{"id":113,"date":"2023-09-16T13:06:27","date_gmt":"2023-09-16T13:06:27","guid":{"rendered":"https:\/\/mathority.org\/nl\/hoe-je-de-hoek-tussen-twee-vectoren-berekent-voorbeelden-opgeloste-oefeningen\/"},"modified":"2023-09-16T13:06:27","modified_gmt":"2023-09-16T13:06:27","slug":"hoe-je-de-hoek-tussen-twee-vectoren-berekent-voorbeelden-opgeloste-oefeningen","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/hoe-je-de-hoek-tussen-twee-vectoren-berekent-voorbeelden-opgeloste-oefeningen\/","title":{"rendered":"Hoe de hoek tussen twee vectoren te berekenen"},"content":{"rendered":"<p>Op deze pagina ontdek je hoe je de hoek tussen twee vectoren berekent. Daarnaast krijg je ook voorbeelden te zien en kun je oefenen met oefeningen en stap voor stap opgeloste problemen. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-angulo-entre-dos-vectores\"><\/span> Formule voor de hoek tussen twee vectoren <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/angle-entre-deux-vecteurs-et-produit-scalaire.webp\" alt=\"hoek tussen twee puntproductvectoren\" class=\"wp-image-583\" width=\"187\" height=\"190\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Als we de <a href=\"https:\/\/mathority.org\/nl\/bereken-het-scalaire-product-tussen-twee-vectoren-voorbeelden-opgeloste-oefeningen\/\">definitie van het puntproduct<\/a> onthouden, kan dit worden berekend met behulp van de volgende vergelijking:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-66d255d6dd1b74f0a5cd6a209c2a9505_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}} \\cdot \\vv{\\text{v}} = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(\\alpha )\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Uit deze gelijkheid kunnen we de formule verkrijgen die ons zal helpen direct de hoek te vinden die wordt gevormd door twee vectoren:<\/p>\n<p> <strong>De cosinus van de hoek gevormd door twee vectoren is gelijk aan het puntproduct tussen de twee vectoren gedeeld door het product van de moduli van de twee vectoren.<\/strong><\/p>\n<p> Met andere woorden, de formule voor het bepalen van de hoek gevormd door twee vectoren is als volgt: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-dangle-entre-deux-vecteurs.webp\" alt=\"hoekformule tussen twee vectoren\" class=\"wp-image-587\" width=\"270\" height=\"128\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Om de hoek te vinden die door twee vectoren wordt gevormd, is het daarom essentieel dat u weet hoe u <a href=\"https:\/\/mathority.org\/nl\/module-van-een-vectorformule-voorbeelden-opgeloste-oefeningen\/\">de grootte van een vector kunt berekenen<\/a> . In deze link vind je de formule, voorbeelden en opgeloste oefeningen voor de module van een vector, dus als je deze vectorbewerking nog niet onder de knie hebt, raden we je aan om er eens een kijkje te nemen.<\/p>\n<p> Deze formule werkt zowel voor het vlak (in R2) als voor de ruimte (in R3). Dat wil zeggen dat we het door elkaar kunnen gebruiken voor vectoren met twee of drie componenten.<\/p>\n<p> Soms is het echter niet nodig om deze formule toe te passen, omdat de hoek tussen de vectoren kan worden afgeleid:<\/p>\n<ul>\n<li> De hoek tussen twee <strong>loodrechte<\/strong> vectoren (die dezelfde richting hebben) is 0\u00ba.<\/li>\n<li> De hoek tussen twee <strong>orthogonale<\/strong> (of loodrechte) vectoren is 90\u00ba. <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-hallar-el-angulo-entre-dos-vectores\"><\/span> Voorbeeld van hoe u de hoek tussen twee vectoren kunt vinden<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Als voorbeeld berekenen we de hoek gevormd door de volgende twee vectoren:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f9ae15324c67998fbf5bb3a1a23a39b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,-1) \\qquad \\vv{\\text{v}} = (2,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> We moeten eerst de module van elke vector berekenen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-407fe81f99d6f328024431ce1264a262_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{4^2+(-1)^2}= \\sqrt{17}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"198\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f8f801f8b137ed446af80104eaeba37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{v}} \\rvert = \\sqrt{2^2+5^2}= \\sqrt{29}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"170\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> We gebruiken nu de formule om de cosinus van de hoek tussen de twee vectoren te berekenen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b1f1619914a14d6cf7117760d651a0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}=\\cfrac{ 4\\cdot 2 + (-1)\\cdot 5}{\\sqrt{17}\\cdot \\sqrt{29}} = \\cfrac{3}{\\sqrt{493}} = 0,14\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"387\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> En ten slotte vinden we de overeenkomstige hoek door de inverse van de cosinus uit te voeren met behulp van de rekenmachine:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26875d19d098689f9f1829db2a0a5b7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos^{-1}(0,14) = \\bm{81,95\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"157\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> De twee vectoren vormen dus een hoek van 81,95\u00ba. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-angulos-entre-vectores\"><\/span> Opgeloste oefeningen over hoeken tussen vectoren<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Bereken de hoek tussen de volgende twee vectoren: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-499126756373eb2239008634feec46c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(5,3) \\qquad  \\vv{\\text{v}} =(1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"180\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Allereerst moeten we de modulus van de twee vectoren berekenen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ec0bd408f2cb730372b129c07447bb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{5^2+3^2}= \\sqrt{34}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"170\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1f75602deaa00b7d68322e42455d4954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{v}} \\rvert = \\sqrt{ 1^2+2^2}= \\sqrt{5}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"161\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We gebruiken de formule om de cosinus van de hoek gevormd door de vectoren te berekenen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26d67b7514a805a0fd3ad15191fbd1c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}=\\cfrac{ 5\\cdot 1 + 3\\cdot 2}{\\sqrt{34}\\cdot \\sqrt{5}} = \\cfrac{11}{\\sqrt{170}} = 0,84\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"360\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ten slotte vinden we de overeenkomstige hoek door de inverse van de cosinus uit te voeren met de rekenmachine: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c08fd9d0bf6afbf2cc5b22809df64d3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos^{-1}(0,84) = \\bm{32,47\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"158\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 2<\/h3>\n<p> Bepaal de hoek die bestaat tussen de volgende twee vectoren: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-608b8fbebe6a76c94f7260810ae1dcdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(-2,-7) \\qquad  \\vv{\\text{v}} =(-1,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"221\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Allereerst moeten we de modules van de vectoren vinden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-751a8c816f55c14f3c48b2c863c3efa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ (-2)^2+(-7)^2}= \\sqrt{53}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"225\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c8a9191c8cc87d5eeb9ba2adef3860f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{v}} \\rvert = \\sqrt{ (-1)^2+5^2}= \\sqrt{26}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"197\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We gebruiken de formule om de cosinus te bepalen van de hoek die de vectoren hebben:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbea7ef49f40b0d70811175b255fc3d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}=\\cfrac{ (-2)\\cdot (-1) + (-7)\\cdot 5}{\\sqrt{53}\\cdot \\sqrt{26}} = \\cfrac{-33}{\\sqrt{1378}} = -0,89\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"465\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte vinden we de overeenkomstige hoek door de inverse van de cosinus met de rekenmachine uit te voeren: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db9c37ff21d0c5d933f0600eb634035f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos^{-1}(-0,89) = \\bm{152,74\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"180\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 3<\/h3>\n<p> Bereken de waarde van<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> zodat de volgende vectoren loodrecht staan: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0c3383b73b240784dc6b044cc5a3b10a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(6,3) \\qquad  \\vv{\\text{v}} =(-4,k)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"195\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Twee loodrechte vectoren vormen een hoek van 90\u00b0. Nog: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e83a6b694c8dfa0975854f1bffec44de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(90\u00ba) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"133\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3ef11c2ecbf7bc8dff4217a761960387_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0=\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"86\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De noemer van de breuk deelt de hele rechterkant van de vergelijking, dus we kunnen deze met de andere kant vermenigvuldigen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-797dd0ce47130f959c984510894f08b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 \\cdot \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert  =\\vv{\\text{u}} \\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b58e54d3d5fa6e123ca5e27a27d77ad1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0  =\\vv{\\text{u}} \\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu lossen we het puntproduct op: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee1d711c5614be091814b91a3ad3affa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =(6,3) \\cdot (-4,k)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2cd46b62546f556478c5a3b070dd0f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =6 \\cdot (-4) + 3\\cdot k\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"143\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a4cc0480b00da3aba9adb1d3aaf37325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-24 +3k\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"104\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En tot slot ontrafelen we het mysterie: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7c13e237f419b6189e700d3b375233c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -3k =-24\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"87\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-28ec14747b40a0cd14929b157215c4e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle k =\\cfrac{-24}{-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"75\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-54d31d581504a9c867c5f29e53acd788_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{k =8}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 4<\/h3>\n<p> Zoek de waarde die de constanten moeten hebben<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> En<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> zodat de volgende vectoren loodrecht staan, en bovendien is het waar <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1f1e9bd6329419b155da0ff40b9d61e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert =10.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"63\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b32214ac021a1b48b4643fea940062ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(-6,a) \\qquad  \\vv{\\text{v}} =(b,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We zullen eerst de modulusvoorwaarde gebruiken om de waarde van te vinden <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4fad90838c7fa310bdbea2364787ced6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a:\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c247d22a0918c32059ba3ddaaff4fbfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert =10\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"59\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d28001c5a9398f2653cdda9fc6ca3070_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{(-6)^2+a^2}=10\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"141\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-43d07a907289c2425d41ef63f6ff1acd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{36+a^2}=10\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We verhogen beide zijden van de vergelijking om de vierkantswortel te verwijderen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-82bc99f0d181db4e2499a41633bafc8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(\\sqrt{36+a^2}\\right)^2=10^2\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"146\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b950dcac0bf08532d7e6f5d41b7de602_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"36+a^2=100\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we ontrafelen het mysterie: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0d2957cd064f117478d48077c1d6a7be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2=100 -36\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"107\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbc56fe6fb2923832590bd589be0c593_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2=64\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16b2f840c377014039359be948ab832b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a=\\sqrt{64}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7897d0ec97cc23b3a386de3efb8e2466_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{a=8}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zodra we de waarde ervan kennen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> , vind de waarde van<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> door de formule voor de hoek van twee vectoren toe te passen, aangezien de verklaring ons vertelt dat ze loodrecht moeten staan, of wat gelijkwaardig is, ze moeten 90 graden vormen. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e83a6b694c8dfa0975854f1bffec44de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(90\u00ba) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"133\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3ef11c2ecbf7bc8dff4217a761960387_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0=\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"86\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De noemer van de breuk deelt de hele rechterkant van de vergelijking, dus we kunnen deze met de andere kant vermenigvuldigen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-797dd0ce47130f959c984510894f08b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 \\cdot \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert  =\\vv{\\text{u}} \\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b58e54d3d5fa6e123ca5e27a27d77ad1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0  =\\vv{\\text{u}} \\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Laten we nu proberen het puntproduct op te lossen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cc125de9cd9b958bb85b7ffb79ef91ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =(-6,8) \\cdot (b,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67a24872ace77b1c8cbc512d361d9866_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-6 \\cdot b +8\\cdot 3\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"128\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dfbbe49209a7e767517fab13634eaf8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-6b +24\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"102\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En tot slot ontrafelen we het mysterie: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b73000a4e9d07e053e2b45eb4522a7aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 6b =24\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33bb24f252697112b44e30bc9f9240ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle b =\\cfrac{24}{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"51\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-13d0b4768ea510b5e1b989f4eeb9deb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{b =4}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 5<\/h3>\n<p> Hoeken berekenen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-28575fb8fa361427b255d8744e982cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha , \\beta\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"30\" style=\"vertical-align: -4px;\"><\/p>\n<p> En<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4de02fc502ed5dbd15f371728ea270a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\gamma\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> die de zijden vormen van de volgende driehoek: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-angle-resolu-entre-vecteurs-produit-scalaire.webp\" alt=\"oefeningen en problemen die stap voor stap worden opgelost van het scalaire product van twee vectoren\" class=\"wp-image-560\" width=\"290\" height=\"226\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De hoekpunten waaruit de driehoek bestaat, zijn de volgende punten:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-75a4919fae29190e3effdeedcec8eb6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(2,1) \\qquad B(4,4) \\qquad C(6,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"230\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om de binnenhoeken van de driehoek te berekenen, kunnen we de vectoren van elk van de zijden berekenen en vervolgens de hoek vinden die ze vormen met behulp van de puntproductformule.<\/p>\n<p class=\"has-text-align-left\"> Bijvoorbeeld om de hoek te vinden<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f0b6b1a01f8fcc2f95be0364c090397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> We berekenen de vectoren van de zijden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a9da14fa9cc4e50b06bdfa76801b083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (4,4)-(2,1)= (2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"287\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e4e2e72bee87bba3e7657a53935e660_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C - A = (6,2)-(2,1)= (4,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we vinden de hoek gevormd door de twee vectoren met behulp van de scalaire productformule: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b6aad49b300d421fc3bb486f051294c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{AB} \\rvert = \\sqrt{2^2+3^2} = \\sqrt{13}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f6657897e68d6b68f79277c89abe6868_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{AC} \\rvert = \\sqrt{4^2+1^2} = \\sqrt{17}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a966db5753cbb53c424c0f962fb27102_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{AB} \\cdot \\vv{AC}}{\\lvert \\vv{AB} \\rvert \\cdot \\lvert \\vv{AC} \\rvert}=\\cfrac{ 2\\cdot 4 + 3\\cdot 1}{\\sqrt{13}\\cdot \\sqrt{17}} = \\cfrac{11}{\\sqrt{221}} =0,74\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"396\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9fac783dc0113263dfb5c31b58231fae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\alpha = 42,27\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"79\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu herhalen we dezelfde procedure om de hoek te bepalen <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea160d5901518098e691e051e6efa4a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\beta:\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"20\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-398c0b2dc840abfc63700a084e9e2956_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{BC} = C - B = (6,2)-(4,4)= (2,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"302\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b7825d9e3b0ceee57e7ecd470e52a242_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{BC} \\rvert = \\sqrt{2^2+(-2)^2} = \\sqrt{8}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"207\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e0e73fd58d6a5b487af9f971fdcdc97f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\beta) =\\cfrac{\\vv{AB} \\cdot \\vv{BC}}{\\lvert \\vv{AB} \\rvert \\cdot \\lvert \\vv{BC} \\rvert}=\\cfrac{ 2\\cdot 2 + 3\\cdot (-2)}{\\sqrt{13}\\cdot \\sqrt{8}} = \\cfrac{-2}{\\sqrt{104}} =-0,20\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"437\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b2f148d28b9679b8267886497e16518_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\beta = 101,31\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om ten slotte de laatste hoek te vinden, kunnen we dezelfde procedure herhalen. Alle hoeken in een driehoek moeten echter opgeteld 180 graden zijn, dus: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-662cae07e8d96d1164dad2b0358302fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\gamma = 180 -42,27-101,31 = \\bm{36,42\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"266\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Op deze pagina ontdek je hoe je de hoek tussen twee vectoren berekent. Daarnaast krijg je ook voorbeelden te zien en kun je oefenen met oefeningen en stap voor stap opgeloste problemen. Formule voor de hoek tussen twee vectoren Als we de definitie van het puntproduct onthouden, kan dit worden berekend met behulp van de &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/hoe-je-de-hoek-tussen-twee-vectoren-berekent-voorbeelden-opgeloste-oefeningen\/\"> <span class=\"screen-reader-text\">Hoe de hoek tussen twee vectoren te berekenen<\/span> Lees meer &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[54],"tags":[],"class_list":["post-113","post","type-post","status-publish","format-standard","hentry","category-vectoren"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Hoe wordt de hoek tussen twee vectoren berekend? 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