{"id":71,"date":"2023-09-16T13:06:27","date_gmt":"2023-09-16T13:06:27","guid":{"rendered":"https:\/\/mathority.org\/de\/wie-man-den-winkel-zwischen-zwei-vektoren-berechnet-beispiele-fur-geloste-ubungen\/"},"modified":"2023-09-16T13:06:27","modified_gmt":"2023-09-16T13:06:27","slug":"wie-man-den-winkel-zwischen-zwei-vektoren-berechnet-beispiele-fur-geloste-ubungen","status":"publish","type":"post","link":"https:\/\/mathority.org\/de\/wie-man-den-winkel-zwischen-zwei-vektoren-berechnet-beispiele-fur-geloste-ubungen\/","title":{"rendered":"So berechnen sie den winkel zwischen zwei vektoren"},"content":{"rendered":"<p>Auf dieser Seite erfahren Sie, wie Sie den Winkel zwischen zwei Vektoren berechnen. Dar\u00fcber hinaus sehen Sie auch Beispiele und k\u00f6nnen mit Schritt f\u00fcr Schritt gel\u00f6sten \u00dcbungen und Problemen \u00fcben. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-angulo-entre-dos-vectores\"><\/span> Formel f\u00fcr den Winkel zwischen zwei Vektoren <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=\"Winkel zwischen zwei Skalarproduktvektoren\" class=\"wp-image-583\" width=\"187\" height=\"190\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Wenn wir uns an die <a href=\"https:\/\/mathority.org\/de\/berechnen-sie-das-skalarprodukt-zwischen-zwei-vektoren.-beispiele-fur-geloste-ubungen\/\">Definition des Skalarprodukts<\/a> erinnern, kann es mit der folgenden Gleichung berechnet werden:<\/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> Aus dieser Gleichheit k\u00f6nnen wir die Formel erhalten, die uns hilft, den Winkel, der von zwei Vektoren gebildet wird, direkt zu ermitteln:<\/p>\n<p> <strong>Der Kosinus des von zwei Vektoren gebildeten Winkels ist gleich dem Skalarprodukt zwischen den beiden Vektoren dividiert durch das Produkt der Moduli der beiden Vektoren.<\/strong><\/p>\n<p> Mit anderen Worten lautet die Formel zur Bestimmung des von zwei Vektoren gebildeten Winkels wie folgt: <\/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=\"Winkelformel zwischen zwei Vektoren\" class=\"wp-image-587\" width=\"270\" height=\"128\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Um den von zwei Vektoren gebildeten Winkel zu ermitteln, ist es daher wichtig, dass Sie wissen, wie <a href=\"https:\/\/mathority.org\/de\/modul-einer-vektorformel-beispiele-geloster-ubungen\/\">man den Betrag eines Vektors berechnet<\/a> . Unter diesem Link finden Sie die Formel, Beispiele und gel\u00f6ste Aufgaben zum Modul eines Vektors. Wenn Sie diese Vektoroperation also noch nicht beherrschen, empfehlen wir Ihnen, einen Blick darauf zu werfen.<\/p>\n<p> Diese Formel funktioniert sowohl f\u00fcr die Ebene (in R2) als auch f\u00fcr den Raum (in R3). Das hei\u00dft, wir k\u00f6nnen es austauschbar f\u00fcr Zwei- oder Dreikomponentenvektoren verwenden.<\/p>\n<p> Manchmal ist es jedoch nicht notwendig, diese Formel anzuwenden, da der Winkel zwischen den Vektoren abgeleitet werden kann:<\/p>\n<ul>\n<li> Der Winkel zwischen zwei <strong>senkrechten<\/strong> Vektoren (die dieselbe Richtung haben) betr\u00e4gt 0\u00b0.<\/li>\n<li> Der Winkel zwischen zwei <strong>orthogonalen<\/strong> (oder senkrechten) Vektoren betr\u00e4gt 90\u00b0. <\/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> Beispiel f\u00fcr die Bestimmung des Winkels zwischen zwei Vektoren<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Als Beispiel berechnen wir den Winkel, der durch die folgenden zwei Vektoren gebildet wird:<\/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> Wir m\u00fcssen zuerst den Modul jedes Vektors berechnen:<\/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> Mit der Formel berechnen wir nun den Kosinus des Winkels zwischen den beiden Vektoren:<\/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> Und schlie\u00dflich ermitteln wir den entsprechenden Winkel, indem wir mit dem Taschenrechner die Umkehrung des Kosinus berechnen:<\/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> Die beiden Vektoren bilden also einen Winkel von 81,95\u00b0. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-angulos-entre-vectores\"><\/span> Aufgaben zu Winkeln zwischen Vektoren gel\u00f6st<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> \u00dcbung 1<\/h3>\n<p> Berechnen Sie den Winkel zwischen den folgenden zwei Vektoren: <\/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>Sehen Sie sich die L\u00f6sung an<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Zun\u00e4chst m\u00fcssen wir den Modul der beiden Vektoren berechnen: <\/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\"> Wir verwenden die Formel, um den Kosinus des von den Vektoren gebildeten Winkels zu berechnen:<\/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\"> Schlie\u00dflich ermitteln wir den entsprechenden Winkel, indem wir mit dem Taschenrechner die Umkehrung des Kosinus berechnen: <\/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\">\u00dcbung 2<\/h3>\n<p> Bestimmen Sie den Winkel, der zwischen den folgenden beiden Vektoren besteht: <\/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>Sehen Sie sich die L\u00f6sung an<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Zun\u00e4chst m\u00fcssen wir die Module der Vektoren finden: <\/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\"> Wir verwenden die Formel, um den Kosinus des Winkels zu ermitteln, den die Vektoren haben:<\/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\"> Und schlie\u00dflich finden wir den entsprechenden Winkel, indem wir mit dem Taschenrechner die Umkehrung des Kosinus berechnen: <\/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\">\u00dcbung 3<\/h3>\n<p> Berechnen Sie den Wert von<\/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> so dass die folgenden Vektoren senkrecht stehen: <\/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>Sehen Sie sich die L\u00f6sung an<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Zwei senkrecht zueinander stehende Vektoren bilden einen Winkel von 90\u00b0. Noch: <\/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\"> Der Nenner des Bruchs teilt die gesamte rechte Seite der Gleichung, sodass wir sie mit der anderen Seite multiplizieren k\u00f6nnen: <\/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\"> Nun l\u00f6sen wir das Skalarprodukt: <\/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\"> Und schlie\u00dflich kl\u00e4ren wir das R\u00e4tsel: <\/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\">\u00dcbung 4<\/h3>\n<p> Finden Sie den Wert, den die Konstanten haben sollten<\/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> Und<\/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> so dass die folgenden Vektoren senkrecht stehen und au\u00dferdem ist es wahr <\/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>Sehen Sie sich die L\u00f6sung an<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Wir werden zun\u00e4chst die Modulbedingung verwenden, um den Wert von zu ermitteln <\/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\"> Wir erh\u00f6hen beide Seiten der Gleichung, um die Quadratwurzel zu entfernen: <\/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\"> Und wir kl\u00e4ren das R\u00e4tsel auf: <\/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\"> Sobald wir den Wert 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> , finden Sie den Wert von<\/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> indem wir die Formel f\u00fcr den Winkel zweier Vektoren anwenden, da die Aussage besagt, dass sie senkrecht sein m\u00fcssen, oder was \u00e4quivalent ist, sie m\u00fcssen 90\u00b0 bilden. <\/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\"> Der Nenner des Bruchs teilt die gesamte rechte Seite der Gleichung, sodass wir sie mit der anderen Seite multiplizieren k\u00f6nnen: <\/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\"> Versuchen wir nun, das Skalarprodukt zu l\u00f6sen: <\/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\"> Und schlie\u00dflich kl\u00e4ren wir das R\u00e4tsel: <\/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\">\u00dcbung 5<\/h3>\n<p> Winkel berechnen<\/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> Und<\/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 die Seiten des folgenden Dreiecks bilden: <\/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=\"\u00dcbungen und Schritt f\u00fcr Schritt gel\u00f6ste Probleme des Skalarprodukts zweier Vektoren\" 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>Sehen Sie sich die L\u00f6sung an<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Die Eckpunkte, aus denen das Dreieck besteht, sind die folgenden Punkte:<\/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\"> Um die Innenwinkel des Dreiecks zu berechnen, k\u00f6nnen wir die Vektoren jeder seiner Seiten berechnen und dann mithilfe der Skalarproduktformel den Winkel ermitteln, den sie bilden.<\/p>\n<p class=\"has-text-align-left\"> Zum Beispiel, um den Winkel zu finden<\/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> Wir berechnen die Vektoren seiner Seiten: <\/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\"> Und wir ermitteln den Winkel, den die beiden Vektoren bilden, mithilfe der Skalarproduktformel: <\/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\"> Jetzt wiederholen wir den gleichen Vorgang, um den Winkel zu bestimmen <\/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\"> Um schlie\u00dflich den letzten Winkel zu finden, k\u00f6nnen wir den gleichen Vorgang wiederholen. Allerdings m\u00fcssen sich alle Winkel in einem Dreieck zu 180 Grad addieren, also: <\/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>Auf dieser Seite erfahren Sie, wie Sie den Winkel zwischen zwei Vektoren berechnen. Dar\u00fcber hinaus sehen Sie auch Beispiele und k\u00f6nnen mit Schritt f\u00fcr Schritt gel\u00f6sten \u00dcbungen und Problemen \u00fcben. Formel f\u00fcr den Winkel zwischen zwei Vektoren Wenn wir uns an die Definition des Skalarprodukts erinnern, kann es mit der folgenden Gleichung berechnet werden: Aus &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/de\/wie-man-den-winkel-zwischen-zwei-vektoren-berechnet-beispiele-fur-geloste-ubungen\/\"> <span class=\"screen-reader-text\">So berechnen sie den winkel zwischen zwei vektoren<\/span> Weiterlesen &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":[22],"tags":[],"class_list":["post-71","post","type-post","status-publish","format-standard","hentry","category-vektoren"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Wie wird der Winkel zwischen zwei Vektoren berechnet? 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