Вектор — Википедија

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Прочитајте ги правилата и вклучете се!\n\u003C/p\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="Мрежно место"> <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="Содржина" 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">Содржина</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">премести во страничникот</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">скриј</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">Почеток</div> </a> </li> <li id="toc-Претставување_на_векторите" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Претставување_на_векторите"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Претставување на векторите</span> </div> </a> <button aria-controls="toc-Претставување_на_векторите-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Префрли на пододделот Претставување на векторите</span> </button> <ul id="toc-Претставување_на_векторите-sublist" class="vector-toc-list"> <li id="toc-Геометриско_претставување_на_векторите" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Геометриско_претставување_на_векторите"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Геометриско претставување на векторите</span> </div> </a> <ul id="toc-Геометриско_претставување_на_векторите-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Аналитичко_претставување_на_векторите" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Аналитичко_претставување_на_векторите"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Аналитичко претставување на векторите</span> </div> </a> <ul id="toc-Аналитичко_претставување_на_векторите-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Операции_со_вектори" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Операции_со_вектори"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Операции со вектори</span> </div> </a> <button aria-controls="toc-Операции_со_вектори-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Префрли на пододделот Операции со вектори</span> </button> <ul id="toc-Операции_со_вектори-sublist" class="vector-toc-list"> <li id="toc-Собирање_на_вектори" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Собирање_на_вектори"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Собирање на вектори</span> </div> </a> <ul id="toc-Собирање_на_вектори-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Одземање_на_вектори" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Одземање_на_вектори"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Одземање на вектори</span> </div> </a> <ul id="toc-Одземање_на_вектори-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Множење_на_вектори" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Множење_на_вектори"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Множење на вектори</span> </div> </a> <ul id="toc-Множење_на_вектори-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Поврзано" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Поврзано"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Поврзано</span> </div> </a> <ul id="toc-Поврзано-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="Содржина" 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="Прик./скр. содржина" > <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">Прик./скр. содржина</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">Вектор</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="Појдете на статија на друг јазик. Достапно на 95 јазици" > <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-95" 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">95 јазици</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/Vektor_(Wiskunde)" title="Vektor (Wiskunde) — африканс" lang="af" hreflang="af" data-title="Vektor (Wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="африканс" 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/Vektor" title="Vektor — швајцарски германски" lang="gsw" hreflang="gsw" data-title="Vektor" data-language-autonym="Alemannisch" data-language-local-name="швајцарски германски" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8C%A8%E1%88%A8%E1%88%AD" title="ጨረር — амхарски" lang="am" hreflang="am" data-title="ጨረር" data-language-autonym="አማርኛ" data-language-local-name="амхарски" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Vektor" title="Vektor — инариски сами" lang="smn" hreflang="smn" data-title="Vektor" data-language-autonym="Anarâškielâ" data-language-local-name="инариски сами" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%AC%D9%87" title="متجه — арапски" lang="ar" hreflang="ar" data-title="متجه" data-language-autonym="العربية" data-language-local-name="арапски" 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/Vector" title="Vector — астурски" lang="ast" hreflang="ast" data-title="Vector" data-language-autonym="Asturianu" data-language-local-name="астурски" 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/Vektor_(h%C9%99nd%C9%99s%C9%99)" title="Vektor (həndəsə) — азербејџански" lang="az" hreflang="az" data-title="Vektor (həndəsə)" data-language-autonym="Azərbaycanca" data-language-local-name="азербејџански" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DB%8C%D8%A4%D9%86%D8%A6%DB%8C_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="یؤنئی (هندسه) — South Azerbaijani" lang="azb" hreflang="azb" data-title="یؤنئی (هندسه)" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%A6%E0%A6%BF%E0%A6%95_%E0%A6%B0%E0%A6%BE%E0%A6%B6%E0%A6%BF" title="সদিক রাশি — бенгалски" lang="bn" hreflang="bn" data-title="সদিক রাশি" data-language-autonym="বাংলা" data-language-local-name="бенгалски" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Hi%C3%B2ng-li%C5%8Dng" title="Hiòng-liōng — јужномински" lang="nan" hreflang="nan" data-title="Hiòng-liōng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="јужномински" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Вектор (геометрия) — башкирски" lang="ba" hreflang="ba" data-title="Вектор (геометрия)" data-language-autonym="Башҡортса" data-language-local-name="башкирски" 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%92%D0%B5%D0%BA%D1%82%D0%B0%D1%80_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Вектар (матэматыка) — белоруски" lang="be" hreflang="be" data-title="Вектар (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="белоруски" 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%92%D1%8D%D0%BA%D1%82%D0%B0%D1%80" title="Вэктар — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Вэктар" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" 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%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор — бугарски" lang="bg" hreflang="bg" data-title="Вектор" data-language-autonym="Български" data-language-local-name="бугарски" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Euklidski_vektor" title="Euklidski vektor — босански" lang="bs" hreflang="bs" data-title="Euklidski vektor" data-language-autonym="Bosanski" data-language-local-name="босански" 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/Vector_(matem%C3%A0tiques)" title="Vector (matemàtiques) — каталонски" lang="ca" hreflang="ca" data-title="Vector (matemàtiques)" data-language-autonym="Català" data-language-local-name="каталонски" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8)" title="Вектор (геометри) — чувашки" lang="cv" hreflang="cv" data-title="Вектор (геометри)" data-language-autonym="Чӑвашла" data-language-local-name="чувашки" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vektor" title="Vektor — чешки" lang="cs" hreflang="cs" data-title="Vektor" data-language-autonym="Čeština" data-language-local-name="чешки" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Fector" title="Fector — велшки" lang="cy" hreflang="cy" data-title="Fector" data-language-autonym="Cymraeg" data-language-local-name="велшки" 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/Vektor_(geometri)" title="Vektor (geometri) — дански" lang="da" hreflang="da" data-title="Vektor (geometri)" data-language-autonym="Dansk" data-language-local-name="дански" 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/Vektor" title="Vektor — германски" lang="de" hreflang="de" data-title="Vektor" data-language-autonym="Deutsch" data-language-local-name="германски" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Vektor" title="Vektor — естонски" lang="et" hreflang="et" data-title="Vektor" data-language-autonym="Eesti" data-language-local-name="естонски" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B5%CE%B9%CE%BF_%CE%B4%CE%B9%CE%AC%CE%BD%CF%85%CF%83%CE%BC%CE%B1" title="Ευκλείδειο διάνυσμα — грчки" lang="el" hreflang="el" data-title="Ευκλείδειο διάνυσμα" data-language-autonym="Ελληνικά" data-language-local-name="грчки" 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_vector" title="Euclidean vector — англиски" lang="en" hreflang="en" data-title="Euclidean vector" data-language-autonym="English" data-language-local-name="англиски" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Вектор (геометрия) — ерзјански" lang="myv" hreflang="myv" data-title="Вектор (геометрия)" data-language-autonym="Эрзянь" data-language-local-name="ерзјански" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Vector" title="Vector — шпански" lang="es" hreflang="es" data-title="Vector" data-language-autonym="Español" data-language-local-name="шпански" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vektoro" title="Vektoro — есперанто" lang="eo" hreflang="eo" data-title="Vektoro" data-language-autonym="Esperanto" data-language-local-name="есперанто" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bektore_(matematika)" title="Bektore (matematika) — баскиски" lang="eu" hreflang="eu" data-title="Bektore (matematika)" data-language-autonym="Euskara" data-language-local-name="баскиски" 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%A8%D8%B1%D8%AF%D8%A7%D8%B1_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="بردار اقلیدسی — персиски" lang="fa" hreflang="fa" data-title="بردار اقلیدسی" data-language-autonym="فارسی" data-language-local-name="персиски" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vecteur_euclidien" title="Vecteur euclidien — француски" lang="fr" hreflang="fr" data-title="Vecteur euclidien" data-language-autonym="Français" data-language-local-name="француски" 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/Veicteoir" title="Veicteoir — ирски" lang="ga" hreflang="ga" data-title="Veicteoir" data-language-autonym="Gaeilge" data-language-local-name="ирски" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Bheactor" title="Bheactor — шкотски гелски" lang="gd" hreflang="gd" data-title="Bheactor" data-language-autonym="Gàidhlig" data-language-local-name="шкотски гелски" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Vector" title="Vector — галициски" lang="gl" hreflang="gl" data-title="Vector" data-language-autonym="Galego" data-language-local-name="галициски" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EB%B2%A1%ED%84%B0" title="유클리드 벡터 — корејски" lang="ko" hreflang="ko" data-title="유클리드 벡터" data-language-autonym="한국어" data-language-local-name="корејски" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6_%E0%A4%B0%E0%A4%BE%E0%A4%B6%E0%A4%BF" title="सदिश राशि — хинди" lang="hi" hreflang="hi" data-title="सदिश राशि" data-language-autonym="हिन्दी" data-language-local-name="хинди" 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/Vektor" title="Vektor — хрватски" lang="hr" hreflang="hr" data-title="Vektor" data-language-autonym="Hrvatski" data-language-local-name="хрватски" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Vektoro" title="Vektoro — идо" lang="io" hreflang="io" data-title="Vektoro" data-language-autonym="Ido" data-language-local-name="идо" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Vektor_Euklides" title="Vektor Euklides — индонезиски" lang="id" hreflang="id" data-title="Vektor Euklides" data-language-autonym="Bahasa Indonesia" data-language-local-name="индонезиски" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vigur_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Vigur (stærðfræði) — исландски" lang="is" hreflang="is" data-title="Vigur (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="исландски" 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/Vettore_(matematica)" title="Vettore (matematica) — италијански" lang="it" hreflang="it" data-title="Vettore (matematica)" data-language-autonym="Italiano" data-language-local-name="италијански" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_%D7%90%D7%95%D7%A7%D7%9C%D7%99%D7%93%D7%99" title="וקטור אוקלידי — хебрејски" lang="he" hreflang="he" data-title="וקטור אוקלידי" data-language-autonym="עברית" data-language-local-name="хебрејски" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%95%E1%83%94%E1%83%A5%E1%83%A2%E1%83%9D%E1%83%A0%E1%83%98" title="ვექტორი — грузиски" lang="ka" hreflang="ka" data-title="ვექტორი" data-language-autonym="ქართული" data-language-local-name="грузиски" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор — казашки" lang="kk" hreflang="kk" data-title="Вектор" data-language-autonym="Қазақша" data-language-local-name="казашки" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Vekt%C3%A8" title="Vektè — хаитски" lang="ht" hreflang="ht" data-title="Vektè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="хаитски" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Vector_(mathematica)" title="Vector (mathematica) — латински" lang="la" hreflang="la" data-title="Vector (mathematica)" data-language-autonym="Latina" data-language-local-name="латински" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Vektors" title="Vektors — латвиски" lang="lv" hreflang="lv" data-title="Vektors" data-language-autonym="Latviešu" data-language-local-name="латвиски" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vektorius" title="Vektorius — литвански" lang="lt" hreflang="lt" data-title="Vektorius" data-language-autonym="Lietuvių" data-language-local-name="литвански" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Vettor_(matematega)" title="Vettor (matematega) — ломбардиски" lang="lmo" hreflang="lmo" data-title="Vettor (matematega)" data-language-autonym="Lombard" data-language-local-name="ломбардиски" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vektor" title="Vektor — унгарски" lang="hu" hreflang="hu" data-title="Vektor" data-language-autonym="Magyar" data-language-local-name="унгарски" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%A6%E0%B4%BF%E0%B4%B6%E0%B4%82_(%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%AE%E0%B4%BF%E0%B4%A4%E0%B4%BF)" title="സദിശം (ജ്യാമിതി) — малајалски" lang="ml" hreflang="ml" data-title="സദിശം (ജ്യാമിതി)" data-language-autonym="മലയാളം" data-language-local-name="малајалски" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Vettur_ewklidju" title="Vettur ewklidju — малтешки" lang="mt" hreflang="mt" data-title="Vettur ewklidju" data-language-autonym="Malti" data-language-local-name="малтешки" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Vektor" title="Vektor — малајски" lang="ms" hreflang="ms" data-title="Vektor" data-language-autonym="Bahasa Melayu" data-language-local-name="малајски" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/Hi%C3%B3ng-li%C3%B4ng" title="Hióng-liông — Mindong" lang="cdo" hreflang="cdo" data-title="Hióng-liông" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%B8%D0%B9%D0%BD_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Евклидийн вектор — монголски" lang="mn" hreflang="mn" data-title="Евклидийн вектор" data-language-autonym="Монгол" data-language-local-name="монголски" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vector_(wiskunde)" title="Vector (wiskunde) — холандски" lang="nl" hreflang="nl" data-title="Vector (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="холандски" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A9%BA%E9%96%93%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB" title="空間ベクトル — јапонски" lang="ja" hreflang="ja" data-title="空間ベクトル" data-language-autonym="日本語" data-language-local-name="јапонски" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Vektor" title="Vektor — севернофризиски" lang="frr" hreflang="frr" data-title="Vektor" data-language-autonym="Nordfriisk" data-language-local-name="севернофризиски" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Vektor_(matematikk)" title="Vektor (matematikk) — норвешки букмол" lang="nb" hreflang="nb" data-title="Vektor (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="норвешки букмол" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Vektor" title="Vektor — норвешки нинорск" lang="nn" hreflang="nn" data-title="Vektor" data-language-autonym="Norsk nynorsk" data-language-local-name="норвешки нинорск" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор — Eastern Mari" lang="mhr" hreflang="mhr" data-title="Вектор" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Kalqabee" title="Kalqabee — оромо" lang="om" hreflang="om" data-title="Kalqabee" data-language-autonym="Oromoo" data-language-local-name="оромо" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Vektor_(matematika)" title="Vektor (matematika) — узбечки" lang="uz" hreflang="uz" data-title="Vektor (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="узбечки" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D8%A7%D9%82%D9%84%D9%8A%D8%AF%D8%B3_%D9%84%D9%88%D8%B1%DB%8C" title="د اقليدس لوری — паштунски" lang="ps" hreflang="ps" data-title="د اقليدس لوری" data-language-autonym="پښتو" data-language-local-name="паштунски" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Vetor" title="Vetor — пиемонтски" lang="pms" hreflang="pms" data-title="Vetor" data-language-autonym="Piemontèis" data-language-local-name="пиемонтски" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Vekter" title="Vekter — долногермански" lang="nds" hreflang="nds" data-title="Vekter" data-language-autonym="Plattdüütsch" data-language-local-name="долногермански" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wektor" title="Wektor — полски" lang="pl" hreflang="pl" data-title="Wektor" data-language-autonym="Polski" data-language-local-name="полски" 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/Vetor_(matem%C3%A1tica)" title="Vetor (matemática) — португалски" lang="pt" hreflang="pt" data-title="Vetor (matemática)" data-language-autonym="Português" data-language-local-name="португалски" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Vector_euclidian" title="Vector euclidian — романски" lang="ro" hreflang="ro" data-title="Vector euclidian" data-language-autonym="Română" data-language-local-name="романски" 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%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Вектор (геометрия) — руски" lang="ru" hreflang="ru" data-title="Вектор (геометрия)" data-language-autonym="Русский" data-language-local-name="руски" 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%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Вектор (геометрия) — јакутски" lang="sah" hreflang="sah" data-title="Вектор (геометрия)" data-language-autonym="Саха тыла" data-language-local-name="јакутски" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vektori" title="Vektori — албански" lang="sq" hreflang="sq" data-title="Vektori" data-language-autonym="Shqip" data-language-local-name="албански" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Vettura_euclideu" title="Vettura euclideu — сицилијански" lang="scn" hreflang="scn" data-title="Vettura euclideu" data-language-autonym="Sicilianu" data-language-local-name="сицилијански" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BA%E0%B7%94%E0%B6%9A%E0%B7%8A%E0%B6%BD%E0%B7%92%E0%B6%A9%E0%B7%92%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%94_%E0%B6%AF%E0%B7%9B%E0%B7%81%E0%B7%92%E0%B6%9A%E0%B6%BA" title="යුක්ලිඩියානු දෛශිකය — синхалски" lang="si" hreflang="si" data-title="යුක්ලිඩියානු දෛශිකය" data-language-autonym="සිංහල" data-language-local-name="синхалски" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Vector" title="Vector — Simple English" lang="en-simple" hreflang="en-simple" data-title="Vector" 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/Vektor_(matematika)" title="Vektor (matematika) — словачки" lang="sk" hreflang="sk" data-title="Vektor (matematika)" data-language-autonym="Slovenčina" data-language-local-name="словачки" 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/Vektor_(matematika)" title="Vektor (matematika) — словенечки" lang="sl" hreflang="sl" data-title="Vektor (matematika)" data-language-autonym="Slovenščina" data-language-local-name="словенечки" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Wekt%C5%AFr" title="Wektůr — шлезиски" lang="szl" hreflang="szl" data-title="Wektůr" data-language-autonym="Ślůnski" data-language-local-name="шлезиски" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%D8%A7%DA%95%D8%A7%D8%B3%D8%AA%DB%95%D8%A8%DA%95%DB%8C_%D8%A6%DB%8C%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="ئاڕاستەبڕی ئیقلیدسی — централнокурдски" lang="ckb" hreflang="ckb" data-title="ئاڕاستەبڕی ئیقلیدسی" data-language-autonym="کوردی" data-language-local-name="централнокурдски" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор — српски" lang="sr" hreflang="sr" data-title="Вектор" data-language-autonym="Српски / srpski" data-language-local-name="српски" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Vektor" title="Vektor — српскохрватски" lang="sh" hreflang="sh" data-title="Vektor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="српскохрватски" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/V%C3%A9ktor_(rohangan)" title="Véktor (rohangan) — сундски" lang="su" hreflang="su" data-title="Véktor (rohangan)" data-language-autonym="Sunda" data-language-local-name="сундски" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vektori" title="Vektori — фински" lang="fi" hreflang="fi" data-title="Vektori" data-language-autonym="Suomi" data-language-local-name="фински" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Vektor" title="Vektor — шведски" lang="sv" hreflang="sv" data-title="Vektor" data-language-autonym="Svenska" data-language-local-name="шведски" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Euclidyanong_bektor" title="Euclidyanong bektor — тагалог" lang="tl" hreflang="tl" data-title="Euclidyanong bektor" data-language-autonym="Tagalog" data-language-local-name="тагалог" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AE%A9%E0%AF%8D" title="திசையன் — тамилски" lang="ta" hreflang="ta" data-title="திசையன்" data-language-autonym="தமிழ்" data-language-local-name="тамилски" 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%B9%80%E0%B8%A7%E0%B8%81%E0%B9%80%E0%B8%95%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="เวกเตอร์ — тајландски" lang="th" hreflang="th" data-title="เวกเตอร์" data-language-autonym="ไทย" data-language-local-name="тајландски" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Vekt%C3%B6r" title="Vektör — турски" lang="tr" hreflang="tr" data-title="Vektör" data-language-autonym="Türkçe" data-language-local-name="турски" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Wektor_ululyklar" title="Wektor ululyklar — туркменски" lang="tk" hreflang="tk" data-title="Wektor ululyklar" data-language-autonym="Türkmençe" data-language-local-name="туркменски" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D1%96%D0%B4%D1%96%D0%B2_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Евклідів вектор — украински" lang="uk" hreflang="uk" data-title="Евклідів вектор" data-language-autonym="Українська" data-language-local-name="украински" 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/%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C_%D8%B3%D9%85%D8%AA%DB%8C%DB%81" title="اقلیدسی سمتیہ — урду" lang="ur" hreflang="ur" data-title="اقلیدسی سمتیہ" data-language-autonym="اردو" data-language-local-name="урду" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Vect%C6%A1" title="Vectơ — виетнамски" lang="vi" hreflang="vi" data-title="Vectơ" data-language-autonym="Tiếng Việt" data-language-local-name="виетнамски" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%90%91%E9%87%8F" title="向量 — ву" lang="wuu" hreflang="wuu" data-title="向量" data-language-autonym="吴语" data-language-local-name="ву" 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%95%D7%95%D7%A2%D7%A7%D7%98%D7%90%D7%A8" title="וועקטאר — јидиш" lang="yi" hreflang="yi" data-title="וועקטאר" data-language-autonym="ייִדיש" data-language-local-name="јидиш" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%90%91%E9%87%8F" title="向量 — кантонски" lang="yue" hreflang="yue" data-title="向量" data-language-autonym="粵語" data-language-local-name="кантонски" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%90%91%E9%87%8F" title="向量 — кинески" lang="zh" hreflang="zh" data-title="向量" data-language-autonym="中文" data-language-local-name="кинески" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q44528#sitelinks-wikipedia" title="Уредување на меѓујазични врски" class="wbc-editpage">Уреди врски</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Именски простори"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Преглед на содржинската страница [c]" accesskey="c"><span>Страница</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%D0%A0%D0%B0%D0%B7%D0%B3%D0%BE%D0%B2%D0%BE%D1%80:%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" rel="discussion" title="Разговор за содржинската страница [t]" accesskey="t"><span>Разговор</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Менување на јазичната варијанта" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">македонски</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Посети"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80"><span>Читај</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit" title="Уредете ја страницава [v]" accesskey="v"><span>Уреди</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit" title="Уредување на изворниот код на страницава [e]" accesskey="e"><span>Уреди извор</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=history" title="Претходни верзии на оваа страница. [h]" accesskey="h"><span>Историја</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Алатки за страници"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="алатник" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">алатник</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Алатки</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">премести во страничникот</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">скриј</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Повеќе можности" > <div class="vector-menu-heading"> Дејства </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80"><span>Читај</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit" title="Уредете ја страницава [v]" accesskey="v"><span>Уреди</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit" title="Уредување на изворниот код на страницава [e]" accesskey="e"><span>Уреди извор</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=history"><span>Историја</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Општо </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%A8%D1%82%D0%BE%D0%92%D0%BE%D0%B4%D0%B8%D0%9E%D0%B2%D0%B4%D0%B5/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Список на сите викистраници што водат овде [j]" accesskey="j"><span>Што води овде</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%BE%D0%B2%D1%80%D0%B7%D0%B0%D0%BD%D0%B8%D0%9F%D1%80%D0%BE%D0%BC%D0%B5%D0%BD%D0%B8/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" rel="nofollow" title="Скорешни промени на страници со врски на оваа страница [k]" accesskey="k"><span>Поврзани промени</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%A1%D0%BB%D1%83%D0%B6%D0%B1%D0%B5%D0%BD%D0%B8%D0%A1%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B8" title="Список на сите службени страници [q]" accesskey="q"><span>Службени страници</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&oldid=4910097" title="Постојана врска до оваа преработка на страницата"><span>Постојана врска</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=info" title="Повеќе информации за страницата"><span>Информации за страницата</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9D%D0%B0%D0%B2%D0%BE%D0%B4&page=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&id=4910097&wpFormIdentifier=titleform" title="Информации како да ја наведете оваа страница"><span>Наведи ја страницава</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%A1%D0%BA%D1%80%D0%B0%D1%82%D1%83%D0%B2%D0%B0%D1%87%D0%9D%D0%B0URL&url=https%3A%2F%2Fmk.wikipedia.org%2Fwiki%2F%25D0%2592%25D0%25B5%25D0%25BA%25D1%2582%25D0%25BE%25D1%2580"><span>Дај скратена URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:QrCode&url=https%3A%2F%2Fmk.wikipedia.org%2Fwiki%2F%25D0%2592%25D0%25B5%25D0%25BA%25D1%2582%25D0%25BE%25D1%2580"><span>Преземи QR-код</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Печати/извези </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9A%D0%BD%D0%B8%D0%B3%D0%B0&bookcmd=book_creator&referer=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80"><span>Создај книга</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:DownloadAsPdf&page=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=show-download-screen"><span>Преземи како PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&printable=yes" title="Верзија за печатење на оваа страница [p]" accesskey="p"><span>Верзија за печатење</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> На други проекти </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Vectors" hreflang="en"><span>Ризница</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q44528" title="Врска до поврзаниот предмет во складиштето за податоци [g]" accesskey="g"><span>Предмет на Википодатоци</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Алатки за страници"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Изглед"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" 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">Изглед</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">премести во страничникот</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">скриј</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 id="mw-indicator-Top_icon_raw" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%98%D0%B0:%D0%9A%D1%80%D0%B8%D1%82%D0%B5%D1%80%D0%B8%D1%83%D0%BC%D0%B8_%D0%B7%D0%B0_%D0%B8%D0%B7%D0%B1%D1%80%D0%B0%D0%BD%D0%B0_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%98%D0%B0" title="Ова е избрана статија. Стиснете тука за повеќе информации."><img alt="Ова е избрана статија. Стиснете тука за повеќе информации." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Cscr-featured.svg/17px-Cscr-featured.svg.png" decoding="async" width="17" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Cscr-featured.svg/26px-Cscr-featured.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Cscr-featured.svg/34px-Cscr-featured.svg.png 2x" data-file-width="466" data-file-height="443" /></a></span></div></div> </div> <div id="siteSub" class="noprint">Од Википедија — слободната енциклопедија</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="mk" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Vector_by_Zureks-mk.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Vector_by_Zureks-mk.svg/250px-Vector_by_Zureks-mk.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Vector_by_Zureks-mk.svg/375px-Vector_by_Zureks-mk.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Vector_by_Zureks-mk.svg/500px-Vector_by_Zureks-mk.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>Елементи на векторот</figcaption></figure> <p><b>Вектор</b> — поим во <a href="/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика">математиката</a> што означува секоја <a href="/wiki/%D0%A4%D0%B8%D0%B7%D0%B8%D1%87%D0%BA%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Физичка величина">величина</a> која во себе носи информација за <b><a href="/wiki/%D0%9A%D0%BE%D0%BB%D0%B8%D1%87%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Количество">количество</a></b> (квантитет) и квалитет. Наспроти векторите, во математиката стојат <a href="/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80" title="Скалар">скаларите</a> кои носат информација само за количество. Така, грубо речено, на пример, величината: <i>три килограми</i> е скаларна, додека величината: <i>три килограми јаболка</i> е векторска. Но, под квалитет во математиката може да се подразбираат и некои својства кои немаат смисла за нематематичарите. Така на пример, отсечките може да ги сметаме за вектори, ако ги <i>насочиме</i>, т.е. ако условно кажеме каде почнува, а каде завршува отсечката. Во овој случај квалитетот е <i>насоката</i>, а за квантитет би ја зеле <i>должината на отсечката</i>. Од друга страна секој <a href="/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80" title="Скалар">скалар</a> може да го сметаме за вектор со <i>квалитет еднаков на нула</i>, при што смислата на квалитетот во овој случај е филозофска, т.е. имплицитна. Во <a href="/wiki/%D0%A4%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0" title="Физика">физиката</a> векторски величини се, на пример, <a href="/wiki/%D0%91%D1%80%D0%B7%D0%B8%D0%BD%D0%B0" title="Брзина">брзината</a>, <a href="/wiki/%D0%97%D0%B0%D0%B1%D1%80%D0%B7%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Забрзување">забрзувањето</a>, <a href="/wiki/%D0%A1%D0%B8%D0%BB%D0%B0" title="Сила">силата</a>, <a href="/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%81_(%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0)" title="Импулс (механика)">импулсот</a> и сл. </p><p>Најчестото толкување на векторите е геометриското - векторот е насочена отсечка од <a href="/wiki/%D0%A0%D0%B0%D0%BC%D0%BD%D0%B8%D0%BD%D0%B0" class="mw-redirect mw-disambig" title="Рамнина">рамнината</a> или <a href="/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Простор">просторот</a>. Ова толкување има многу практична примена во математиката и особено во <a href="/wiki/%D0%A4%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0" title="Физика">физиката</a>. </p><p>За разлика од скаларите, кај векторите важат поинакви правила за извршување на операциите. </p><p>Сите вектори во математиката се разгледуваат во рамките на <a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Векторски простор">теоријата на векторски простори</a>, која пак сама по себе е дел од <a href="/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линеарна алгебра">линеарната алгебра</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Претставување_на_векторите"><span id=".D0.9F.D1.80.D0.B5.D1.82.D1.81.D1.82.D0.B0.D0.B2.D1.83.D0.B2.D0.B0.D1.9A.D0.B5_.D0.BD.D0.B0_.D0.B2.D0.B5.D0.BA.D1.82.D0.BE.D1.80.D0.B8.D1.82.D0.B5"></span>Претставување на векторите</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit&section=1" title="Уреди го одделот „Претставување на векторите“" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&section=1" title="Уреди изворен код на одделот: Претставување на векторите"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Математичката апстракција дозволува елементите на <a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Векторски простор">векторскиот простор</a> да се наречат вектори иако директно од нив не се отчитуваат количественоста и качественоста. Така, на пример, <a href="/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество">множеството</a> од сите <a href="/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BD%D0%BE%D0%BC" title="Полином">полиноми</a> со <a href="/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реален број">реални коефициенти</a> со степен не поголем од некој <a href="/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Природен број">природен број</a> <i>n</i> претставува векторски простор, па следствено секој полином претставува вектор. </p><p>При ваквото сфаќање на векторите се јавува потребата за нивно <i>претставување</i>, слично како кај скаларите. Но претставувањето на полиномот, на пример, како вектор е невозможно со „геометрискиот модел на вектор“, т.е. со насочена отсечка. Затоа се применуваат други, поапстрактни, методи кои важат за сите вектори подеднакво. </p> <div class="mw-heading mw-heading3"><h3 id="Геометриско_претставување_на_векторите"><span id=".D0.93.D0.B5.D0.BE.D0.BC.D0.B5.D1.82.D1.80.D0.B8.D1.81.D0.BA.D0.BE_.D0.BF.D1.80.D0.B5.D1.82.D1.81.D1.82.D0.B0.D0.B2.D1.83.D0.B2.D0.B0.D1.9A.D0.B5_.D0.BD.D0.B0_.D0.B2.D0.B5.D0.BA.D1.82.D0.BE.D1.80.D0.B8.D1.82.D0.B5"></span>Геометриско претставување на векторите</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit&section=2" title="Уреди го одделот „Геометриско претставување на векторите“" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&section=2" title="Уреди изворен код на одделот: Геометриско претставување на векторите"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Vector_AB_from_A_to_B.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Vector_AB_from_A_to_B.svg/220px-Vector_AB_from_A_to_B.svg.png" decoding="async" width="220" height="86" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Vector_AB_from_A_to_B.svg/330px-Vector_AB_from_A_to_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Vector_AB_from_A_to_B.svg/440px-Vector_AB_from_A_to_B.svg.png 2x" data-file-width="342" data-file-height="134" /></a><figcaption>Вектор со почеток во точка <i><b>A</b></i> и крај во точка <i><b>B</b></i></figcaption></figure> <p>Векторите како насочени отсечки во рамнината или просторот може да ги разгледуваме само во ограничен број случаи. Така во <a href="/w/index.php?title=%D0%A0%D0%B5%D0%B0%D0%BB%D0%B5%D0%BD_%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80&action=edit&redlink=1" class="new" title="Реален Евклидов простор (страницата не постои)">реалниот Евклидов простор</a>, а тоа е просторот како што човекот го восприема, векторите може да ги <i>нацртаме</i> како <i>стрелки</i>. Ова може да го направиме и во рамнината (две димензии) и во просторот (три димензии). „Цртањето“ може да продолжи и во четири димензии, но визуелната репрезентација сега ќе биде несфатлива за човековиот мозок. Затоа се преминува кон <i>аналитичко претставување</i> на векторите од векторскиот простор. </p><p>Нека избереме произволен вектор од рамнината или просторот. За него знаеме <i>каде почнува</i>, а <i>каде завршува</i>. Нека сега го земеме векторот кој е потполно ист со претходно избраниот, но така што ги промениме местата на крајот и почетокот, т.е. она што кај првиот вектор било почеток, кај вториот нека биде крај. Тогаш ваквиот вектор се вика <i>спротивен вектор</i> на избраниот. Ако избраниот вектор го обележиме со <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span>, тогаш спротивниот ќе го бележиме со <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 -{\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861c1f4c6464113ddda6059d9b42834c48bfca1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.509ex;" alt="{\displaystyle -{\vec {a}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Аналитичко_претставување_на_векторите"><span id=".D0.90.D0.BD.D0.B0.D0.BB.D0.B8.D1.82.D0.B8.D1.87.D0.BA.D0.BE_.D0.BF.D1.80.D0.B5.D1.82.D1.81.D1.82.D0.B0.D0.B2.D1.83.D0.B2.D0.B0.D1.9A.D0.B5_.D0.BD.D0.B0_.D0.B2.D0.B5.D0.BA.D1.82.D0.BE.D1.80.D0.B8.D1.82.D0.B5"></span>Аналитичко претставување на векторите</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit&section=3" title="Уреди го одделот „Аналитичко претставување на векторите“" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&section=3" title="Уреди изворен код на одделот: Аналитичко претставување на векторите"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Linearna_kombinacija.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/mk/thumb/3/37/Linearna_kombinacija.png/300px-Linearna_kombinacija.png" decoding="async" width="300" height="274" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/mk/thumb/3/37/Linearna_kombinacija.png/450px-Linearna_kombinacija.png 1.5x, //upload.wikimedia.org/wikipedia/mk/3/37/Linearna_kombinacija.png 2x" data-file-width="456" data-file-height="416" /></a><figcaption>Произволен вектор од рамнината како комбинација на два базни вектори</figcaption></figure> <p>Во теоријата на векторските простори имаме <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема">тврдење</a> кое вели дека секој <a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80#База_и_димензија_на_векторски_простор" title="Векторски простор">векторски простор има база</a>. База е најмалото <a href="/w/index.php?title=%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B7%D0%B0%D0%B2%D0%B8%D1%81%D0%BD%D0%BE%D1%81%D1%82&action=edit&redlink=1" class="new" title="Линеарна зависност (страницата не постои)">линеарно независно множество</a> такво што сите вектори од просторот можат да се претстават како <a href="/w/index.php?title=%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%BA%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%86%D0%B8%D1%98%D0%B0&action=edit&redlink=1" class="new" title="Линеарна комбинација (страницата не постои)">комбинација на елементите од базата</a>. Така ако во <a href="/wiki/%D0%A0%D0%B0%D0%BC%D0%BD%D0%B8%D0%BD%D0%B0" class="mw-redirect mw-disambig" title="Рамнина">рамнината</a> воведеме правоаголен <a href="/wiki/%D0%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D0%BE%D0%B2_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Декартов координатен систем">Декартов координатен систем</a>, и избереме два вектора такви што секој од нив лежи на различна координатна оска и двата за почеток го имаат <a href="/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5%D0%BD_%D0%BF%D0%BE%D1%87%D0%B5%D1%82%D0%BE%D0%BA" title="Координатен почеток">координатниот почеток</a>, тогаш овие вектори чинат база за дводимензионалниот реален Евклидов простор - <a href="/wiki/%D0%A0%D0%B0%D0%BC%D0%BD%D0%B8%D0%BD%D0%B0" class="mw-redirect mw-disambig" title="Рамнина">рамнината</a>. Слично е и за просторот, само што во тој случај ќе имаме три такви вектори. Нека земеме 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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> од рамнината и нека векторите <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 {\vec {e}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad1143107d7fbfc245bd9fcdcc422d341ec852e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{1}}"></span> и <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 {\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c09c5782e951663cb81ed18fcfe715315d4e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{2}}"></span> ја чинат базата за <i>просторот</i> (во овој случај под <i>простор</i> се подразбира <i>рамнината</i>!). Тогаш постојат реални броеви (<a href="/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80" title="Скалар">скалари</a>) <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/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> така што важи: </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 {\vec {v}}=a\cdot {\vec {e}}_{1}+b\cdot {\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=a\cdot {\vec {e}}_{1}+b\cdot {\vec {e}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0f9a09e42a7561569a3a9f521925d2f30b57ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.254ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}=a\cdot {\vec {e}}_{1}+b\cdot {\vec {e}}_{2}}"></span></dd></dl> <p>Тие реални броеви ги нарекуваме координати на векторот <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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> во однос на базата <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 \{{\vec {e}}_{1},{\vec {e}}_{2}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\vec {e}}_{1},{\vec {e}}_{2}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20aef78080faba73d6cbb05534a7db44df58267b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.913ex; height:2.843ex;" alt="{\displaystyle \{{\vec {e}}_{1},{\vec {e}}_{2}\}}"></span> и запишуваме: </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 {\vec {v}}=(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8303087f9f1d7ff6f118e2628adbf4f0db9df4e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.344ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}=(a,b)}"></span></dd></dl> <p>што всушност претставува <b>аналитички</b> (<i>координатен</i>) запис за векторот кој го избравме. Бидејќи векторскиот простор има бесконечно многу бази, секој вектор не мора да има ист аналитички запис во однос на различни бази. Всушност во пракса ретко се случува ист вектор во однос на различни бази да има ист аналитички запис. </p><p>Ова што го направивме за рамнината можеме да го направиме и за просторот и за векторскиот простор од полиноми и за n-димензионалниот реален Евклидов простор - едноставно за секој векторски простор. Значи може да кажеме дека аналитичкото бележење на векторите е универзално и не зависи од изборот на просторот, туку само може да варира во зависност од избраната база. </p><p>Кога е воведено аналитичкото претставување на векторите може да се разгледуваат и посложени простори од познатиот тридимензионален простор, но нивното разгледување е лишено од визуелизацијата која е речиси пресудна кај рамнината и тридимензионалниот простор. </p><p>Ако е даден векторот <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 {\vec {v}}=(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8303087f9f1d7ff6f118e2628adbf4f0db9df4e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.344ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}=(a,b)}"></span>, тогаш негов спротивен ќе биде векторот <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 -{\vec {v}}=(-a,-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo>,</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\vec {v}}=(-a,-b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7638148f54fc87f4a0005dd374ce2fbd58e58496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.769ex; height:2.843ex;" alt="{\displaystyle -{\vec {v}}=(-a,-b)}"></span>. </p><p><b>Модулот на векторот</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 {\vec {v}}=(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8303087f9f1d7ff6f118e2628adbf4f0db9df4e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.344ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}=(a,b)}"></span> (т.е. неговата „должина“),со ознака <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|{\vec {v}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\vec {v}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e28dd3f7749b52b4f61beb83bf0e05b3dfce4caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.469ex; height:2.843ex;" alt="{\displaystyle \left|{\vec {v}}\right|}"></span>, може да се пресмета како: </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|{\vec {v}}\right|={\sqrt {a^{2}+b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <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> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\vec {v}}\right|={\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e2a9920b2a925e6d59b715fc5c9f7a5cec4fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.067ex; height:3.509ex;" alt="{\displaystyle \left|{\vec {v}}\right|={\sqrt {a^{2}+b^{2}}}}"></span></dd></dl> <p>Во општ случај, за <i>n</i>-димензионален вектор (<b><i>n</i>-вектор</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 {\vec {a}}=(a_{1},a_{2},...,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=(a_{1},a_{2},...,a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a6a3bd062c5a4c63435ea8bd94c556b7eecc18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.358ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}=(a_{1},a_{2},...,a_{n})}"></span>, модулот се пресметува како: </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|{\vec {a}}\right|={\sqrt {a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}}}={\sqrt {\sum _{k=1}^{n}a_{k}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\vec {a}}\right|={\sqrt {a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}}}={\sqrt {\sum _{k=1}^{n}a_{k}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4686f82fbe6757a752d70db01fea0819606c919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.003ex; height:7.509ex;" alt="{\displaystyle \left|{\vec {a}}\right|={\sqrt {a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}}}={\sqrt {\sum _{k=1}^{n}a_{k}^{2}}}}"></span></dd></dl> <p>Ако се работи за вектори во смисла на насочени отсечки, тогаш модулот е всушност должината. Во општ случај не може да се говори за должина на вектор во потесна смисла. </p> <div class="mw-heading mw-heading2"><h2 id="Операции_со_вектори"><span id=".D0.9E.D0.BF.D0.B5.D1.80.D0.B0.D1.86.D0.B8.D0.B8_.D1.81.D0.BE_.D0.B2.D0.B5.D0.BA.D1.82.D0.BE.D1.80.D0.B8"></span>Операции со вектори</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit&section=4" title="Уреди го одделот „Операции со вектори“" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&section=4" title="Уреди изворен код на одделот: Операции со вектори"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Како што напоменавме операциите со вектори се разликуваат од операциите со скалари. Според самата <a href="/wiki/%D0%94%D0%B5%D1%84%D0%B8%D0%BD%D0%B8%D1%86%D0%B8%D1%98%D0%B0" title="Дефиниција">дефиниција</a> на поимот <i>операција</i> во математиката, <i>исходот</i> од оперирањето со вектори треба и самиот да е вектор. </p> <div class="mw-heading mw-heading3"><h3 id="Собирање_на_вектори"><span id=".D0.A1.D0.BE.D0.B1.D0.B8.D1.80.D0.B0.D1.9A.D0.B5_.D0.BD.D0.B0_.D0.B2.D0.B5.D0.BA.D1.82.D0.BE.D1.80.D0.B8"></span>Собирање на вектори</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit&section=5" title="Уреди го одделот „Собирање на вектори“" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&section=5" title="Уреди изворен код на одделот: Собирање на вектори"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Vector_addition.svg" class="mw-file-description" title="Собирање вектори"><img alt="Собирање вектори" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Vector_addition.svg/200px-Vector_addition.svg.png" decoding="async" width="200" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Vector_addition.svg/300px-Vector_addition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Vector_addition.svg/400px-Vector_addition.svg.png 2x" data-file-width="445" data-file-height="235" /></a><figcaption>Собирање вектори</figcaption></figure> <p>Собирањето на геометриските вектори (насочените отсечки) се врши на следниов начин: треба да се пресмета збирот на векторите <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> и <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 {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span>. За таа цел постапуваме вака: го нанесуваме векторот <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> со почеток во некоја избрана точка (при ова ги запазуваме насоката и должината на векторот!), а потоа во крајната точка на векторот <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> (при врвот) го нанесуваме векторот <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 {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> (исто така запазувајќи ги неговите насока и должина). Векторот <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 {\vec {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}}"></span> кој има почеток во почетната точка (почетокот на <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span>) и крај во последната точка (врвот на <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 {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span>) се вика збир на векторите <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> и <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 {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> и се бележи исто како и кај скаларите: </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 {\vec {c}}={\vec {a}}+{\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}={\vec {a}}+{\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606f0086204fe0b7db7cd5f7d25baa80de992a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.486ex; height:3.009ex;" alt="{\displaystyle {\vec {c}}={\vec {a}}+{\vec {b}}}"></span></dd></dl> <p>Ако векторите се зададени аналитички т.е. координатно, тогаш собирањето се врши „по координати“. Нека се дадени векторите (во општ случај со <i>n</i>-координати): </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 {\vec {a}}=(a_{1},a_{2},...,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=(a_{1},a_{2},...,a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a6a3bd062c5a4c63435ea8bd94c556b7eecc18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.358ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}=(a_{1},a_{2},...,a_{n})}"></span> и</dd></dl> <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 {\vec {b}}=(b_{1},b_{2},...,b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}=(b_{1},b_{2},...,b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b99a2aab2270709b5c4aaf25ed4529d3210ad363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.525ex; height:3.343ex;" alt="{\displaystyle {\vec {b}}=(b_{1},b_{2},...,b_{n})}"></span></dd></dl> <p>тогаш за збирот <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 {\vec {c}}={\vec {a}}+{\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}={\vec {a}}+{\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606f0086204fe0b7db7cd5f7d25baa80de992a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.486ex; height:3.009ex;" alt="{\displaystyle {\vec {c}}={\vec {a}}+{\vec {b}}}"></span> имаме: </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 {\vec {c}}=(a_{1}+b_{1},a_{2}+b_{2},...,a_{n}+b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <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>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}=(a_{1}+b_{1},a_{2}+b_{2},...,a_{n}+b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7b21e9fe6e10353e40d39dc4be4f24a34d8a7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.192ex; height:2.843ex;" alt="{\displaystyle {\vec {c}}=(a_{1}+b_{1},a_{2}+b_{2},...,a_{n}+b_{n})}"></span></dd></dl> <p>За собирањето на вектори важат: </p> <ul><li><b><a href="/wiki/%D0%9A%D0%BE%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%B5%D0%BD_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD" class="mw-redirect" title="Комутативен закон">комутативност</a>:</b></li></ul> <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 {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2b7482da2847013fa5de8900757c561b98c815" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.427ex; height:3.009ex;" alt="{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}"></span></dd></dl></dd></dl> <ul><li><b><a href="/w/index.php?title=%D0%90%D1%81%D0%BE%D1%86%D0%B8%D1%98%D0%B0%D1%82%D0%B8%D0%B2%D0%B5%D0%BD_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD&action=edit&redlink=1" class="new" title="Асоцијативен закон (страницата не постои)">асоцијативност</a>:</b></li></ul> <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 {\vec {a}}+\left({\vec {b}}+{\vec {c}}\right)=\left({\vec {a}}+{\vec {b}}\right)+{\vec {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}+\left({\vec {b}}+{\vec {c}}\right)=\left({\vec {a}}+{\vec {b}}\right)+{\vec {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22c7af05b24fcd00a0bd4ac5f350c56710cc0123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.105ex; height:4.843ex;" alt="{\displaystyle {\vec {a}}+\left({\vec {b}}+{\vec {c}}\right)=\left({\vec {a}}+{\vec {b}}\right)+{\vec {c}}}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Одземање_на_вектори"><span id=".D0.9E.D0.B4.D0.B7.D0.B5.D0.BC.D0.B0.D1.9A.D0.B5_.D0.BD.D0.B0_.D0.B2.D0.B5.D0.BA.D1.82.D0.BE.D1.80.D0.B8"></span>Одземање на вектори</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit&section=6" title="Уреди го одделот „Одземање на вектори“" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&section=6" title="Уреди изворен код на одделот: Одземање на вектори"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Vector_subtraction.svg" class="mw-file-description" title="Одземање на вектори"><img alt="Одземање на вектори" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Vector_subtraction.svg/200px-Vector_subtraction.svg.png" decoding="async" width="200" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Vector_subtraction.svg/300px-Vector_subtraction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Vector_subtraction.svg/400px-Vector_subtraction.svg.png 2x" data-file-width="206" data-file-height="149" /></a><figcaption>Одземање на вектори</figcaption></figure> <p>Одземањето на вектори се извршува на ист начин како и собирањето, така што разликата на векторите <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> и <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 {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> е всушност збир на векторот <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> и векторот <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 -{\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd2f8046171316a2108b92251cf4639ee195daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.902ex; height:3.009ex;" alt="{\displaystyle -{\vec {b}}}"></span>. Истото важи и за векторите зададени во координатна форма: </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 {\vec {a}}-{\vec {b}}=(a_{1}-b_{1},a_{2}-b_{2},...,a_{n}-b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <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>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}-{\vec {b}}=(a_{1}-b_{1},a_{2}-b_{2},...,a_{n}-b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5546bd96d5c064a09ed7afe14edbc950b6847188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.133ex; height:3.343ex;" alt="{\displaystyle {\vec {a}}-{\vec {b}}=(a_{1}-b_{1},a_{2}-b_{2},...,a_{n}-b_{n})}"></span></dd></dl> <p>ако се зададени векторите: </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 {\vec {a}}=(a_{1},a_{2},...,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=(a_{1},a_{2},...,a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a6a3bd062c5a4c63435ea8bd94c556b7eecc18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.358ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}=(a_{1},a_{2},...,a_{n})}"></span> и</dd></dl> <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 {\vec {b}}=(b_{1},b_{2},...,b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}=(b_{1},b_{2},...,b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b99a2aab2270709b5c4aaf25ed4529d3210ad363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.525ex; height:3.343ex;" alt="{\displaystyle {\vec {b}}=(b_{1},b_{2},...,b_{n})}"></span></dd></dl> <p>Ако на векторот <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> му го додадеме неговиот спротивен 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 -{\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861c1f4c6464113ddda6059d9b42834c48bfca1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.509ex;" alt="{\displaystyle -{\vec {a}}}"></span>, тогаш се добива: </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 {\vec {a}}+(-{\vec {a}})={\vec {a}}-{\vec {a}}=(a_{1}-a_{1},a_{2}-a_{2},...,a_{n}-a_{n})=(0,0,...,0)=\operatorname {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}+(-{\vec {a}})={\vec {a}}-{\vec {a}}=(a_{1}-a_{1},a_{2}-a_{2},...,a_{n}-a_{n})=(0,0,...,0)=\operatorname {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70675ff3fe20090360a36cd405bae3237cb25429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.487ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}+(-{\vec {a}})={\vec {a}}-{\vec {a}}=(a_{1}-a_{1},a_{2}-a_{2},...,a_{n}-a_{n})=(0,0,...,0)=\operatorname {O} }"></span></dd></dl> <p>Вака добиениот вектор (кој е збир на било кои два спротивни вектори) се нарекува <b>нулти вектор</b>, <i>нула-вектор</i> или само <i>нула</i> (кога не води до забуна со скаларната нула!). Овој вектор во однос на сите операции со вектори се однесува како и нулата во однос на сите операции со скалари, па може да кажеме дека нултиот вектор во векторскиот простор ѝ соодветствува на нулата во скаларното поле. За да не се меша (во ознаката) со скаларната нула, се бележи со <i>големо о</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 \operatorname {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/145578612c5ddd9c84c1ab413731c69664a6ad5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \operatorname {O} }"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Множење_на_вектори"><span id=".D0.9C.D0.BD.D0.BE.D0.B6.D0.B5.D1.9A.D0.B5_.D0.BD.D0.B0_.D0.B2.D0.B5.D0.BA.D1.82.D0.BE.D1.80.D0.B8"></span>Множење на вектори</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit&section=7" title="Уреди го одделот „Множење на вектори“" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&section=7" title="Уреди изворен код на одделот: Множење на вектори"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Scalar_multiplication_by_r%3D3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/mk/thumb/f/fa/Scalar_multiplication_by_r%3D3.svg/200px-Scalar_multiplication_by_r%3D3.svg.png" decoding="async" width="200" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/mk/thumb/f/fa/Scalar_multiplication_by_r%3D3.svg/300px-Scalar_multiplication_by_r%3D3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/mk/thumb/f/fa/Scalar_multiplication_by_r%3D3.svg/400px-Scalar_multiplication_by_r%3D3.svg.png 2x" data-file-width="622" data-file-height="345" /></a><figcaption>Множење на вектор со скалар.</figcaption></figure> <table align="right" class="infobox" style="width: 200px; margin: 0 0 1em 1em"> <tbody><tr style="background:#ccccff" align="center"> <td style="border-bottom: 2px solid #303060"><b>Статии поврзани со <a href="/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линеарна алгебра">линеарната алгебра</a></b> </td></tr> <tr style="background:#ccccff" align="center"> <td><b>Теорија на матрици</b> </td></tr> <tr> <td align="center"> <p><a href="/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" class="mw-redirect" title="Матрица">Матрица</a> <br /> <a href="/wiki/%D0%94%D0%B5%D1%82%D0%B5%D1%80%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82%D0%B0" title="Детерминанта">Детерминанта</a> <br /> </p> </td></tr> <tr style="background:#ccccff" align="center"> <td><b>Системи линеарни равенки</b> </td></tr> <tr> <td align="center"> <p><a href="/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B0" title="Линеарна равенка">Линеарна равенка</a> <br /> <a href="/wiki/%D0%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B8_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B8" class="mw-redirect" title="Систем линеарни равенки">Систем линеарни равенки</a> <br /> <a href="/w/index.php?title=%D0%9A%D1%80%D0%B0%D0%BC%D0%B5%D1%80%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D0%BE&action=edit&redlink=1" class="new" title="Крамерово правило (страницата не постои)">Крамерово правило</a> <br /> <a href="/w/index.php?title=%D0%9A%D1%80%D0%BE%D0%BD%D0%B5%D0%BA%D0%B5%D1%80-%D0%9A%D0%B0%D0%BF%D0%B5%D0%BB%D0%B8%D0%B5%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0&action=edit&redlink=1" class="new" title="Кронекер-Капелиева теорема (страницата не постои)">Кронекер-Капелиева теорема</a> <br /> </p> </td></tr> <tr style="background:#ccccff" align="center"> <td><b>Линеарни пресликувања и векторски простори</b> </td></tr> <tr> <td align="center"> <p><a class="mw-selflink selflink">Вектор</a>, <a href="/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80" title="Скалар">Скалар</a> <br /> <a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Векторски простор">Векторски простор</a> <br /> <a href="/w/index.php?title=%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B7%D0%B0%D0%B2%D0%B8%D1%81%D0%BD%D0%BE%D1%81%D1%82&action=edit&redlink=1" class="new" title="Линеарна зависност (страницата не постои)">Линеарна зависност</a> <br /> <a href="/wiki/%D0%9F%D1%80%D0%B5%D1%81%D0%BB%D0%B8%D0%BA%D1%83%D0%B2%D0%B0%D1%9A%D0%B5#Линеарни_пресликувања" title="Пресликување">Линеарно пресликување</a> <br /> <a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D0%BA%D1%82%D1%80%D0%B0%D0%BB%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0&action=edit&redlink=1" class="new" title="Спектрална теорема (страницата не постои)">Спектрална теорема</a> <br /> </p> </td></tr> <tr style="background:#ccccff" align="center"> <td><b>Останати статии</b> </td></tr> <tr> <td align="center"> <p><a href="/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4" title="Скаларен производ">Скаларен производ</a> <br /> <a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4&action=edit&redlink=1" class="new" title="Векторски производ (страницата не постои)">Векторски производ</a> <br /> <a href="/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Аналитичка геометрија">Аналитичка геометрија</a><br /> </p> <div class="tright" style="margin: 0 0 1em 1em; border: solid #aaa 1px; background: #f9f9f9; padding: 1ex; font-size: 90%;"> <table style="background: transparent;"> <tbody><tr> <td><div style="position: relative; width: 36px; height: 32px; overflow: hidden"> <div style="position: absolute; top: 0px; left: 0px; z-index: 2"><span typeof="mw:File"><a href="/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description" title="Портал:Математика"><img alt="Портал:Математика" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/48px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/64px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></div> </div> </td> <td><i><b><a href="/wiki/%D0%9F%D0%BE%D1%80%D1%82%D0%B0%D0%BB:%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Портал:Математика">Портал: Математика</a></b></i> </td></tr></tbody></table> </div> </td></tr></tbody></table> <p>Кога се множат вектори често настанува следнава забуна: множењето вектори се меша со множењето на вектор со <a href="/wiki/%D0%91%D1%80%D0%BE%D1%98" title="Број">број</a> (т.е. <a href="/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80" title="Скалар">скалар</a>). Множењето на 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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> со скалар <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> се врши на следниов начин: </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 k\cdot {\vec {a}}=k\cdot (a_{1},a_{2},...,a_{n})=(ka_{1},ka_{2},...,ka_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>k</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>k</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>k</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>k</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot {\vec {a}}=k\cdot (a_{1},a_{2},...,a_{n})=(ka_{1},ka_{2},...,ka_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf486d2b490865da910b3f2e42250b296da4f17f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.9ex; height:2.843ex;" alt="{\displaystyle k\cdot {\vec {a}}=k\cdot (a_{1},a_{2},...,a_{n})=(ka_{1},ka_{2},...,ka_{n})}"></span></dd></dl> <p>Ова геометриски може да го толкуваме на следниот начин: векторот <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 k\cdot {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6dfdd6be1fa138983448e8d3a0d05aecbb80f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.12ex; height:2.343ex;" alt="{\displaystyle k\cdot {\vec {a}}}"></span> ја има истата насока како и векторот <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span>, со таа разлика што има должина (модул) за <i>k</i> пати поголема (или помала, ако <i>k</i><1) од него. </p><p>„Вистинското“ множење на вектори во математиката се нарекува <a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4&action=edit&redlink=1" class="new" title="Векторски производ (страницата не постои)">векторски производ</a> на вектори и се бележи со симболот <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 \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span>. Околу дефиницијата и оперирањето со векторските производи, <a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4&action=edit&redlink=1" class="new" title="Векторски производ (страницата не постои)">видете на соодветната статија</a>. Векторскиот производ на два вектора <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 {\vec {a}}=(a_{1},a_{2},a_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=(a_{1},a_{2},a_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/335b867595a49b7a5fc01dbf39f93a1ceac5715f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.058ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}=(a_{1},a_{2},a_{3})}"></span> и <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 {\vec {b}}=(b_{1},b_{2},b_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}=(b_{1},b_{2},b_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08b5f6d48d3d872ac8e7dfe3f275652dfb40ca46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.225ex; height:3.343ex;" alt="{\displaystyle {\vec {b}}=(b_{1},b_{2},b_{3})}"></span> е вектор кој е нормален на обата вектора и истовремено има модул: </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|{\vec {a}}\times {\vec {b}}\right|=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cdot \sin {\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mo>⋅</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\vec {a}}\times {\vec {b}}\right|=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cdot \sin {\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/286f2b7b41d4e50b3b93186469e28dd8a9379ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:22.556ex; height:4.009ex;" alt="{\displaystyle \left|{\vec {a}}\times {\vec {b}}\right|=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cdot \sin {\alpha }}"></span></dd></dl> <p>каде со <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> е означен аголот меѓу почетните вектори, а со <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|{\vec {a}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\vec {a}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb1c5421b5d64ad749a12c356c6af28b6ae2cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.523ex; height:2.843ex;" alt="{\displaystyle \left|{\vec {a}}\right|}"></span> и <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|{\vec {b}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\vec {b}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b150af37507773547081e357cad7f61fd6a2e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:2.388ex; height:4.009ex;" alt="{\displaystyle \left|{\vec {b}}\right|}"></span> се означени нивните модули, додека неговиот координатен облик е: </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 {\vec {a}}\times {\vec {b}}={\begin{vmatrix}{\vec {e}}_{1}&{\vec {e}}_{2}&{\vec {e}}_{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}=\left({\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}},{\begin{vmatrix}a_{3}&a_{1}\\b_{3}&b_{1}\end{vmatrix}},{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\times {\vec {b}}={\begin{vmatrix}{\vec {e}}_{1}&{\vec {e}}_{2}&{\vec {e}}_{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}=\left({\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}},{\begin{vmatrix}a_{3}&a_{1}\\b_{3}&b_{1}\end{vmatrix}},{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba0ce7abc7f3e64ab9ee671b4eb785b920421e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:57.2ex; height:9.176ex;" alt="{\displaystyle {\vec {a}}\times {\vec {b}}={\begin{vmatrix}{\vec {e}}_{1}&{\vec {e}}_{2}&{\vec {e}}_{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}=\left({\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}},{\begin{vmatrix}a_{3}&a_{1}\\b_{3}&b_{1}\end{vmatrix}},{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\right)}"></span></dd></dl> <p>Постои и друг начин на множење вектори, т.н. <i>скаларно множење на вектори</i> (<a href="/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4" title="Скаларен производ">скаларен производ</a>), но при скаларно множење на два вектора се добива резултат скалар (од таму и името) што, математички значи дека <i>операцијата не е затворена</i> во однос на векторскиот простор, т.е., на некој начин, <i>не е добро дефинирана</i>. Скаларниот производ се бележи со <i>точка</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 \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"></span>. Скаларниот производ на истите два вектора изнесува: </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 {\vec {a}}\cdot {\vec {b}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\cdot {\vec {b}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5840f614304f6275963639f6b87877f9e39693e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.79ex; height:3.176ex;" alt="{\displaystyle {\vec {a}}\cdot {\vec {b}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}}"></span></dd></dl> <p>или </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 {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\left|{\vec {b}}\right|\cos {\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\left|{\vec {b}}\right|\cos {\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf45da5fafaf850f19dfe485cd405b91ed6f3f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:17.772ex; height:4.009ex;" alt="{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\left|{\vec {b}}\right|\cos {\alpha }}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Поврзано"><span id=".D0.9F.D0.BE.D0.B2.D1.80.D0.B7.D0.B0.D0.BD.D0.BE"></span>Поврзано</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&veaction=edit&section=8" title="Уреди го одделот „Поврзано“" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&section=8" title="Уреди изворен код на одделот: Поврзано"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80" title="Скалар">Скалар</a></li> <li><a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Векторски простор">Векторски простор</a></li> <li><a href="/w/index.php?title=%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4&action=edit&redlink=1" class="new" title="Векторски производ (страницата не постои)">Векторски производ</a></li> <li><a href="/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4" title="Скаларен производ">Скаларен производ</a></li> <li><a href="/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линеарна алгебра">Линеарна алгебра</a></li> <li><a href="/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Аналитичка геометрија">Аналитичка геометрија</a></li> <li><a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B0_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B0" title="Векторска графика">Векторска графика</a></li></ul> <p><br /> </p> <table width="55%" align="center" cellspacing="3" style="border: 1px solid #708090; -moz-border-radius: 10px; background-color: white; margin-bottom: 3px;"> <tbody><tr> <td align="center"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Cscr-featured.svg/50px-Cscr-featured.svg.png" decoding="async" width="50" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Cscr-featured.svg/75px-Cscr-featured.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Cscr-featured.svg/100px-Cscr-featured.svg.png 2x" data-file-width="466" data-file-height="443" /></span></span> </td> <td align="center">Статијата „<b>Вектор</b>“ е <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%98%D0%B0:%D0%9A%D1%80%D0%B8%D1%82%D0%B5%D1%80%D0%B8%D1%83%D0%BC%D0%B8_%D0%B7%D0%B0_%D0%B8%D0%B7%D0%B1%D1%80%D0%B0%D0%BD%D0%B0_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%98%D0%B0" title="Википедија:Критериуми за избрана статија">избрана статија</a>. Ве повикуваме и Вас да <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%98%D0%B0:%D0%92%D0%B0%D1%88%D0%B0%D1%82%D0%B0_%D0%BF%D1%80%D0%B2%D0%B0_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%98%D0%B0" title="Википедија:Вашата прва статија">напишете</a> и <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%98%D0%B0:%D0%9A%D0%B0%D0%BD%D0%B4%D0%B8%D0%B4%D0%B0%D1%82%D0%B8_%D0%B7%D0%B0_%D0%B8%D0%B7%D0%B1%D1%80%D0%B0%D0%BD%D0%B0_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%98%D0%B0" title="Википедија:Кандидати за избрана статија">предложите</a> избрана статија (останати <a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%98%D0%B7%D0%B1%D1%80%D0%B0%D0%BD%D0%B8_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%B8" title="Категорија:Избрани статии">избрани статии</a>). </td></tr></tbody></table></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Преземено од „<a dir="ltr" href="https://mk.wikipedia.org/w/index.php?title=Вектор&oldid=4910097">https://mk.wikipedia.org/w/index.php?title=Вектор&oldid=4910097</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8" title="Специјална:Категории">Категории</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8" title="Категорија:Вектори">Вектори</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%90%D0%BF%D1%81%D1%82%D1%80%D0%B0%D0%BA%D1%82%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Категорија:Апстрактна алгебра">Апстрактна алгебра</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Категорија:Линеарна алгебра">Линеарна алгебра</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Категорија:Векторска анализа">Векторска анализа</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%9A%D0%BE%D0%BD%D1%86%D0%B5%D0%BF%D1%82%D0%B8_%D0%B2%D0%BE_%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0%D1%82%D0%B0" title="Категорија:Концепти во физиката">Концепти во физиката</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Категорија:Аналитичка геометрија">Аналитичка геометрија</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Скриена категорија: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%98%D0%B7%D0%B1%D1%80%D0%B0%D0%BD%D0%B8_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%B8" title="Категорија:Избрани статии">Избрани статии</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"> Последната промена на страницава е извршена на 24 декември 2022 г. во 08:01 ч.</li> <li id="footer-info-copyright">Текстот е достапен под условите на лиценцата <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/">Криејтив комонс Наведи извор-Сподели под исти услови</a>. 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Вектор — Википедија