739436673" ] ] ] #original: array:6 [ "id" => 941655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "745061" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "646-1.jpg" "alt" => null "width" => "1386" "height" => 954 "path" => "/media/geometrija_10/atanasjan-u23/4-00/646-1.webp?ts=1739436673" ] 1 => array:5 [ "name" => "646-2.jpg" "alt" => null "width" => "1386" "height" => 245 "path" => "/media/geometrija_10/atanasjan-u23/4-00/646-2.webp?ts=1739436673" ] ] ] #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 {#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" => 941808 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Edition {#1092 #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" => "745061" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "646-1.jpg" "alt" => null "width" => "416" "height" => 470 "path" => "/media/geometrija_10/atanasjan-u23/5-00/646-1.webp?ts=1739436689" ] ] ] #original: array:6 [ "id" => 941808 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1080} "task" => array:2 [ "refs" => "745061" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "646-1.jpg" "alt" => null "width" => "416" "height" => 470 "path" => "/media/geometrija_10/atanasjan-u23/5-00/646-1.webp?ts=1739436689" ] ] ] #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 {#1178 #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" => 1352374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ 0 => App\Models\Edition {#1179 #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" => "745061" "type" => "task" ] "text" => "<p>Для выполнения операций с векторами необходимо производить соответствующие арифметические действия над их координатами. Исходные данные векторов:</p><p>$\vec{a}\{5; -1; 1\}$</p><p>$\vec{b}\{-2; 1; 0\}$</p><p>$\vec{c}\{0; 0,2; 0\}$</p><p>$\vec{d}\{\frac{1}{3}; 2\frac{2}{5}; -\frac{1}{7}\}$</p><p><strong>а)</strong> Чтобы найти координаты вектора $\vec{a} - \vec{b}$, нужно из координат вектора $\vec{a}$ вычесть соответствующие координаты вектора $\vec{b}$.</p><p>$\vec{a} - \vec{b} = \{5 - (-2); -1 - 1; 1 - 0\} = \{5 + 2; -2; 1\} = \{7; -2; 1\}$.</p><p>Ответ: $\{7; -2; 1\}$.</p><p><strong>б)</strong> Чтобы найти координаты вектора $\vec{b} - \vec{a}$, нужно из координат вектора $\vec{b}$ вычесть соответствующие координаты вектора $\vec{a}$.</p><p>$\vec{b} - \vec{a} = \{-2 - 5; 1 - (-1); 0 - 1\} = \{-7; 1 + 1; -1\} = \{-7; 2; -1\}$.</p><p>Ответ: $\{-7; 2; -1\}$.</p><p><strong>в)</strong> Вычтем из координат вектора $\vec{a}$ соответствующие координаты вектора $\vec{c}$.</p><p>$\vec{a} - \vec{c} = \{5 - 0; -1 - 0,2; 1 - 0\} = \{5; -1,2; 1\}$.</p><p>Ответ: $\{5; -1,2; 1\}$.</p><p><strong>г)</strong> Для нахождения разности $\vec{d} - \vec{a}$ сначала преобразуем смешанную дробь в координатах вектора $\vec{d}$ в неправильную: $2\frac{2}{5} = \frac{12}{5}$. Таким образом, $\vec{d}\{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\}$.</p><p>$\vec{d} - \vec{a} = \{\frac{1}{3} - 5; \frac{12}{5} - (-1); -\frac{1}{7} - 1\} = \{\frac{1}{3} - \frac{15}{3}; \frac{12}{5} + \frac{5}{5}; -\frac{1}{7} - \frac{7}{7}\} = \{-\frac{14}{3}; \frac{17}{5}; -\frac{8}{7}\}$.</p><p>Ответ: $\{-\frac{14}{3}; \frac{17}{5}; -\frac{8}{7}\}$.</p><p><strong>д)</strong> Для нахождения разности $\vec{c} - \vec{d}$ вычтем из координат вектора $\vec{c}$ координаты вектора $\vec{d}$, используя $\vec{d}\{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\}$.</p><p>$\vec{c} - \vec{d} = \{0 - \frac{1}{3}; 0,2 - \frac{12}{5}; 0 - (-\frac{1}{7})\}$. Преобразуем $0,2$ в дробь: $0,2 = \frac{1}{5}$.</p><p>$\{0 - \frac{1}{3}; \frac{1}{5} - \frac{12}{5}; 0 + \frac{1}{7}\} = \{-\frac{1}{3}; -\frac{11}{5}; \frac{1}{7}\}$.</p><p>Ответ: $\{-\frac{1}{3}; -\frac{11}{5}; \frac{1}{7}\}$.</p><p><strong>е)</strong> Для нахождения координат вектора $\vec{a} - \vec{b} + \vec{c}$ выполним соответствующие операции над координатами.</p><p>$\vec{a} - \vec{b} + \vec{c} = \{5 - (-2) + 0; -1 - 1 + 0,2; 1 - 0 + 0\} = \{7; -2 + 0,2; 1\} = \{7; -1,8; 1\}$.</p><p>Ответ: $\{7; -1,8; 1\}$.</p><p><strong>ж)</strong> Для нахождения координат вектора $\vec{a} - \vec{b} - \vec{c}$ выполним соответствующие вычитания над координатами.</p><p>$\vec{a} - \vec{b} - \vec{c} = \{5 - (-2) - 0; -1 - 1 - 0,2; 1 - 0 - 0\} = \{7; -2 - 0,2; 1\} = \{7; -2,2; 1\}$.</p><p>Ответ: $\{7; -2,2; 1\}$.</p><p><strong>з)</strong> Для умножения вектора на число (скаляр) необходимо каждую координату вектора умножить на это число.</p><p>$2\vec{a} = 2 \cdot \{5; -1; 1\} = \{2 \cdot 5; 2 \cdot (-1); 2 \cdot 1\} = \{10; -2; 2\}$.</p><p>Ответ: $\{10; -2; 2\}$.</p><p><strong>и)</strong> Умножим каждую координату вектора $\vec{b}$ на скаляр $-3$.</p><p>$-3\vec{b} = -3 \cdot \{-2; 1; 0\} = \{-3 \cdot (-2); -3 \cdot 1; -3 \cdot 0\} = \{6; -3; 0\}$.</p><p>Ответ: $\{6; -3; 0\}$.</p><p><strong>к)</strong> Умножим каждую координату вектора $\vec{c}$ на скаляр $-6$.</p><p>$-6\vec{c} = -6 \cdot \{0; 0,2; 0\} = \{-6 \cdot 0; -6 \cdot 0,2; -6 \cdot 0\} = \{0; -1,2; 0\}$.</p><p>Ответ: $\{0; -1,2; 0\}$.</p><p><strong>л)</strong> Умножим каждую координату вектора $\vec{d}$ на скаляр $-\frac{1}{3}$, используя $\vec{d}\{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\}$.</p><p>$-\frac{1}{3}\vec{d} = -\frac{1}{3} \cdot \{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\} = \{-\frac{1}{3} \cdot \frac{1}{3}; -\frac{1}{3} \cdot \frac{12}{5}; -\frac{1}{3} \cdot (-\frac{1}{7})\} = \{-\frac{1}{9}; -\frac{12}{15}; \frac{1}{21}\}$.</p><p>Сократим вторую координату: $-\frac{12}{15} = -\frac{4}{5}$.</p><p>Результат: $\{-\frac{1}{9}; -\frac{4}{5}; \frac{1}{21}\}$.</p><p>Ответ: $\{-\frac{1}{9}; -\frac{4}{5}; \frac{1}{21}\}$.</p><p><strong>м)</strong> Умножим каждую координату вектора $\vec{b}$ на скаляр $0,2$.</p><p>$0,2\vec{b} = 0,2 \cdot \{-2; 1; 0\} = \{0,2 \cdot (-2); 0,2 \cdot 1; 0,2 \cdot 0\} = \{-0,4; 0,2; 0\}$.</p><p>Ответ: $\{-0,4; 0,2; 0\}$.</p>" ] #original: array:6 [ "id" => 1352374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188} "task" => array:2 [ "refs" => "745061" "type" => "task" ] "text" => "<p>Для выполнения операций с векторами необходимо производить соответствующие арифметические действия над их координатами. Исходные данные векторов:</p><p>$\vec{a}\{5; -1; 1\}$</p><p>$\vec{b}\{-2; 1; 0\}$</p><p>$\vec{c}\{0; 0,2; 0\}$</p><p>$\vec{d}\{\frac{1}{3}; 2\frac{2}{5}; -\frac{1}{7}\}$</p><p><strong>а)</strong> Чтобы найти координаты вектора $\vec{a} - \vec{b}$, нужно из координат вектора $\vec{a}$ вычесть соответствующие координаты вектора $\vec{b}$.</p><p>$\vec{a} - \vec{b} = \{5 - (-2); -1 - 1; 1 - 0\} = \{5 + 2; -2; 1\} = \{7; -2; 1\}$.</p><p>Ответ: $\{7; -2; 1\}$.</p><p><strong>б)</strong> Чтобы найти координаты вектора $\vec{b} - \vec{a}$, нужно из координат вектора $\vec{b}$ вычесть соответствующие координаты вектора $\vec{a}$.</p><p>$\vec{b} - \vec{a} = \{-2 - 5; 1 - (-1); 0 - 1\} = \{-7; 1 + 1; -1\} = \{-7; 2; -1\}$.</p><p>Ответ: $\{-7; 2; -1\}$.</p><p><strong>в)</strong> Вычтем из координат вектора $\vec{a}$ соответствующие координаты вектора $\vec{c}$.</p><p>$\vec{a} - \vec{c} = \{5 - 0; -1 - 0,2; 1 - 0\} = \{5; -1,2; 1\}$.</p><p>Ответ: $\{5; -1,2; 1\}$.</p><p><strong>г)</strong> Для нахождения разности $\vec{d} - \vec{a}$ сначала преобразуем смешанную дробь в координатах вектора $\vec{d}$ в неправильную: $2\frac{2}{5} = \frac{12}{5}$. Таким образом, $\vec{d}\{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\}$.</p><p>$\vec{d} - \vec{a} = \{\frac{1}{3} - 5; \frac{12}{5} - (-1); -\frac{1}{7} - 1\} = \{\frac{1}{3} - \frac{15}{3}; \frac{12}{5} + \frac{5}{5}; -\frac{1}{7} - \frac{7}{7}\} = \{-\frac{14}{3}; \frac{17}{5}; -\frac{8}{7}\}$.</p><p>Ответ: $\{-\frac{14}{3}; \frac{17}{5}; -\frac{8}{7}\}$.</p><p><strong>д)</strong> Для нахождения разности $\vec{c} - \vec{d}$ вычтем из координат вектора $\vec{c}$ координаты вектора $\vec{d}$, используя $\vec{d}\{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\}$.</p><p>$\vec{c} - \vec{d} = \{0 - \frac{1}{3}; 0,2 - \frac{12}{5}; 0 - (-\frac{1}{7})\}$. Преобразуем $0,2$ в дробь: $0,2 = \frac{1}{5}$.</p><p>$\{0 - \frac{1}{3}; \frac{1}{5} - \frac{12}{5}; 0 + \frac{1}{7}\} = \{-\frac{1}{3}; -\frac{11}{5}; \frac{1}{7}\}$.</p><p>Ответ: $\{-\frac{1}{3}; -\frac{11}{5}; \frac{1}{7}\}$.</p><p><strong>е)</strong> Для нахождения координат вектора $\vec{a} - \vec{b} + \vec{c}$ выполним соответствующие операции над координатами.</p><p>$\vec{a} - \vec{b} + \vec{c} = \{5 - (-2) + 0; -1 - 1 + 0,2; 1 - 0 + 0\} = \{7; -2 + 0,2; 1\} = \{7; -1,8; 1\}$.</p><p>Ответ: $\{7; -1,8; 1\}$.</p><p><strong>ж)</strong> Для нахождения координат вектора $\vec{a} - \vec{b} - \vec{c}$ выполним соответствующие вычитания над координатами.</p><p>$\vec{a} - \vec{b} - \vec{c} = \{5 - (-2) - 0; -1 - 1 - 0,2; 1 - 0 - 0\} = \{7; -2 - 0,2; 1\} = \{7; -2,2; 1\}$.</p><p>Ответ: $\{7; -2,2; 1\}$.</p><p><strong>з)</strong> Для умножения вектора на число (скаляр) необходимо каждую координату вектора умножить на это число.</p><p>$2\vec{a} = 2 \cdot \{5; -1; 1\} = \{2 \cdot 5; 2 \cdot (-1); 2 \cdot 1\} = \{10; -2; 2\}$.</p><p>Ответ: $\{10; -2; 2\}$.</p><p><strong>и)</strong> Умножим каждую координату вектора $\vec{b}$ на скаляр $-3$.</p><p>$-3\vec{b} = -3 \cdot \{-2; 1; 0\} = \{-3 \cdot (-2); -3 \cdot 1; -3 \cdot 0\} = \{6; -3; 0\}$.</p><p>Ответ: $\{6; -3; 0\}$.</p><p><strong>к)</strong> Умножим каждую координату вектора $\vec{c}$ на скаляр $-6$.</p><p>$-6\vec{c} = -6 \cdot \{0; 0,2; 0\} = \{-6 \cdot 0; -6 \cdot 0,2; -6 \cdot 0\} = \{0; -1,2; 0\}$.</p><p>Ответ: $\{0; -1,2; 0\}$.</p><p><strong>л)</strong> Умножим каждую координату вектора $\vec{d}$ на скаляр $-\frac{1}{3}$, используя $\vec{d}\{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\}$.</p><p>$-\frac{1}{3}\vec{d} = -\frac{1}{3} \cdot \{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\} = \{-\frac{1}{3} \cdot \frac{1}{3}; -\frac{1}{3} \cdot \frac{12}{5}; -\frac{1}{3} \cdot (-\frac{1}{7})\} = \{-\frac{1}{9}; -\frac{12}{15}; \frac{1}{21}\}$.</p><p>Сократим вторую координату: $-\frac{12}{15} = -\frac{4}{5}$.</p><p>Результат: $\{-\frac{1}{9}; -\frac{4}{5}; \frac{1}{21}\}$.</p><p>Ответ: $\{-\frac{1}{9}; -\frac{4}{5}; \frac{1}{21}\}$.</p><p><strong>м)</strong> Умножим каждую координату вектора $\vec{b}$ на скаляр $0,2$.</p><p>$0,2\vec{b} = 0,2 \cdot \{-2; 1; 0\} = \{0,2 \cdot (-2); 0,2 \cdot 1; 0,2 \cdot 0\} = \{-0,4; 0,2; 0\}$.</p><p>Ответ: $\{-0,4; 0,2; 0\}$.</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" => "745062" "type" => "task" ] "previous" => array:2 [ "refs" => "745060" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1131} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\BookPage {#1114 #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" => 933949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "167" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-167" "field_display_title" => "167" "field_folder" => "1" "field_image_name" => "167" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1115 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Book {#1131} ] #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 {#1116 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Element {#1194 #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" => "933950" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933948" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1333 #items: array:10 [ 0 => App\Models\Task {#1349 #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 {#1356 #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 {#1359 #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 {#1358 #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 {#1360 #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 {#1380 #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 {#1400 #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 {#1420 #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 {#1483 #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 {#1486 #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" => 933949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "167" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-167" "field_display_title" => "167" "field_folder" => "1" "field_image_name" => "167" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1115} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1081} "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 {#1116} "content" => Illuminate\Database\Eloquent\Collection {#1195} "next" => array:2 [ "refs" => "933950" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933948" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1333} ] #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" => 745061 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "167" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-646" "field_display_title" => "646" "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 {#1079} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1120} "content" => Illuminate\Database\Eloquent\Collection {#1077} "next" => array:2 [ "refs" => "745062" "type" => "task" ] "previous" => array:2 [ "refs" => "745060" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1110} "page" => array:2 [ "refs" => "933949" "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 => "*" ] }
№646 (с. 167)
Условие. №646 (с. 167)
скриншот условия
646. Даны векторы
Найдите координаты векторов:
Решение 2. №646 (с. 167)
Решение 3. №646 (с. 167)
Решение 4. №646 (с. 167)
Решение 5. №646 (с. 167)
Решение 6. №646 (с. 167)
Для выполнения операций с векторами необходимо производить соответствующие арифметические действия над их координатами. Исходные данные векторов:
$\vec{a}\{5; -1; 1\}$
$\vec{b}\{-2; 1; 0\}$
$\vec{c}\{0; 0,2; 0\}$
$\vec{d}\{\frac{1}{3}; 2\frac{2}{5}; -\frac{1}{7}\}$
а) Чтобы найти координаты вектора $\vec{a} - \vec{b}$, нужно из координат вектора $\vec{a}$ вычесть соответствующие координаты вектора $\vec{b}$.
$\vec{a} - \vec{b} = \{5 - (-2); -1 - 1; 1 - 0\} = \{5 + 2; -2; 1\} = \{7; -2; 1\}$.
Ответ: $\{7; -2; 1\}$.
б) Чтобы найти координаты вектора $\vec{b} - \vec{a}$, нужно из координат вектора $\vec{b}$ вычесть соответствующие координаты вектора $\vec{a}$.
$\vec{b} - \vec{a} = \{-2 - 5; 1 - (-1); 0 - 1\} = \{-7; 1 + 1; -1\} = \{-7; 2; -1\}$.
Ответ: $\{-7; 2; -1\}$.
в) Вычтем из координат вектора $\vec{a}$ соответствующие координаты вектора $\vec{c}$.
$\vec{a} - \vec{c} = \{5 - 0; -1 - 0,2; 1 - 0\} = \{5; -1,2; 1\}$.
Ответ: $\{5; -1,2; 1\}$.
г) Для нахождения разности $\vec{d} - \vec{a}$ сначала преобразуем смешанную дробь в координатах вектора $\vec{d}$ в неправильную: $2\frac{2}{5} = \frac{12}{5}$. Таким образом, $\vec{d}\{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\}$.
$\vec{d} - \vec{a} = \{\frac{1}{3} - 5; \frac{12}{5} - (-1); -\frac{1}{7} - 1\} = \{\frac{1}{3} - \frac{15}{3}; \frac{12}{5} + \frac{5}{5}; -\frac{1}{7} - \frac{7}{7}\} = \{-\frac{14}{3}; \frac{17}{5}; -\frac{8}{7}\}$.
Ответ: $\{-\frac{14}{3}; \frac{17}{5}; -\frac{8}{7}\}$.
д) Для нахождения разности $\vec{c} - \vec{d}$ вычтем из координат вектора $\vec{c}$ координаты вектора $\vec{d}$, используя $\vec{d}\{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\}$.
$\vec{c} - \vec{d} = \{0 - \frac{1}{3}; 0,2 - \frac{12}{5}; 0 - (-\frac{1}{7})\}$. Преобразуем $0,2$ в дробь: $0,2 = \frac{1}{5}$.
$\{0 - \frac{1}{3}; \frac{1}{5} - \frac{12}{5}; 0 + \frac{1}{7}\} = \{-\frac{1}{3}; -\frac{11}{5}; \frac{1}{7}\}$.
Ответ: $\{-\frac{1}{3}; -\frac{11}{5}; \frac{1}{7}\}$.
е) Для нахождения координат вектора $\vec{a} - \vec{b} + \vec{c}$ выполним соответствующие операции над координатами.
$\vec{a} - \vec{b} + \vec{c} = \{5 - (-2) + 0; -1 - 1 + 0,2; 1 - 0 + 0\} = \{7; -2 + 0,2; 1\} = \{7; -1,8; 1\}$.
Ответ: $\{7; -1,8; 1\}$.
ж) Для нахождения координат вектора $\vec{a} - \vec{b} - \vec{c}$ выполним соответствующие вычитания над координатами.
$\vec{a} - \vec{b} - \vec{c} = \{5 - (-2) - 0; -1 - 1 - 0,2; 1 - 0 - 0\} = \{7; -2 - 0,2; 1\} = \{7; -2,2; 1\}$.
Ответ: $\{7; -2,2; 1\}$.
з) Для умножения вектора на число (скаляр) необходимо каждую координату вектора умножить на это число.
$2\vec{a} = 2 \cdot \{5; -1; 1\} = \{2 \cdot 5; 2 \cdot (-1); 2 \cdot 1\} = \{10; -2; 2\}$.
Ответ: $\{10; -2; 2\}$.
и) Умножим каждую координату вектора $\vec{b}$ на скаляр $-3$.
$-3\vec{b} = -3 \cdot \{-2; 1; 0\} = \{-3 \cdot (-2); -3 \cdot 1; -3 \cdot 0\} = \{6; -3; 0\}$.
Ответ: $\{6; -3; 0\}$.
к) Умножим каждую координату вектора $\vec{c}$ на скаляр $-6$.
$-6\vec{c} = -6 \cdot \{0; 0,2; 0\} = \{-6 \cdot 0; -6 \cdot 0,2; -6 \cdot 0\} = \{0; -1,2; 0\}$.
Ответ: $\{0; -1,2; 0\}$.
л) Умножим каждую координату вектора $\vec{d}$ на скаляр $-\frac{1}{3}$, используя $\vec{d}\{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\}$.
$-\frac{1}{3}\vec{d} = -\frac{1}{3} \cdot \{\frac{1}{3}; \frac{12}{5}; -\frac{1}{7}\} = \{-\frac{1}{3} \cdot \frac{1}{3}; -\frac{1}{3} \cdot \frac{12}{5}; -\frac{1}{3} \cdot (-\frac{1}{7})\} = \{-\frac{1}{9}; -\frac{12}{15}; \frac{1}{21}\}$.
Сократим вторую координату: $-\frac{12}{15} = -\frac{4}{5}$.
Результат: $\{-\frac{1}{9}; -\frac{4}{5}; \frac{1}{21}\}$.
Ответ: $\{-\frac{1}{9}; -\frac{4}{5}; \frac{1}{21}\}$.
м) Умножим каждую координату вектора $\vec{b}$ на скаляр $0,2$.
$0,2\vec{b} = 0,2 \cdot \{-2; 1; 0\} = \{0,2 \cdot (-2); 0,2 \cdot 1; 0,2 \cdot 0\} = \{-0,4; 0,2; 0\}$.
Ответ: $\{-0,4; 0,2; 0\}$.
Другие задания:
Помогло решение? Оставьте отзыв в комментариях ниже.
Присоединяйтесь к Телеграм-группе @top_gdz
ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по геометрии за 10-11 класс, для упражнения номер 646 расположенного на странице 167 к учебнику 2019 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по геометрии к упражнению №646 (с. 167), авторов: Атанасян (Левон Сергеевич), Бутузов (Валентин Фёдорович), Кадомцев (Сергей Борисович), Позняк (Эдуард Генрихович), Киселёва (Людмила Сергеевна), ФГОС (старый) базовый и углублённый уровень обучения учебного пособия издательства Просвещение.