Яссылык (математика)

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decoding=\"async\" width=\"12\" height=\"11\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/WikiProject_Council_with_transparent_background.svg/18px-WikiProject_Council_with_transparent_background.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/WikiProject_Council_with_transparent_background.svg/24px-WikiProject_Council_with_transparent_background.svg.png 2x\" data-file-width=\"512\" data-file-height=\"459\" /\u003E\u003C/a\u003E\u003C/span\u003E \u003Cb\u003E\u003Ca href=\"/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D1%80%D0%BE%D0%B5%D0%BA%D1%82%D0%BB%D0%B0%D1%80\" title=\"Википедия:Проектлар\"\u003EПроектлар\u003C/a\u003E\u003C/b\u003E: \u003Cspan typeof=\"mw:File\"\u003E\u003Ca href=\"/wiki/2024_%D0%B5%D0%BB#Вафатлар\" title=\"2024 ел\"\u003E\u003Cimg src=\"//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Kerze.png/5px-Kerze.png\" decoding=\"async\" width=\"5\" height=\"19\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Kerze.png/8px-Kerze.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Kerze.png/10px-Kerze.png 2x\" data-file-width=\"218\" data-file-height=\"822\" /\u003E\u003C/a\u003E\u003C/span\u003E \u003Cu\u003E\u003Ci\u003E\u003Ca href=\"/wiki/2024_%D0%B5%D0%BB#Вафатлар\" title=\"2024 ел\"\u003EХәтерТактасы\u003C/a\u003E\u003C/i\u003E\u003C/u\u003E \u003Cspan typeof=\"mw:File\"\u003E\u003Ca href=\"/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%90%D0%BA%D1%8B%D0%BB%D0%BB%D1%8B_%D0%A2%D0%B0%D1%82%D0%B0%D1%80%D1%81%D1%82%D0%B0%D0%BD/%D0%95%D0%BB_%D0%BF%D1%80%D0%BE%D0%B5%D0%BA%D1%82%D1%8B\" title=\"Википедия:Акыллы Татарстан/Ел проекты\"\u003E\u003Cimg src=\"//upload.wikimedia.org/wikipedia/commons/thumb/5/56/2021TatarstanYearLogo-1.jpg/12px-2021TatarstanYearLogo-1.jpg\" decoding=\"async\" width=\"12\" height=\"12\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/5/56/2021TatarstanYearLogo-1.jpg/18px-2021TatarstanYearLogo-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/2021TatarstanYearLogo-1.jpg/24px-2021TatarstanYearLogo-1.jpg 2x\" data-file-width=\"1626\" data-file-height=\"1626\" /\u003E\u003C/a\u003E\u003C/span\u003E \u003Ca href=\"/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%90%D0%BA%D1%8B%D0%BB%D0%BB%D1%8B_%D0%A2%D0%B0%D1%82%D0%B0%D1%80%D1%81%D1%82%D0%B0%D0%BD/%D0%95%D0%BB_%D0%BF%D1%80%D0%BE%D0%B5%D0%BA%D1%82%D1%8B\" title=\"Википедия:Акыллы Татарстан/Ел проекты\"\u003EЕлПроекты\u003C/a\u003E \u003Cspan typeof=\"mw:File\"\u003E\u003Ca href=\"/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9C%D0%B0%D1%80%D0%B0%D1%84%D0%BE%D0%BD%D0%BD%D0%B0%D1%80/%D0%90%D0%B3%D1%8B%D0%BC%D0%B4%D0%B0%D0%B3%D1%8B%D0%BB%D0%B0%D1%80\" title=\"Википедия:Марафоннар/Агымдагылар\"\u003E\u003Cimg src=\"//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Olympic_pictogram_Athletics.png/12px-Olympic_pictogram_Athletics.png\" decoding=\"async\" width=\"12\" height=\"12\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Olympic_pictogram_Athletics.png/18px-Olympic_pictogram_Athletics.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Olympic_pictogram_Athletics.png/24px-Olympic_pictogram_Athletics.png 2x\" data-file-width=\"300\" data-file-height=\"300\" /\u003E\u003C/a\u003E\u003C/span\u003E \u003Cb\u003E\u003Ca href=\"/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9C%D0%B0%D1%80%D0%B0%D1%84%D0%BE%D0%BD%D0%BD%D0%B0%D1%80/%D0%90%D0%B3%D1%8B%D0%BC%D0%B4%D0%B0%D0%B3%D1%8B%D0%BB%D0%B0%D1%80\" title=\"Википедия:Марафоннар/Агымдагылар\"\u003EАгымдагыМарафоннар\u003C/a\u003E\u003C/b\u003E: \u003Ci\u003E\u003Cspan typeof=\"mw:File\"\u003E\u003Ca href=\"/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D1%80%D0%BE%D0%B5%D0%BA%D1%82:%D0%A2%D3%A9%D0%B1%D3%99%D0%BA_%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA_%D0%B0%D1%82%D0%BD%D0%B0%D0%BB%D1%8B%D0%B3%D1%8B\" title=\"Википедия:Проект:Төбәк тематик атналыгы\"\u003E\u003Cimg src=\"//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/World_Map_%28political%29.svg/20px-World_Map_%28political%29.svg.png\" decoding=\"async\" width=\"20\" height=\"10\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/World_Map_%28political%29.svg/30px-World_Map_%28political%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/World_Map_%28political%29.svg/40px-World_Map_%28political%29.svg.png 2x\" data-file-width=\"1920\" data-file-height=\"975\" /\u003E\u003C/a\u003E\u003C/span\u003E \u003Cu\u003E\u003Ca href=\"/wiki/%D0%9F%D1%80%D0%BE%D0%B5%D0%BA%D1%82:%D0%A2%D3%A9%D0%B1%D3%99%D0%BA_%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA_%D0%B0%D1%82%D0%BD%D0%B0%D0%BB%D1%8B%D0%B3%D1%8B\" title=\"Проект:Төбәк тематик атналыгы\"\u003EТөбәкАтналыгы\u003C/a\u003E\u003C/u\u003E\u003C/i\u003E \u003Ci\u003E\u003Cspan typeof=\"mw:File\"\u003E\u003Ca href=\"/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D1%80%D0%BE%D0%B5%D0%BA%D1%82:%D0%90%D0%B7%D0%B8%D1%8F_%D0%B0%D0%B5_2024\" title=\"Википедия:Проект:Азия ае 2024\"\u003E\u003Cimg src=\"//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Wikipedia_Asian_Month_Logo.svg/20px-Wikipedia_Asian_Month_Logo.svg.png\" decoding=\"async\" width=\"20\" height=\"22\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Wikipedia_Asian_Month_Logo.svg/30px-Wikipedia_Asian_Month_Logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Wikipedia_Asian_Month_Logo.svg/40px-Wikipedia_Asian_Month_Logo.svg.png 2x\" data-file-width=\"512\" data-file-height=\"555\" /\u003E\u003C/a\u003E\u003C/span\u003E \u003Cu\u003E\u003Ca href=\"/wiki/%D0%9F%D1%80%D0%BE%D0%B5%D0%BA%D1%82:%D0%90%D0%B7%D0%B8%D1%8F_%D0%B0%D0%B5_2024\" title=\"Проект:Азия ае 2024\"\u003EАзия ае 2024\u003C/a\u003E\u003C/u\u003E\u003C/i\u003E\u003C/small\u003E\n\u003C/small\u003E\u003C/div\u003E\u003Csmall\u003E\n\u003Cdiv style=\"clear: both;\"\u003E\u003C/div\u003E\n\u003C/small\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Сайт"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Эчтәлек" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Эчтәлек</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">переместить в боковую панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">яшерү</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Башы</div> </a> </li> <li id="toc-Яссылыкның_төп_үзлекләренең_кайберләре" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Яссылыкның_төп_үзлекләренең_кайберләре"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Яссылыкның төп үзлекләренең кайберләре</span> </div> </a> <ul id="toc-Яссылыкның_төп_үзлекләренең_кайберләре-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Яссылык_тигезләмәләре" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Яссылык_тигезләмәләре"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Яссылык тигезләмәләре</span> </div> </a> <ul id="toc-Яссылык_тигезләмәләре-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ноктадан_яссылыкка_кадәр_ераклык" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ноктадан_яссылыкка_кадәр_ераклык"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ноктадан яссылыкка кадәр ераклык</span> </div> </a> <ul id="toc-Ноктадан_яссылыкка_кадәр_ераклык-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Параллель_яссылыклар_арасындагы_ераклык" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Параллель_яссылыклар_арасындагы_ераклык"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Параллель яссылыклар арасындагы ераклык</span> </div> </a> <ul id="toc-Параллель_яссылыклар_арасындагы_ераклык-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Бәйләнгән_төшенчәләр" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Бәйләнгән_төшенчәләр"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Бәйләнгән төшенчәләр</span> </div> </a> <ul id="toc-Бәйләнгән_төшенчәләр-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Дүртүлчәмле_киңлектә_яссылык_тигезләмәләре" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Дүртүлчәмле_киңлектә_яссылык_тигезләмәләре"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Дүртүлчәмле киңлектә яссылык тигезләмәләре</span> </div> </a> <ul id="toc-Дүртүлчәмле_киңлектә_яссылык_тигезләмәләре-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Чыганак" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Чыганак"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Чыганак</span> </div> </a> <ul id="toc-Чыганак-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Эчтәлек" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Отобразить/Скрыть содержание" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Отобразить/Скрыть содержание</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Яссылык (математика)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Перейти к статье на другом языке. Доступно на 88 языках" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-88" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">88 тел</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vlak" title="Vlak — африкаанс" lang="af" hreflang="af" data-title="Vlak" data-language-autonym="Afrikaans" data-language-local-name="африкаанс" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik) — швейцарский немецкий" lang="gsw" hreflang="gsw" data-title="Ebene (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="швейцарский немецкий" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%AA%D9%88_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مستو (رياضيات) — гарәп" lang="ar" hreflang="ar" data-title="مستو (رياضيات)" data-language-autonym="العربية" data-language-local-name="гарәп" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Planu_(xeometr%C3%ADa)" title="Planu (xeometría) — астурийский" lang="ast" hreflang="ast" data-title="Planu (xeometría)" data-language-autonym="Asturianu" data-language-local-name="астурийский" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C3%BCst%C9%99vi" title="Müstəvi — әзәрбайҗан" lang="az" hreflang="az" data-title="Müstəvi" data-language-autonym="Azərbaycanca" data-language-local-name="әзәрбайҗан" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AF%D2%AB%D1%8B%D0%BB%D1%8B%D2%A1" title="Яҫылыҡ — башкорт" lang="ba" hreflang="ba" data-title="Яҫылыҡ" data-language-autonym="Башҡортса" data-language-local-name="башкорт" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%81%D0%BA%D0%B0%D1%81%D1%86%D1%8C" title="Плоскасць — белорус" lang="be" hreflang="be" data-title="Плоскасць" data-language-autonym="Беларуская" data-language-local-name="белорус" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A0%D0%BE%D1%9E%D0%BD%D1%96%D1%86%D0%B0" title="Роўніца — белорусский (тарашкевица)" lang="be-tarask" hreflang="be-tarask" data-title="Роўніца" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="белорусский (тарашкевица)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B2%D0%BD%D0%B8%D0%BD%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Равнина (математика) — болгар" lang="bg" hreflang="bg" data-title="Равнина (математика)" data-language-autonym="Български" data-language-local-name="болгар" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%AE%E0%A6%A4%E0%A6%B2" title="সমতল — бенгали" lang="bn" hreflang="bn" data-title="সমতল" data-language-autonym="বাংলা" data-language-local-name="бенгали" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ravan" title="Ravan — босния" lang="bs" hreflang="bs" data-title="Ravan" data-language-autonym="Bosanski" data-language-local-name="босния" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Pla" title="Pla — каталан" lang="ca" hreflang="ca" data-title="Pla" data-language-autonym="Català" data-language-local-name="каталан" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8E%AD%E1%8F%AB%E1%8E%BE%E1%8F%97%E1%8F%A2_%E1%8F%97%E1%8F%8E%E1%8F%8D%E1%8F%97_%E1%8E%A4%E1%8E%AC%E1%8F%A9%E1%8E%B5" title="ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ — чероки" lang="chr" hreflang="chr" data-title="ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="чероки" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D9%88%D9%88%D8%AA%DB%95%D8%AE%D8%AA" title="ڕووتەخت — үзәк көрд" lang="ckb" hreflang="ckb" data-title="ڕووتەخت" data-language-autonym="کوردی" data-language-local-name="үзәк көрд" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rovina" title="Rovina — чех" lang="cs" hreflang="cs" data-title="Rovina" data-language-autonym="Čeština" data-language-local-name="чех" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BF%D1%82%D0%B0%D0%BA" title="Лаптак — чувашский" lang="cv" hreflang="cv" data-title="Лаптак" data-language-autonym="Чӑвашла" data-language-local-name="чувашский" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Pl%C3%A2n_geometraidd" title="Plân geometraidd — уэльс" lang="cy" hreflang="cy" data-title="Plân geometraidd" data-language-autonym="Cymraeg" data-language-local-name="уэльс" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Plan_(matematik)" title="Plan (matematik) — дания" lang="da" hreflang="da" data-title="Plan (matematik)" data-language-autonym="Dansk" data-language-local-name="дания" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik) — алман" lang="de" hreflang="de" data-title="Ebene (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="алман" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%80%CE%AF%CF%80%CE%B5%CE%B4%CE%BF" title="Επίπεδο — грек" lang="el" hreflang="el" data-title="Επίπεδο" data-language-autonym="Ελληνικά" data-language-local-name="грек" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euclidean_plane" title="Euclidean plane — инглиз" lang="en" hreflang="en" data-title="Euclidean plane" data-language-autonym="English" data-language-local-name="инглиз" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ebeno_(matematiko)" title="Ebeno (matematiko) — эсперанто" lang="eo" hreflang="eo" data-title="Ebeno (matematiko)" data-language-autonym="Esperanto" data-language-local-name="эсперанто" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Plano_(geometr%C3%ADa)" title="Plano (geometría) — испан" lang="es" hreflang="es" data-title="Plano (geometría)" data-language-autonym="Español" data-language-local-name="испан" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tasand" title="Tasand — эстон" lang="et" hreflang="et" data-title="Tasand" data-language-autonym="Eesti" data-language-local-name="эстон" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Plano" title="Plano — баск" lang="eu" hreflang="eu" data-title="Plano" data-language-autonym="Euskara" data-language-local-name="баск" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B5%D9%81%D8%AD%D9%87_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="صفحه (هندسه) — фарсы" lang="fa" hreflang="fa" data-title="صفحه (هندسه)" data-language-autonym="فارسی" data-language-local-name="фарсы" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Taso" title="Taso — фин" lang="fi" hreflang="fi" data-title="Taso" data-language-autonym="Suomi" data-language-local-name="фин" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Plan_(math%C3%A9matiques)" title="Plan (mathématiques) — француз" lang="fr" hreflang="fr" data-title="Plan (mathématiques)" data-language-autonym="Français" data-language-local-name="француз" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Pl%C3%A1na_(matamaitic)" title="Plána (matamaitic) — ирланд" lang="ga" hreflang="ga" data-title="Plána (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="ирланд" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%B9%B3%E9%9D%A2" title="平面 — гань" lang="gan" hreflang="gan" data-title="平面" data-language-autonym="贛語" data-language-local-name="гань" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Plan_(mat%C3%A9matik)" title="Plan (matématik) — Guianan Creole" lang="gcr" hreflang="gcr" data-title="Plan (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Plano_(xeometr%C3%ADa)" title="Plano (xeometría) — галисия" lang="gl" hreflang="gl" data-title="Plano (xeometría)" data-language-autonym="Galego" data-language-local-name="галисия" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%99%D7%A9%D7%95%D7%A8_(%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94)" title="מישור (גאומטריה) — яһүд" lang="he" hreflang="he" data-title="מישור (גאומטריה)" data-language-autonym="עברית" data-language-local-name="яһүд" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%A4%E0%A4%B2_(%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF)" title="समतल (ज्यामिति) — һинд" lang="hi" hreflang="hi" data-title="समतल (ज्यामिति)" data-language-autonym="हिन्दी" data-language-local-name="һинд" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ravnina" title="Ravnina — хорват" lang="hr" hreflang="hr" data-title="Ravnina" data-language-autonym="Hrvatski" data-language-local-name="хорват" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/S%C3%ADk_(geometria)" title="Sík (geometria) — венгр" lang="hu" hreflang="hu" data-title="Sík (geometria)" data-language-autonym="Magyar" data-language-local-name="венгр" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D6%80%D5%A9%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Հարթություն — әрмән" lang="hy" hreflang="hy" data-title="Հարթություն" data-language-autonym="Հայերեն" data-language-local-name="әрмән" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Plano" title="Plano — интерлингва" lang="ia" hreflang="ia" data-title="Plano" data-language-autonym="Interlingua" data-language-local-name="интерлингва" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bidang_(geometri)" title="Bidang (geometri) — индонезия" lang="id" hreflang="id" data-title="Bidang (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="индонезия" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Plano" title="Plano — идо" lang="io" hreflang="io" data-title="Plano" data-language-autonym="Ido" data-language-local-name="идо" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sl%C3%A9tta_(r%C3%BAmfr%C3%A6%C3%B0i)" title="Slétta (rúmfræði) — исланд" lang="is" hreflang="is" data-title="Slétta (rúmfræði)" data-language-autonym="Íslenska" data-language-local-name="исланд" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Piano_(geometria)" title="Piano (geometria) — итальян" lang="it" hreflang="it" data-title="Piano (geometria)" data-language-autonym="Italiano" data-language-local-name="итальян" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B3%E9%9D%A2" title="平面 — япон" lang="ja" hreflang="ja" data-title="平面" data-language-autonym="日本語" data-language-local-name="япон" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Plien_(matimatix)" title="Plien (matimatix) — Jamaican Creole English" lang="jam" hreflang="jam" data-title="Plien (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%91%E1%83%A0%E1%83%A2%E1%83%A7%E1%83%94" title="სიბრტყე — грузин" lang="ka" hreflang="ka" data-title="სიბრტყე" data-language-autonym="ქართული" data-language-local-name="грузин" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Agni_(tanzeggit)" title="Agni (tanzeggit) — кабильский" lang="kab" hreflang="kab" data-title="Agni (tanzeggit)" data-language-autonym="Taqbaylit" data-language-local-name="кабильский" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D0%B0%D0%B7%D1%8B%D2%9B%D1%82%D1%8B%D2%9B_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Жазықтық (геометрия) — казакъ" lang="kk" hreflang="kk" data-title="Жазықтық (геометрия)" data-language-autonym="Қазақша" data-language-local-name="казакъ" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%94%E1%9F%92%E1%9E%9B%E1%9E%84%E1%9F%8B" title="ប្លង់ — кхмер" lang="km" hreflang="km" data-title="ប្លង់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="кхмер" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%89%EB%A9%B4" title="평면 — корея" lang="ko" hreflang="ko" data-title="평면" data-language-autonym="한국어" data-language-local-name="корея" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A2%D0%B5%D0%B3%D0%B8%D0%B7%D0%B4%D0%B8%D0%BA" title="Тегиздик — кыргыз" lang="ky" hreflang="ky" data-title="Тегиздик" data-language-autonym="Кыргызча" data-language-local-name="кыргыз" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Planum_(geometria)" title="Planum (geometria) — латин" lang="la" hreflang="la" data-title="Planum (geometria)" data-language-autonym="Latina" data-language-local-name="латин" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Plok%C5%A1tuma" title="Plokštuma — литва" lang="lt" hreflang="lt" data-title="Plokštuma" data-language-autonym="Lietuvių" data-language-local-name="литва" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Plakne" title="Plakne — латыш" lang="lv" hreflang="lv" data-title="Plakne" data-language-autonym="Latviešu" data-language-local-name="латыш" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Lemaka_(je%C3%B4metria)" title="Lemaka (jeômetria) — малагаси" lang="mg" hreflang="mg" data-title="Lemaka (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="малагаси" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%BC%D0%BD%D0%B8%D0%BD%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Рамнина (математика) — македон" lang="mk" hreflang="mk" data-title="Рамнина (математика)" data-language-autonym="Македонски" data-language-local-name="македон" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Satah" title="Satah — малай" lang="ms" hreflang="ms" data-title="Satah" data-language-autonym="Bahasa Melayu" data-language-local-name="малай" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Flach_(Mathematik)" title="Flach (Mathematik) — нижненемецкий" lang="nds" hreflang="nds" data-title="Flach (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="нижненемецкий" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vlak_(meetkunde)" title="Vlak (meetkunde) — голланд" lang="nl" hreflang="nl" data-title="Vlak (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="голланд" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Plan_i_matematikk" title="Plan i matematikk — нюнорск" lang="nn" hreflang="nn" data-title="Plan i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="нюнорск" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Plan_(matematikk)" title="Plan (matematikk) — норвежский букмол" lang="nb" hreflang="nb" data-title="Plan (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="норвежский букмол" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/P%C5%82aszczyzna" title="Płaszczyzna — поляк" lang="pl" hreflang="pl" data-title="Płaszczyzna" data-language-autonym="Polski" data-language-local-name="поляк" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Plano_(geometria)" title="Plano (geometria) — португал" lang="pt" hreflang="pt" data-title="Plano (geometria)" data-language-autonym="Português" data-language-local-name="португал" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/P%27allta" title="P'allta — кечуа" lang="qu" hreflang="qu" data-title="P'allta" data-language-autonym="Runa Simi" data-language-local-name="кечуа" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Plan_(geometrie)" title="Plan (geometrie) — румын" lang="ro" hreflang="ro" data-title="Plan (geometrie)" data-language-autonym="Română" data-language-local-name="румын" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%81%D0%BA%D0%BE%D1%81%D1%82%D1%8C" title="Плоскость — рус" lang="ru" hreflang="ru" data-title="Плоскость" data-language-autonym="Русский" data-language-local-name="рус" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A5%D0%B0%D0%BF%D1%82%D0%B0%D0%BB" title="Хаптал — саха" lang="sah" hreflang="sah" data-title="Хаптал" data-language-autonym="Саха тыла" data-language-local-name="саха" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ravan" title="Ravan — сербскохорватский" lang="sh" hreflang="sh" data-title="Ravan" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="сербскохорватский" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Plane_(mathematics)" title="Plane (mathematics) — Simple English" lang="en-simple" hreflang="en-simple" data-title="Plane (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Rovina_(geometria)" title="Rovina (geometria) — словак" lang="sk" hreflang="sk" data-title="Rovina (geometria)" data-language-autonym="Slovenčina" data-language-local-name="словак" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ravnina" title="Ravnina — словен" lang="sl" hreflang="sl" data-title="Ravnina" data-language-autonym="Slovenščina" data-language-local-name="словен" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Mutsendo" title="Mutsendo — шона" lang="sn" hreflang="sn" data-title="Mutsendo" data-language-autonym="ChiShona" data-language-local-name="шона" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rrafshi_Euklidian" title="Rrafshi Euklidian — албан" lang="sq" hreflang="sq" data-title="Rrafshi Euklidian" data-language-autonym="Shqip" data-language-local-name="албан" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B2%D0%B0%D0%BD" title="Раван — серб" lang="sr" hreflang="sr" data-title="Раван" data-language-autonym="Српски / srpski" data-language-local-name="серб" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Plan_(geometri)" title="Plan (geometri) — швед" lang="sv" hreflang="sv" data-title="Plan (geometri)" data-language-autonym="Svenska" data-language-local-name="швед" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%B3%E0%AE%AE%E0%AF%8D_(%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="தளம் (வடிவவியல்) — тамил" lang="ta" hreflang="ta" data-title="தளம் (வடிவவியல்)" data-language-autonym="தமிழ்" data-language-local-name="тамил" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%99%E0%B8%B2%E0%B8%9A" title="ระนาบ — тай" lang="th" hreflang="th" data-title="ระนาบ" data-language-autonym="ไทย" data-language-local-name="тай" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Plano_(heometriya)" title="Plano (heometriya) — тагалог" lang="tl" hreflang="tl" data-title="Plano (heometriya)" data-language-autonym="Tagalog" data-language-local-name="тагалог" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/D%C3%BCzlem" title="Düzlem — төрек" lang="tr" hreflang="tr" data-title="Düzlem" data-language-autonym="Türkçe" data-language-local-name="төрек" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B8%D0%BD%D0%B0" title="Площина — украин" lang="uk" hreflang="uk" data-title="Площина" data-language-autonym="Українська" data-language-local-name="украин" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B3%D8%AA%D9%88%DB%8C_(%DB%81%D9%86%D8%AF%D8%B3%DB%81)" title="مستوی (ہندسہ) — урду" lang="ur" hreflang="ur" data-title="مستوی (ہندسہ)" data-language-autonym="اردو" data-language-local-name="урду" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Tekislik" title="Tekislik — үзбәк" lang="uz" hreflang="uz" data-title="Tekislik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="үзбәк" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BA%B7t_ph%E1%BA%B3ng_(to%C3%A1n_h%E1%BB%8Dc)" title="Mặt phẳng (toán học) — вьетнам" lang="vi" hreflang="vi" data-title="Mặt phẳng (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="вьетнам" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Plano_(heyometriya)" title="Plano (heyometriya) — варай" lang="war" hreflang="war" data-title="Plano (heyometriya)" data-language-autonym="Winaray" data-language-local-name="варай" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%B9%B3%E9%9D%A2" title="平面 — у" lang="wuu" hreflang="wuu" data-title="平面" data-language-autonym="吴语" data-language-local-name="у" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%9C%D7%90%D7%9A" title="פלאך — идиш" lang="yi" hreflang="yi" data-title="פלאך" data-language-autonym="ייִדיש" data-language-local-name="идиш" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学) — кытай" lang="zh" hreflang="zh" data-title="平面 (数学)" data-language-autonym="中文" data-language-local-name="кытай" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/P%C3%AE%E2%81%BF-b%C4%ABn" title="Pîⁿ-bīn — миньнань" lang="nan" hreflang="nan" data-title="Pîⁿ-bīn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="миньнань" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9D%A2_(%E5%B9%BE%E4%BD%95)" 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data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Внешний вид</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">переместить в боковую панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">яшерү</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Wikipedia — ирекле энциклопедия проектыннан ([http://tt.wikipedia.org.ttcysuttlart1999.aylandirow.tmf.org.ru/wiki/Яссылык (математика) latin yazuında])</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="tt" dir="ltr"><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:PlaneIntersection.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/PlaneIntersection.png/220px-PlaneIntersection.png" decoding="async" width="220" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/PlaneIntersection.png/330px-PlaneIntersection.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/PlaneIntersection.png/440px-PlaneIntersection.png 2x" data-file-width="497" data-file-height="539" /></a><figcaption>Ике кисешүче яссылык</figcaption></figure> <p><b>Яссылы́к</b> — <a href="/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия">геометриянең</a> төп төшенчәләреннән берсе. Геометрияне систематик рәвештә аңлатканда, яссылык төшенчәсен беренчел төшенчә буларак кулланалар. Яссылык тигезләмәсе беренче тапкыр <a href="/w/index.php?title=%D0%9A%D0%BB%D0%B5%D1%80%D0%BE,_%D0%90%D0%BB%D0%B5%D0%BA%D1%81%D0%B8_%D0%9A%D0%BB%D0%BE%D0%B4&action=edit&redlink=1" class="new" title="Клеро, Алекси Клод (бит юк)">А. К. Клеро</a> хезмәтләрендә (<a href="/wiki/1731" class="mw-redirect" title="1731">1731</a>), яссылыкның турылар аша бирелгән тигезләмәсе <a href="/w/index.php?title=%D0%9B%D0%B0%D0%BC%D0%B5,_%D0%93%D0%B0%D0%B1%D1%80%D0%B8%D1%8D%D0%BB%D1%8C&action=edit&redlink=1" class="new" title="Ламе, Габриэль (бит юк)">Г.Ламеда</a> очрый (<a href="/wiki/1816" class="mw-redirect" title="1816">1816</a>—<a href="/wiki/1818" class="mw-redirect" title="1818">1818</a>), нормаль тигезләмәне <a href="/w/index.php?title=%D0%93%D0%B5%D1%81%D1%81%D0%B5,_%D0%9B%D1%8E%D0%B4%D0%B2%D0%B8%D0%B3_%D0%9E%D1%82%D1%82%D0%BE&action=edit&redlink=1" class="new" title="Гессе, Людвиг Отто (бит юк)">Л. О. Гессе</a> кертә (<a href="/wiki/1861" class="mw-redirect" title="1861">1861</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Яссылыкның_төп_үзлекләренең_кайберләре"><span id=".D0.AF.D1.81.D1.81.D1.8B.D0.BB.D1.8B.D0.BA.D0.BD.D1.8B.D2.A3_.D1.82.D3.A9.D0.BF_.D2.AF.D0.B7.D0.BB.D0.B5.D0.BA.D0.BB.D3.99.D1.80.D0.B5.D0.BD.D0.B5.D2.A3_.D0.BA.D0.B0.D0.B9.D0.B1.D0.B5.D1.80.D0.BB.D3.99.D1.80.D0.B5"></span>Яссылыкның төп үзлекләренең кайберләре</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=1" title="«Яссылыкның төп үзлекләренең кайберләре» исемле бүлекне үзгәртү" class="mw-editsection-visualeditor"><span>үзгәртү</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=1" title="Редактировать код раздела «Яссылыкның төп үзлекләренең кайберләре»"><span>вики-текстны үзгәртү</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Яссылык — үзенең ике теләсә нинди ноктасын тоташтыручы бөтен кисемтәләрне үз эченә алган өслек;</li> <li>Яссылык — ике бирелгән ноктадан тигез ераклыкта урнашкан нокталар күплеге.</li></ul> <p><a href="/wiki/%D0%9A%D0%B8%D1%81%D0%B5%D0%BC%D1%82%D3%99" title="Кисемтә">Кисемтә</a> һәм <a href="/w/index.php?title=%D0%98%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)&action=edit&redlink=1" class="new" title="Интервал (геометрия) (бит юк)">интервалга</a> ошатып, кырый нокталарга ия булмаган яссылыкны интерваль яки ачык яссылык дип атарга мөмкин. </p> <div class="mw-heading mw-heading2"><h2 id="Яссылык_тигезләмәләре"><span id=".D0.AF.D1.81.D1.81.D1.8B.D0.BB.D1.8B.D0.BA_.D1.82.D0.B8.D0.B3.D0.B5.D0.B7.D0.BB.D3.99.D0.BC.D3.99.D0.BB.D3.99.D1.80.D0.B5"></span>Яссылык тигезләмәләре</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=2" title="«Яссылык тигезләмәләре» исемле бүлекне үзгәртү" class="mw-editsection-visualeditor"><span>үзгәртү</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=2" title="Редактировать код раздела «Яссылык тигезләмәләре»"><span>вики-текстны үзгәртү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Яссылык</b> — беренче дәрәҗәдәге алгебраик өслек: <a href="/w/index.php?title=%D0%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D1%81%D1%8B&action=edit&redlink=1" class="new" title="Декарт координаталар системасы (бит юк)">декарт координаталар системасында</a> яссылык беренче дәрәҗәдәге <a href="/wiki/%D0%A2%D0%B8%D0%B3%D0%B5%D0%B7%D0%BB%D3%99%D0%BC%D3%99" title="Тигезләмә">тигезләмә</a> белән бирелергә мөмкин. </p> <ul><li><b>Яссылыкның гомуми (тулы) тигезләмәсе</b></li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax+By+Cz+D=0\qquad (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mi>z</mi> <mo>+</mo> <mi>D</mi> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax+By+Cz+D=0\qquad (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2968f5bdf712411953451c07ff3af73a2175212d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.17ex; height:2.843ex;" alt="{\displaystyle Ax+By+Cz+D=0\qquad (1)}"></span></dd></dl> <p>монда <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> — даимиләр, шунда ук <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> бер үк вакытта нульгә тигез түгелләр; <a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&redlink=1" class="new" title="Вектор (бит юк)">векторлы</a> формада: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {r} ,\mathbf {N} )+D=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>D</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {r} ,\mathbf {N} )+D=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a63b9bedfbfbdcaa8bba3133aeb519a72f68de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.062ex; height:2.843ex;" alt="{\displaystyle (\mathbf {r} ,\mathbf {N} )+D=0}"></span></dd></dl> <p>монда <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490cecb2172d0679a32adb9e538bcbc1959a2d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.893ex; height:2.843ex;" alt="{\displaystyle M(x,y,z)}"></span> ноктасының радиус-векторы, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} =(A,B,C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} =(A,B,C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d6ea6579821c950219958bce4368a75c6139c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.341ex; height:2.843ex;" alt="{\displaystyle \mathbf {N} =(A,B,C)}"></span> векторы яссылыкка перпендикуляр (нормаль вектор). <i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f63b6cd6d63ee9b7be0b7e4d14099d7153bd43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {N} }"></span> векторының </i>юнәлтүче <a href="/w/index.php?title=%D0%9A%D0%BE%D1%81%D0%B8%D0%BD%D1%83%D1%81&action=edit&redlink=1" class="new" title="Косинус (бит юк)">косинуслары</a><i>:</i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \alpha ={\frac {A}{\sqrt {A^{2}+B^{2}+C^{2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \alpha ={\frac {A}{\sqrt {A^{2}+B^{2}+C^{2}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d00867efb0d804ff822751fe796818880b77c58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.039ex; height:6.676ex;" alt="{\displaystyle \cos \alpha ={\frac {A}{\sqrt {A^{2}+B^{2}+C^{2}}}},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \beta ={\frac {B}{\sqrt {A^{2}+B^{2}+C^{2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \beta ={\frac {B}{\sqrt {A^{2}+B^{2}+C^{2}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf81bca109801c8aac8b19b2da4c63583c9663f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.884ex; height:6.509ex;" alt="{\displaystyle \cos \beta ={\frac {B}{\sqrt {A^{2}+B^{2}+C^{2}}}},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \gamma ={\frac {C}{\sqrt {A^{2}+B^{2}+C^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \gamma ={\frac {C}{\sqrt {A^{2}+B^{2}+C^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5939a17c840da3a4e04028b81fa905b0c24625a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.814ex; height:6.676ex;" alt="{\displaystyle \cos \gamma ={\frac {C}{\sqrt {A^{2}+B^{2}+C^{2}}}}.}"></span></dd></dl> <p>Әгәр Я. тигезләмәсендә берәр коэффициент нульгә тигез булса, тигезләмәне <i>тулы булмаган</i> дип атыйлар. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d375dfda80ee8df1d1d7aa8b962114044e464305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.185ex; height:2.176ex;" alt="{\displaystyle D=0}"></span> булганда, Я. <a href="/w/index.php?title=%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_%D0%B1%D0%B0%D1%88%D0%BB%D0%B0%D0%BD%D0%B3%D1%8B%D1%87%D1%8B&action=edit&redlink=1" class="new" title="Координаталар башлангычы (бит юк)">координаталар башлангычы</a> аша үтә, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c34024483e6fb7c89e45aff3882ebf11d95a00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.004ex; height:2.176ex;" alt="{\displaystyle A=0}"></span> (яки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af1d79a64f775646f76d79a452ca8e1082fb1f17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle B=0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f750a7094a396d89a81974cdf35783db2bb287b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C=0}"></span>) булган вакытта, Я. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ox}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ox}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d69bd718df9e6a0005953a22f0af2b9794ee70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.103ex; height:2.176ex;" alt="{\displaystyle Ox}"></span> күчәренә параллель (аналогия буенча <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Oy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Oy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0971aed4330d41f75a2f97f4b11dd883958a3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.929ex; height:2.509ex;" alt="{\displaystyle Oy}"></span> яки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Oz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Oz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ade7218877bd912fd376e9b0a1664470d0c7c24e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.861ex; height:2.176ex;" alt="{\displaystyle Oz}"></span> күчәрләренә).<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=B=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>B</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f86f736b242670afc5462dc46af0d72b58e99f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.866ex; height:2.176ex;" alt="{\displaystyle A=B=0}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b2a06976382ad46a09a8e8da3485ff0d089fd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.869ex; height:2.176ex;" alt="{\displaystyle A=C=0}"></span>, яки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d966abd402b0d20318abe297b7c201a657e99034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.89ex; height:2.176ex;" alt="{\displaystyle B=C=0}"></span>) булган вакытта, Я. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Oxy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Oxy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb767e241f4141a4a4b39e0c81fc02f06537329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.258ex; height:2.509ex;" alt="{\displaystyle Oxy}"></span> яссылыгына параллель (аналогия буенча <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Oxz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>x</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Oxz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891818c2696562c9a59507f3ed2bcbebd737123c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.191ex; height:2.176ex;" alt="{\displaystyle Oxz}"></span> яки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Oyz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>y</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Oyz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b1dcb6901fe4e733f928a2fb5241f3a935a0d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.017ex; height:2.509ex;" alt="{\displaystyle Oyz}"></span> яссылыкларына). </p> <ul><li><b>Кисемтәләр аша яссылык тигезләмәсе:</b></li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>b</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966c73950eb3a0c89260caeaa6b8ec567049881c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.67ex; height:5.009ex;" alt="{\displaystyle {\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}=1,}"></span></dd></dl> <p>монда <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-D/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-D/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02aca8ee440be1540429321ac849d1f3ca208601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.966ex; height:2.843ex;" alt="{\displaystyle a=-D/A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=-D/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=-D/B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb7b22e84138a7c662ecb57bb200b5c687ec36f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.755ex; height:2.843ex;" alt="{\displaystyle b=-D/B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=-D/C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=-D/C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f632ad686cc8bf26875bf458166cc4635c5d077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.766ex; height:2.843ex;" alt="{\displaystyle c=-D/C}"></span> — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ox,Oy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>x</mi> <mo>,</mo> <mi>O</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ox,Oy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fa3ac2d20cedf1b9a529fba40f43f8e586057c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.066ex; height:2.509ex;" alt="{\displaystyle Ox,Oy}"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Oz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Oz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ade7218877bd912fd376e9b0a1664470d0c7c24e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.861ex; height:2.176ex;" alt="{\displaystyle Oz}"></span> күчәрләрендә Я. белән кисешә торган кисемтәләр. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x_{0},y_{0},z_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c592a83d9a7f916b48e0e0701bea025142eb2219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.032ex; height:2.843ex;" alt="{\displaystyle M(x_{0},y_{0},z_{0})}"></span> <b>ноктасы аша,</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} (A,B,C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} (A,B,C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc4bbe54c92576352e2381b777f05c42a069e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.242ex; height:2.843ex;" alt="{\displaystyle \mathbf {N} (A,B,C)}"></span> <b>нормаль векторына перпендикуляр рәвештә үтүче яссылык тигезләмәсе</b>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b7850eb07855ee9bf7f5c8e4e4df6c3e3e95473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.097ex; height:2.843ex;" alt="{\displaystyle A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0;}"></span></dd></dl> <p>векторлы формада: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((\mathbf {r} -\mathbf {r_{0}} ),\mathbf {N} )=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((\mathbf {r} -\mathbf {r_{0}} ),\mathbf {N} )=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776e5bb6e914488d806647193cd22ff2aca7d32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.874ex; height:2.843ex;" alt="{\displaystyle ((\mathbf {r} -\mathbf {r_{0}} ),\mathbf {N} )=0.}"></span></dd></dl> <ul><li><b>Бер турыда ятмаучы</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(x_{i},y_{i},z_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x_{i},y_{i},z_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5548d8501d9746ff6c0fec33f40daffa2364e6f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.268ex; height:2.843ex;" alt="{\displaystyle M(x_{i},y_{i},z_{i})}"></span> <b>өч нокта аша үтүче яссылык тигезләмәсе</b>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((\mathbf {r} -\mathbf {r_{1}} ),(\mathbf {r_{2}} -\mathbf {r_{1}} ),(\mathbf {r_{3}} -\mathbf {r_{1}} ))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((\mathbf {r} -\mathbf {r_{1}} ),(\mathbf {r_{2}} -\mathbf {r_{1}} ),(\mathbf {r_{3}} -\mathbf {r_{1}} ))=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ab8a1527926bf6df813c672fc095237eedd878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.587ex; height:2.843ex;" alt="{\displaystyle ((\mathbf {r} -\mathbf {r_{1}} ),(\mathbf {r_{2}} -\mathbf {r_{1}} ),(\mathbf {r_{3}} -\mathbf {r_{1}} ))=0}"></span></dd></dl> <p>(векторларның кушма тапкырчыгышы), ягъни </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\begin{matrix}x-x_{1}&y-y_{1}&z-z_{1}\\x_{2}-x_{1}&y_{2}-y_{1}&z_{2}-z_{1}\\x_{3}-x_{1}&y_{3}-y_{1}&z_{3}-z_{1}\\\end{matrix}}\right|=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\begin{matrix}x-x_{1}&y-y_{1}&z-z_{1}\\x_{2}-x_{1}&y_{2}-y_{1}&z_{2}-z_{1}\\x_{3}-x_{1}&y_{3}-y_{1}&z_{3}-z_{1}\\\end{matrix}}\right|=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b85a670312b20dbec839dc2e5f1f72cd68e3db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:33.545ex; height:9.509ex;" alt="{\displaystyle \left|{\begin{matrix}x-x_{1}&y-y_{1}&z-z_{1}\\x_{2}-x_{1}&y_{2}-y_{1}&z_{2}-z_{1}\\x_{3}-x_{1}&y_{3}-y_{1}&z_{3}-z_{1}\\\end{matrix}}\right|=0.}"></span></dd></dl> <ul><li><b>Яссылыкның нормаль (нормага китерелгән) тигезләмәсе</b></li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cos \alpha +y\cos \beta +z\cos \gamma -p=0\qquad (2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>y</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>z</mi> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo>−</mo> <mi>p</mi> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cos \alpha +y\cos \beta +z\cos \gamma -p=0\qquad (2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ccacfd34242a2c70865dfcf7f8492402b9807d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.88ex; height:2.843ex;" alt="{\displaystyle x\cos \alpha +y\cos \beta +z\cos \gamma -p=0\qquad (2)}"></span></dd></dl> <p>векторлы формада: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {r} ,\mathbf {N^{0}} )\mathbf {-p} =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {r} ,\mathbf {N^{0}} )\mathbf {-p} =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4cbc02a226936da0f360a1b837d7feaa891fd85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.685ex; height:3.176ex;" alt="{\displaystyle (\mathbf {r} ,\mathbf {N^{0}} )\mathbf {-p} =0,}"></span></dd></dl> <p>монда <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N^{0}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N^{0}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8888f52c5c18fe60aef25da606ef57664999654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.269ex; height:2.676ex;" alt="{\displaystyle \mathbf {N^{0}} }"></span>- берле вектор, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> — яссылыкның координаталар башлангычыннан ераклыгы. (2) тигезләмәсе (1) тигезләмәсеннән, түбәндәге нормалаштыручы тапкырлаучыга тапкырлап, табылырга мөмкин: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\pm {\frac {1}{\sqrt {A^{2}+B^{2}+C^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\pm {\frac {1}{\sqrt {A^{2}+B^{2}+C^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cfce7c8bfde8623d9c41c94050aeb69ac9a1a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.617ex; height:6.509ex;" alt="{\displaystyle \mu =\pm {\frac {1}{\sqrt {A^{2}+B^{2}+C^{2}}}}}"></span></dd></dl> <p>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> тамгалары капма-каршылар). </p> <div class="mw-heading mw-heading2"><h2 id="Ноктадан_яссылыкка_кадәр_ераклык"><span id=".D0.9D.D0.BE.D0.BA.D1.82.D0.B0.D0.B4.D0.B0.D0.BD_.D1.8F.D1.81.D1.81.D1.8B.D0.BB.D1.8B.D0.BA.D0.BA.D0.B0_.D0.BA.D0.B0.D0.B4.D3.99.D1.80_.D0.B5.D1.80.D0.B0.D0.BA.D0.BB.D1.8B.D0.BA"></span>Ноктадан яссылыкка кадәр ераклык</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=3" title="«Ноктадан яссылыкка кадәр ераклык» исемле бүлекне үзгәртү" class="mw-editsection-visualeditor"><span>үзгәртү</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=3" title="Редактировать код раздела «Ноктадан яссылыкка кадәр ераклык»"><span>вики-текстны үзгәртү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ноктадан яссылыкка кадәр ераклык — әлеге ноктадан яссылык нокталарына кадәр иң кечкенә ераклык. Ноктадан яссылыкка кадәр <a href="/w/index.php?title=%D0%95%D1%80%D0%B0%D0%BA%D0%BB%D1%8B%D0%BA&action=edit&redlink=1" class="new" title="Ераклык (бит юк)">ераклык</a> шул ноктадан яссылыкка төшерелгән перпендикуляр озынлыгына тигез икәнлеге билгеле. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}(x_{1},y_{1},z_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}(x_{1},y_{1},z_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a815236550356c5d8c0be7bb958bfab2b32432" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.898ex; height:2.843ex;" alt="{\displaystyle M_{1}(x_{1},y_{1},z_{1})}"></span> <b>ноктасының</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f88fdd4acbb57a291f9eb9f23ae23a1e492b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (2)}"></span> нормаль тигезләмә белән бирелгән яссылыктан <b>тайпылышы</b></li></ul> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =x_{1}\cos \alpha +y_{1}\cos \beta +z_{1}\cos \gamma -p;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo>−</mo> <mi>p</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta =x_{1}\cos \alpha +y_{1}\cos \beta +z_{1}\cos \gamma -p;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe9ac05acceefe3a91ca2df94ca5eaccb71e8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.935ex; height:2.843ex;" alt="{\displaystyle \delta =x_{1}\cos \alpha +y_{1}\cos \beta +z_{1}\cos \gamma -p;}"></span></center> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/595d5cea06fdcaf2642caf549eda2cfc537958a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}"></span>, әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8fd06f1cd5de22ed07385a0f8aa19773b2de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.054ex; height:2.509ex;" alt="{\displaystyle M_{i}}"></span> һәм координаталар башлангычы яссылыктан төрле якта урнашкан булсалар, башка очракта <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c6e67daff412fd88869d292c618a1140ff42da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta <0}"></span>. Ноктадан яссылыккача ераклык <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\delta |.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\delta |.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a100f6338b6ce950beef1cdc1bde60a63b4eeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.989ex; height:2.843ex;" alt="{\displaystyle |\delta |.}"></span> тигез.</dd></dl></dd></dl> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{0}(x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{0}(x_{0},y_{0},z_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c493fcce540bdd4c359c40ddc8ba24f50d54951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.898ex; height:2.843ex;" alt="{\displaystyle M_{0}(x_{0},y_{0},z_{0})}"></span> ноктасыннан, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax+by+cz+d=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mi>z</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax+by+cz+d=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/746e9e9c5e6a562485cbc5ed6a9375ec67bad26f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.805ex; height:2.509ex;" alt="{\displaystyle ax+by+cz+d=0}"></span> тигезләмәсе белән бирелгән яссылыкка кадәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> ераклыгы түбәндәге формула буенча исәпләнә:</li></ul> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {\mid ax_{0}+by_{0}+cz_{0}+d\mid }{\sqrt {a^{2}+b^{2}+c^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">∣</mo> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>c</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mo stretchy="false">∣</mo> </mrow> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {\mid ax_{0}+by_{0}+cz_{0}+d\mid }{\sqrt {a^{2}+b^{2}+c^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05335649255574f5c4aaa2f54cff0105df2e9eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.114ex; height:7.009ex;" alt="{\displaystyle \rho ={\frac {\mid ax_{0}+by_{0}+cz_{0}+d\mid }{\sqrt {a^{2}+b^{2}+c^{2}}}}}"></span></center> <div class="mw-heading mw-heading2"><h2 id="Параллель_яссылыклар_арасындагы_ераклык"><span id=".D0.9F.D0.B0.D1.80.D0.B0.D0.BB.D0.BB.D0.B5.D0.BB.D1.8C_.D1.8F.D1.81.D1.81.D1.8B.D0.BB.D1.8B.D0.BA.D0.BB.D0.B0.D1.80_.D0.B0.D1.80.D0.B0.D1.81.D1.8B.D0.BD.D0.B4.D0.B0.D0.B3.D1.8B_.D0.B5.D1.80.D0.B0.D0.BA.D0.BB.D1.8B.D0.BA"></span>Параллель яссылыклар арасындагы ераклык</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=4" title="«Параллель яссылыклар арасындагы ераклык» исемле бүлекне үзгәртү" class="mw-editsection-visualeditor"><span>үзгәртү</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=4" title="Редактировать код раздела «Параллель яссылыклар арасындагы ераклык»"><span>вики-текстны үзгәртү</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax+By+Cz+D_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mi>z</mi> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax+By+Cz+D_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ab2bec8302684312d1c3c287277f736de055a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.346ex; height:2.509ex;" alt="{\displaystyle Ax+By+Cz+D_{1}}"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax+By+Cz+D_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mi>z</mi> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax+By+Cz+D_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba0a26a8afeeb094689c5ae56135fb0163e73fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.346ex; height:2.509ex;" alt="{\displaystyle Ax+By+Cz+D_{2}}"></span> тигезләмәләре аша бирелгән яссылыклар арасындагы ераклык:</li></ul> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {\mid D_{2}-D_{1}\mid }{\sqrt {A^{2}+B^{2}+C^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">∣</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">∣</mo> </mrow> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {\mid D_{2}-D_{1}\mid }{\sqrt {A^{2}+B^{2}+C^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0c72a3e8d0f78d6e0c48d6b273dc429dbe56a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.623ex; height:7.009ex;" alt="{\displaystyle d={\frac {\mid D_{2}-D_{1}\mid }{\sqrt {A^{2}+B^{2}+C^{2}}}}}"></span></center> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {n}}({\bar {r}}-{\bar {r_{1}}})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {n}}({\bar {r}}-{\bar {r_{1}}})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb2c18e2e7e4fbef4b6e32caef72ef96f066331" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.7ex; height:2.843ex;" alt="{\displaystyle {\bar {n}}({\bar {r}}-{\bar {r_{1}}})=0}"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {n}}({\bar {r}}-{\bar {r_{2}}})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {n}}({\bar {r}}-{\bar {r_{2}}})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e84acc71d1d3cb0775adad29af2f4ce16435a54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.7ex; height:2.843ex;" alt="{\displaystyle {\bar {n}}({\bar {r}}-{\bar {r_{2}}})=0}"></span> тигезләмәләре аша бирелгән яссылыклар арасындагы ераклык:</li></ul> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {\mid [{\bar {r}}_{2}-{\bar {r}}_{1},{\bar {n}}]\mid }{\mid {\bar {n}}\mid }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">∣</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">∣</mo> </mrow> <mrow> <mo stretchy="false">∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">∣</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {\mid [{\bar {r}}_{2}-{\bar {r}}_{1},{\bar {n}}]\mid }{\mid {\bar {n}}\mid }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf3b79c1fa73b510197080a2db0d744b6d48080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.699ex; height:6.509ex;" alt="{\displaystyle d={\frac {\mid [{\bar {r}}_{2}-{\bar {r}}_{1},{\bar {n}}]\mid }{\mid {\bar {n}}\mid }}}"></span></center> <div class="mw-heading mw-heading2"><h2 id="Бәйләнгән_төшенчәләр"><span id=".D0.91.D3.99.D0.B9.D0.BB.D3.99.D0.BD.D0.B3.D3.99.D0.BD_.D1.82.D3.A9.D1.88.D0.B5.D0.BD.D1.87.D3.99.D0.BB.D3.99.D1.80"></span>Бәйләнгән төшенчәләр</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=5" title="«Бәйләнгән төшенчәләр» исемле бүлекне үзгәртү" class="mw-editsection-visualeditor"><span>үзгәртү</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=5" title="Редактировать код раздела «Бәйләнгән төшенчәләр»"><span>вики-текстны үзгәртү</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Ике яссылык арасындагы почмак.</b> Әгәр яссылык тигезләмәсе (1) төрендә бирелгән булса:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \varphi ={\frac {A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}}{\sqrt {(A_{1}^{2}+B_{1}^{2}+C_{1}^{2})(A_{2}^{2}+B_{2}^{2}+C_{2}^{2})}}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msqrt> <mo stretchy="false">(</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi ={\frac {A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}}{\sqrt {(A_{1}^{2}+B_{1}^{2}+C_{1}^{2})(A_{2}^{2}+B_{2}^{2}+C_{2}^{2})}}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c0592368fb9dfd46e39d150600057dfec479038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:43.839ex; height:8.176ex;" alt="{\displaystyle \cos \varphi ={\frac {A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}}{\sqrt {(A_{1}^{2}+B_{1}^{2}+C_{1}^{2})(A_{2}^{2}+B_{2}^{2}+C_{2}^{2})}}};}"></span></dd></dl> <p>Векторлы формада: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \varphi ={\frac {(\mathbf {N_{1}} ,\mathbf {N_{2}} )}{|\mathbf {N_{1}} ||\mathbf {N_{2}} |}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi ={\frac {(\mathbf {N_{1}} ,\mathbf {N_{2}} )}{|\mathbf {N_{1}} ||\mathbf {N_{2}} |}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea87516b5f78920a6fd5e131fd4334eb5ffbeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.981ex; height:6.509ex;" alt="{\displaystyle \cos \varphi ={\frac {(\mathbf {N_{1}} ,\mathbf {N_{2}} )}{|\mathbf {N_{1}} ||\mathbf {N_{2}} |}}.}"></span></dd></dl> <ul><li>Түбәндәге шарт үтәлгәндә <b><a href="/wiki/%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA%D0%BB%D0%B0%D1%80%D0%BD%D1%8B%D2%A3_%D0%BF%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D1%8C%D0%BB%D0%B5%D0%B3%D0%B5" title="Яссылыкларның параллельлеге">Яссылыклар параллель</a></b>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A_{1}}{A_{2}}}={\frac {B_{1}}{B_{2}}}={\frac {C_{1}}{C_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A_{1}}{A_{2}}}={\frac {B_{1}}{B_{2}}}={\frac {C_{1}}{C_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894efe46d42353de00212f5513288240ef83930c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.037ex; height:5.843ex;" alt="{\displaystyle {\frac {A_{1}}{A_{2}}}={\frac {B_{1}}{B_{2}}}={\frac {C_{1}}{C_{2}}}}"></span> яки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbf {N_{1}} ,\mathbf {N_{2}} ]=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mathbf {N_{1}} ,\mathbf {N_{2}} ]=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c4d8baa08e5ef8fcbb9a88761b1d6a6232517f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.773ex; height:2.843ex;" alt="{\displaystyle [\mathbf {N_{1}} ,\mathbf {N_{2}} ]=0.}"></span></dd></dl> <ul><li>Түбәндәге шарт үтәлгәндә <b>Яссылыклар үзара перпендикуляр</b>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6055c08672e034aff8c2c81e84b738ed552c5621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.605ex; height:2.509ex;" alt="{\displaystyle A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}=0}"></span> яки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {N_{1}} ,\mathbf {N_{2}} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {N_{1}} ,\mathbf {N_{2}} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2d3b376a4c238b3c970ffb608b3a8cd0c1b00d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.642ex; height:2.843ex;" alt="{\displaystyle (\mathbf {N_{1}} ,\mathbf {N_{2}} )=0}"></span>.</dd></dl> <ul><li><b>Яссылыклар бәйләме</b> — ике яссылыкның кисешү юлы аша үтүче теләсә нинди яссылыкның тигезләмәсе:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (A_{1}x+B_{1}y+C_{1}z)+\beta (A_{2}x+B_{2}y+C_{2}z)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (A_{1}x+B_{1}y+C_{1}z)+\beta (A_{2}x+B_{2}y+C_{2}z)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ae6d12d2066584ec1aed6b6862b479779226fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.358ex; height:2.843ex;" alt="{\displaystyle \alpha (A_{1}x+B_{1}y+C_{1}z)+\beta (A_{2}x+B_{2}y+C_{2}z)=0,}"></span></dd></dl> <p>монда <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> — бер үк вакытта нульгә тигез булмаган теләсә нинди саннар. </p> <div class="mw-heading mw-heading2"><h2 id="Дүртүлчәмле_киңлектә_яссылык_тигезләмәләре"><span id=".D0.94.D2.AF.D1.80.D1.82.D2.AF.D0.BB.D1.87.D3.99.D0.BC.D0.BB.D0.B5_.D0.BA.D0.B8.D2.A3.D0.BB.D0.B5.D0.BA.D1.82.D3.99_.D1.8F.D1.81.D1.81.D1.8B.D0.BB.D1.8B.D0.BA_.D1.82.D0.B8.D0.B3.D0.B5.D0.B7.D0.BB.D3.99.D0.BC.D3.99.D0.BB.D3.99.D1.80.D0.B5"></span>Дүртүлчәмле киңлектә яссылык тигезләмәләре</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=6" title="«Дүртүлчәмле киңлектә яссылык тигезләмәләре» исемле бүлекне үзгәртү" class="mw-editsection-visualeditor"><span>үзгәртү</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=6" title="Редактировать код раздела «Дүртүлчәмле киңлектә яссылык тигезләмәләре»"><span>вики-текстны үзгәртү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Әгәр дүртүлчәмле киңлектә ике яссылык бер <a href="/w/index.php?title=%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D1%8A%D1%8F%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA&action=edit&redlink=1" class="new" title="Гиперъяссылык (бит юк)">гиперъяссылыкта</a> ятсалар, алар параллель (аерым очракта туры килергә), яисә юл линия буенча кисешергә мөмкиннәр. </p><p>Әгәр дүртүлчәмле киңлектә ике яссылык бер <a href="/w/index.php?title=%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D1%8A%D1%8F%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA&action=edit&redlink=1" class="new" title="Гиперъяссылык (бит юк)">гиперъяссылыкта</a> ятмасалар, алар я кисешмиләр (өчүлчәмле киңлектә турылар кисешкән кебек), я <i>бердәнбер уртак ноктага ияләр</i>. </p><p>Ике яссылыкның бер ноктада (өчүлчәмле киңлектәге кебек, линия буенча түгел) кисешүләрен түбәндәгечә күрсәтергә мөмкин. <i>x y z t</i> <a href="/w/index.php?title=%D0%A2%D1%83%D1%80%D1%8B%D0%BF%D0%BE%D1%87%D0%BC%D0%B0%D0%BA%D0%BB%D1%8B_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D1%81%D1%8B&action=edit&redlink=1" class="new" title="Турыпочмаклы координаталар системасы (бит юк)">декарт координаталары</a> системасы бирелгән булсын. Ике α һәм β яссылыклары координаталар башлангычы аша үтсеннәр, һәм α <i>x</i> һәм <i>y</i> координаталы турыларга, ә β яссылыгы <i>z</i> һәм <i>t</i> координаталы турыларга ия булсын. α яссылыгының бөтен нокталарында <i>z</i> һәм <i>t</i> нульгә тигезләр, β яссылыгының бөтен нокталарында <i>x</i> һәм <i>y</i>. Шул очракта, (0,0,0,0) ноктасының гына ике яссылыкка керүе күренә. </p> <div class="mw-heading mw-heading2"><h2 id="Чыганак"><span id=".D0.A7.D1.8B.D0.B3.D0.B0.D0.BD.D0.B0.D0.BA"></span>Чыганак</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=7" title="«Чыганак» исемле бүлекне үзгәртү" class="mw-editsection-visualeditor"><span>үзгәртү</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=7" title="Редактировать код раздела «Чыганак»"><span>вики-текстны үзгәртү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a class="external text" href="https://ru.wikipedia.org/wiki/Плоскость_(математика)">Рус википедиясе</a> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Чыганак — <a dir="ltr" href="https://tt.wikipedia.org/w/index.php?title=Яссылык_(математика)&oldid=2699627">https://tt.wikipedia.org/w/index.php?title=Яссылык_(математика)&oldid=2699627</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:%D0%A2%D3%A9%D1%80%D0%BA%D0%B5%D0%BC%D0%BD%D3%99%D1%80" title="Махсус:Төркемнәр">Төркемнәр</a>: <ul><li><a href="/w/index.php?title=%D0%A2%D3%A9%D1%80%D0%BA%D0%B5%D0%BC:%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F%D1%81%D0%B5&action=edit&redlink=1" class="new" title="Төркем:Евклид геометриясе (бит юк)">Евклид геометриясе</a></li><li><a href="/wiki/%D0%A2%D3%A9%D1%80%D0%BA%D0%B5%D0%BC:%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0%D1%82%D0%B8%D0%BA_%D1%82%D3%A9%D1%88%D0%B5%D0%BD%D1%87%D3%99%D0%BB%D3%99%D1%80" title="Төркем:Аксиоматик төшенчәләр">Аксиоматик төшенчәләр</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Бу бит соңгы тапкыр 26 окт 2020 көнендә 16:31 вакытында үзгәртелде.</li> <li id="footer-info-copyright">Текст <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.ru">Creative Commons Attribution-ShareAlike</a> лицензиясе буенча бирелә, аерым очракларда өстәмә шартлар булырга мөмкин. 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