Deprecated: Creation of dynamic property wpdb::$categories is deprecated in /home/calculadsa/www/wp-includes/wp-db.php on line 760

Deprecated: Creation of dynamic property wpdb::$post2cat is deprecated in /home/calculadsa/www/wp-includes/wp-db.php on line 760

Deprecated: Creation of dynamic property wpdb::$link2cat is deprecated in /home/calculadsa/www/wp-includes/wp-db.php on line 760

Deprecated: Using ${var} in strings is deprecated, use {$var} instead in /home/calculadsa/www/wp-includes/comment-template.php on line 1745

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Return type of Pimple\Container::offsetExists($id) should either be compatible with ArrayAccess::offsetExists(mixed $offset): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/vendor-prod/pimple/pimple/src/Pimple/Container.php on line 133

Deprecated: Return type of Pimple\Container::offsetGet($id) should either be compatible with ArrayAccess::offsetGet(mixed $offset): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/vendor-prod/pimple/pimple/src/Pimple/Container.php on line 98

Deprecated: Return type of Pimple\Container::offsetSet($id, $value) should either be compatible with ArrayAccess::offsetSet(mixed $offset, mixed $value): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/vendor-prod/pimple/pimple/src/Pimple/Container.php on line 79

Deprecated: Return type of Pimple\Container::offsetUnset($id) should either be compatible with ArrayAccess::offsetUnset(mixed $offset): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/vendor-prod/pimple/pimple/src/Pimple/Container.php on line 143

Deprecated: Return type of iThemesSecurity\Lib\Lockout\Execute_Lock\Context::offsetExists($offset) should either be compatible with ArrayAccess::offsetExists(mixed $offset): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/lockout/execute-lock/abstract-context.php on line 60

Deprecated: Return type of iThemesSecurity\Lib\Lockout\Execute_Lock\Context::offsetGet($offset) should either be compatible with ArrayAccess::offsetGet(mixed $offset): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/lockout/execute-lock/abstract-context.php on line 66

Deprecated: Return type of iThemesSecurity\Lib\Lockout\Execute_Lock\Context::offsetSet($offset, $value) should either be compatible with ArrayAccess::offsetSet(mixed $offset, mixed $value): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/lockout/execute-lock/abstract-context.php on line 72

Deprecated: Return type of iThemesSecurity\Lib\Lockout\Execute_Lock\Context::offsetUnset($offset) should either be compatible with ArrayAccess::offsetUnset(mixed $offset): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/lockout/execute-lock/abstract-context.php on line 76

Deprecated: Return type of iThemesSecurity\Lib\Lockout\Execute_Lock\Context::getIterator() should either be compatible with IteratorAggregate::getIterator(): Traversable, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/lockout/execute-lock/abstract-context.php on line 56

Deprecated: Return type of ITSEC_Lib_Distributed_Storage_Cursor::current() should either be compatible with Iterator::current(): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/class-itsec-lib-distributed-storage.php on line 578

Deprecated: Return type of ITSEC_Lib_Distributed_Storage_Cursor::next() should either be compatible with Iterator::next(): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/class-itsec-lib-distributed-storage.php on line 585

Deprecated: Return type of ITSEC_Lib_Distributed_Storage_Cursor::key() should either be compatible with Iterator::key(): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/class-itsec-lib-distributed-storage.php on line 606

Deprecated: Return type of ITSEC_Lib_Distributed_Storage_Cursor::valid() should either be compatible with Iterator::valid(): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/class-itsec-lib-distributed-storage.php on line 613

Deprecated: Return type of ITSEC_Lib_Distributed_Storage_Cursor::rewind() should either be compatible with Iterator::rewind(): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/lib/class-itsec-lib-distributed-storage.php on line 620

Deprecated: Using ${var} in strings is deprecated, use {$var} instead in /home/calculadsa/www/wp-content/plugins/contact-form-7/includes/mail.php on line 221

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Return type of WPGMZA\CustomFeatureFields::getIterator() should either be compatible with IteratorAggregate::getIterator(): Traversable, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/wp-google-maps-pro/includes/custom-fields/class.custom-map-object-fields.php on line 120

Deprecated: Return type of WPGMZA\CustomFeatureFields::jsonSerialize() should either be compatible with JsonSerializable::jsonSerialize(): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/wp-google-maps-pro/includes/custom-fields/class.custom-map-object-fields.php on line 129

Deprecated: Return type of WPGMZA\CustomFeatureFields::count() should either be compatible with Countable::count(): int, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/wp-google-maps-pro/includes/custom-fields/class.custom-map-object-fields.php on line 111

Deprecated: Return type of WPGMZA\CustomFields::getIterator() should either be compatible with IteratorAggregate::getIterator(): Traversable, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/wp-google-maps-pro/includes/custom-fields/class.custom-fields.php on line 121

Deprecated: Return type of WPGMZA\CustomFields::jsonSerialize() should either be compatible with JsonSerializable::jsonSerialize(): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/wp-google-maps-pro/includes/custom-fields/class.custom-fields.php on line 130

Deprecated: Return type of WPGMZA\CustomFields::count() should either be compatible with Countable::count(): int, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /home/calculadsa/www/wp-content/plugins/wp-google-maps-pro/includes/custom-fields/class.custom-fields.php on line 136

Deprecated: Using ${var} in strings is deprecated, use {$var} instead in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/modules/core/class-itsec-admin-notices.php on line 141

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property WPGMZA\ProDatabase::$version is deprecated in /home/calculadsa/www/wp-content/plugins/wp-google-maps-pro/includes/class.pro-database.php on line 12

Deprecated: Creation of dynamic property WPGMZA\ProDatabase::$charset_collate is deprecated in /home/calculadsa/www/wp-content/plugins/wp-google-maps-pro/includes/class.pro-database.php on line 13

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property WP_Rocket\Engine\Preload\SitemapPreloadSubscriber::$options is deprecated in /home/calculadsa/www/wp-content/plugins/wp-rocket/inc/Engine/Preload/SitemapPreloadSubscriber.php on line 26

Deprecated: Creation of dynamic property WP_Rocket\Engine\Preload\SitemapPreloadSubscriber::$sitemap_preload is deprecated in /home/calculadsa/www/wp-content/plugins/wp-rocket/inc/Engine/Preload/SitemapPreloadSubscriber.php on line 27

Deprecated: Automatic conversion of false to array is deprecated in /home/calculadsa/www/wp-content/plugins/wp-rocket/inc/3rd-party/plugins/seo/yoast-seo.php on line 10

Deprecated: Creation of dynamic property Yoast\WP\SEO\Main::$helpers is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/lib/abstract-main.php on line 65

Deprecated: trim(): Passing null to parameter #1 ($string) of type string is deprecated in /home/calculadsa/www/wp-content/plugins/better-wp-security/core/core.php on line 1092

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property WP_Block_Type::$skip_inner_blocks is deprecated in /home/calculadsa/www/wp-includes/class-wp-block-type.php on line 357

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Constant FILTER_SANITIZE_STRING is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/conditionals/third-party/elementor-edit-conditional.php on line 22

Deprecated: Constant FILTER_SANITIZE_STRING is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/conditionals/third-party/elementor-edit-conditional.php on line 28

Deprecated: Creation of dynamic property POMO_FileReader::$is_overloaded is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 26

Deprecated: Creation of dynamic property POMO_FileReader::$_pos is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 29

Deprecated: Creation of dynamic property POMO_FileReader::$_f is deprecated in /home/calculadsa/www/wp-includes/pomo/streams.php on line 160

Deprecated: Creation of dynamic property MO::$_gettext_select_plural_form is deprecated in /home/calculadsa/www/wp-includes/pomo/translations.php on line 293

Deprecated: Creation of dynamic property WP_Term::$object_id is deprecated in /home/calculadsa/www/wp-includes/class-wp-term.php on line 198

Deprecated: Creation of dynamic property WP_Term::$object_id is deprecated in /home/calculadsa/www/wp-includes/class-wp-term.php on line 198

Deprecated: Creation of dynamic property Yoast\WP\SEO\Surfaces\Classes_Surface::$container is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/surfaces/classes-surface.php on line 20

Deprecated: Creation of dynamic property Yoast\WP\SEO\Main::$classes is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/lib/abstract-main.php on line 65

Deprecated: Creation of dynamic property Yoast\WP\SEO\Context\Meta_Tags_Context::$page_type is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 43

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$robots is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$title is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$source is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64
Producto vectorial de dos vectores | Calculadora, fórmulas, ejercicios
Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$meta_description is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$canonical is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$rel_prev is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$rel_next is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$open_graph_locale is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$open_graph_type is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$open_graph_title is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$open_graph_description is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Presentations\Indexable_Post_Type_Presentation::$open_graph_url is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

Deprecated: Creation of dynamic property Yoast\WP\SEO\Context\Meta_Tags_Context::$wordpress_site_name is deprecated in /home/calculadsa/www/wp-content/plugins/wordpress-seo/src/presentations/abstract-presentation.php on line 64

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Producto vectorial

Resumen

Calcula el producto vectorial de dos vectores con nuestra calculadora. Te contamos la fórmula a usar, ejercicios resueltos, propiedades y más!

¿No sabes cómo calcular el producto vectorial de dos vectores? Usa nuestra calculadora online y averigua automáticamente el resultado, sin hacer operaciones.

Sólo tienes que introducir las componentes i, j y k de los dos vectores y pulsar el botón de calcular para obtener su producto vectorial. Por ejemplo, si te dan el vector Vector u= (1, 2, 3) y vector v= (4, 5, 6), los valores de cada una de sus componentes son los que debes escribir en la herramienta. Si no sabes cómo se calcula el producto vectorial, sigue leyendo y te lo explicamos.

¿Qué es el producto vectorial?

El producto vectorial de dos vectores nos da como resultado otro vector con:

  • Dirección perpendicular a los dos vectores
  • Sentido en base a la regla del sacacorchos o de la mano derecha.

Producto vectorial

¿Qué dice la regla de la mano derecha? La regla de la mano derecha o del sacacorchos nos dice el sentido que tendrá el vector según el movimiento de un sacacorchos (o destornillador, tornillo...etc).

En el caso de la figura que tenéis encima de estas líneas podéis ver que si hacemos girar el sacacorchos hacia la izquierda (sentido antihorario), el sacacorchos sube hacia arriba y, por lo tanto, esa será la dirección del vector "c" resultante.

Es muy importante que no lo confundas el vectorial con el producto escalar.

Cómo se hace el producto vectorial de dos vectores

Para resolver el producto vectorial de dos vectores Vector u= (a1, a2, a3) y vector v= (b1, b2, b3), tenemos que aplicar la siguiente fórmula que nos dará las componentes del vector "c":

Fórmula del producto vectorial de dos vectores

Básicamente hay que escribir en un determinante de 3x3 y descomponerlo en tres determinantes 2x2 para cada componente i, j, k.

No hace falta que recuerdes los elementos del determinante asociados a cada elemento ya que se pueden deducir de forma sencilla. Simplemente tienes que calcular el determinante adjunto, es decir, tienes que eliminar la fila y columna en la que se encuentra cada elemento i, j, k de tal forma que con los elementos restantes formas un determinante 2x2. Aquí te enseñamos cómo resolver un determinante de una matriz 2x2.

Por ejemplo, vamos a ver un ejercicio resuelto en el que nos piden calcular el producto vectorial de los vectores  Vector u= (2, 0, -1) y vector v= (1, 1, -2):

Ejercicio resuelto de producto vectorial de dos vectores

Como resultado del producto vectorial, tenemos este vector:

Resultado de ejercicio del producto vectorial de dos vectores

Existe un segundo método para calcular el producto vectorial de dos vectores a partir de la siguiente fórmula matemática:

Fórmula del producto vectorial de dos vectores

El método consiste en multiplicar sus módulos por el seno del ángulo que forman y el vector unitario "n" que es ortogonal a los vectores y cuyo sentido y dirección se rige por la regla de la mano derecha.

Fórmula del vector unitario

Si no sabes cómo se hace, aquí te explicamos cómo calcular el módulo de un vector.

Módulo del producto vectorial

Para calcular el módulo del producto vectorial de dos vectores tienes que usar esta fórmula:

Fórmula para calcular el módulo de un producto vectorial

En este caso, el resultado será un número escalar que nos representa la longitud que tendrá el segmento del vector resultante.

Producto vectorial de dos vectores en R2

Si nos piden calcular el producto vectorial de dos vectores en R2, el proceso a seguir es exactamente igual que en R3. Simplemente tendremos que completar con un 0 la componente del vector que nos falte.

Por ejemplo, vamos a calcular el producto vectorial de los vectores en R2 Vector u= (3, 2) y vector v= (2, -1).

Como puedes ver, no tenemos componente "k", así que los elementos a3 y b3 serán cero. Teniendo en cuenta este detalle, resolvemos el ejercicio y nos queda lo siguiente:

Ejemplo de producto vectorial de dos vectores en R2

Triple producto vectorial

El triple producto vectorial (también conocido como doble producto vectorial) consiste en multiplicar vectorialmente dos vectores y con el resultado obtenido realizar otro producto vectorial con el vector que nos falta. Por ejemplo:

A × (B × C) = B (A · C) – C (A · B)

Al multiplicar los vectores B x C generamos un vector que posteriormente tenemos que multiplicar vectorialmente por A. El resultado sería un vector contenido en el plano definido por B y C.

Como el doble producto vectorial tiene propiedad anticonmutativa, también podemos expresarlo así:

A × (B × C) = - C x (A x B)

Para terminar, destacar que el triple producto no tiene propiedad asociativa.

Propiedades del producto vectorial

A continuación tienes las propiedades del producto vectorial que aún no han sido mencionadas:

  • Anticonmutativa: a x b = - (b x a)
  • a · (a x b) = 0
  • Si a x b = 0 siendo a≠0 y b≠0, esto quiere decir que los vectores son paralelos y por eso su producto vectorial es cero.
  • (a + b) x c = a x c + b x c
  • a x (b x c) + c x (a x b) + b x (c x a) = 0 según la identidad de Jacobi

Aplicaciones del producto vectorial

Son muchas las aplicaciones que tenemos para el producto vectorial en el mundo de las matemáticas, física o astronomía.

Algunas de las que siempre se suelen poner como ejemplo y que nos permiten calcular el área y volumen de algunas figuras geométricas como el triángulo o un paralelogramo.


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Calculadora Conversor > Calculadoras matemáticas > Producto vectorial

Otras calculadoras y conversores que te pueden interesar:


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