{"id":71,"date":"2023-09-16T13:06:27","date_gmt":"2023-09-16T13:06:27","guid":{"rendered":"https:\/\/mathority.org\/ru\/%d0%ba%d0%b0%d0%ba-%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b8%d1%82%d1%8c-%d1%83%d0%b3%d0%be%d0%bb-%d0%bc%d0%b5%d0%b6%d0%b4%d1%83-%d0%b4%d0%b2%d1%83%d0%bc%d1%8f-%d0%b2%d0%b5%d0%ba%d1%82%d0%be%d1%80\/"},"modified":"2023-09-16T13:06:27","modified_gmt":"2023-09-16T13:06:27","slug":"%d0%ba%d0%b0%d0%ba-%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b8%d1%82%d1%8c-%d1%83%d0%b3%d0%be%d0%bb-%d0%bc%d0%b5%d0%b6%d0%b4%d1%83-%d0%b4%d0%b2%d1%83%d0%bc%d1%8f-%d0%b2%d0%b5%d0%ba%d1%82%d0%be%d1%80","status":"publish","type":"post","link":"https:\/\/mathority.org\/ru\/%d0%ba%d0%b0%d0%ba-%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b8%d1%82%d1%8c-%d1%83%d0%b3%d0%be%d0%bb-%d0%bc%d0%b5%d0%b6%d0%b4%d1%83-%d0%b4%d0%b2%d1%83%d0%bc%d1%8f-%d0%b2%d0%b5%d0%ba%d1%82%d0%be%d1%80\/","title":{"rendered":"\u041a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438"},"content":{"rendered":"<p>\u041d\u0430 \u044d\u0442\u043e\u0439 \u0441\u0442\u0440\u0430\u043d\u0438\u0446\u0435 \u0432\u044b \u0443\u0437\u043d\u0430\u0435\u0442\u0435, \u043a\u0430\u043a \u0440\u0430\u0441\u0441\u0447\u0438\u0442\u0430\u0442\u044c \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0432\u044b \u0442\u0430\u043a\u0436\u0435 \u0443\u0432\u0438\u0434\u0438\u0442\u0435 \u043f\u0440\u0438\u043c\u0435\u0440\u044b \u0438 \u0441\u043c\u043e\u0436\u0435\u0442\u0435 \u043f\u043e\u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0432\u0430\u0442\u044c\u0441\u044f, \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u044f \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u0438 \u0437\u0430\u0434\u0430\u0447\u0438, \u0440\u0435\u0448\u0430\u0435\u043c\u044b\u0435 \u0448\u0430\u0433 \u0437\u0430 \u0448\u0430\u0433\u043e\u043c. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-angulo-entre-dos-vectores\"><\/span> \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u0443\u0433\u043b\u0430 \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 <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=\"\u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f\" class=\"wp-image-583\" width=\"187\" height=\"190\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> \u0415\u0441\u043b\u0438 \u043c\u044b \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u043c  \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f  , \u0435\u0433\u043e \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u0447\u0438\u0442\u0430\u0442\u044c \u043f\u043e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u043c\u0443 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044e:<\/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> \u0418\u0437 \u044d\u0442\u043e\u0433\u043e \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430 \u043c\u043e\u0436\u043d\u043e \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u0444\u043e\u0440\u043c\u0443\u043b\u0443, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u043f\u043e\u043c\u043e\u0436\u0435\u0442 \u043d\u0430\u043c \u043d\u0435\u043f\u043e\u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0435\u043d\u043d\u043e \u043d\u0430\u0439\u0442\u0438 \u0443\u0433\u043e\u043b, \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u043d\u044b\u0439 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438:<\/p>\n<p> <strong>\u041a\u043e\u0441\u0438\u043d\u0443\u0441 \u0443\u0433\u043b\u0430, \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u043d\u043e\u0433\u043e \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438, \u0440\u0430\u0432\u0435\u043d \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u043c\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044e \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432, \u0434\u0435\u043b\u0435\u043d\u043d\u043e\u043c\u0443 \u043d\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043c\u043e\u0434\u0443\u043b\u0435\u0439 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432.<\/strong><\/p>\n<p> \u0414\u0440\u0443\u0433\u0438\u043c\u0438 \u0441\u043b\u043e\u0432\u0430\u043c\u0438, \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0443\u0433\u043b\u0430, \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u043d\u043e\u0433\u043e \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438, \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c: <\/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=\"\u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u0443\u0433\u043b\u0430 \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438\" class=\"wp-image-587\" width=\"270\" height=\"128\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> \u041f\u043e\u044d\u0442\u043e\u043c\u0443, \u0447\u0442\u043e\u0431\u044b \u043d\u0430\u0439\u0442\u0438 \u0443\u0433\u043e\u043b, \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u043d\u044b\u0439 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438, \u0432\u0430\u0436\u043d\u043e \u0437\u043d\u0430\u0442\u044c, \u043a\u0430\u043a  \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0443 \u0432\u0435\u043a\u0442\u043e\u0440\u0430  . \u041f\u043e \u044d\u0442\u043e\u0439 \u0441\u0441\u044b\u043b\u043a\u0435 \u0432\u044b \u043d\u0430\u0439\u0434\u0435\u0442\u0435 \u0444\u043e\u0440\u043c\u0443\u043b\u044b, \u043f\u0440\u0438\u043c\u0435\u0440\u044b \u0438 \u0440\u0435\u0448\u0435\u043d\u043d\u044b\u0435 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043c\u043e\u0434\u0443\u043b\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430, \u043f\u043e\u044d\u0442\u043e\u043c\u0443, \u0435\u0441\u043b\u0438 \u0432\u044b \u0435\u0449\u0435 \u043d\u0435 \u043e\u0441\u0432\u043e\u0438\u043b\u0438 \u044d\u0442\u0443 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044e \u0441 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u043c, \u0440\u0435\u043a\u043e\u043c\u0435\u043d\u0434\u0443\u0435\u043c \u043f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c.<\/p>\n<p> \u042d\u0442\u0430 \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u043a\u0430\u043a \u0434\u043b\u044f \u043f\u043b\u043e\u0441\u043a\u043e\u0441\u0442\u0438 (\u0432 R2), \u0442\u0430\u043a \u0438 \u0434\u043b\u044f \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 (\u0432 R3). \u0422\u043e \u0435\u0441\u0442\u044c \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u0435\u0433\u043e \u0432\u0437\u0430\u0438\u043c\u043e\u0437\u0430\u043c\u0435\u043d\u044f\u0435\u043c\u043e \u0434\u043b\u044f \u0434\u0432\u0443\u0445- \u0438\u043b\u0438 \u0442\u0440\u0435\u0445\u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442\u043d\u044b\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432.<\/p>\n<p> \u041e\u0434\u043d\u0430\u043a\u043e \u0438\u043d\u043e\u0433\u0434\u0430 \u043d\u0435\u0442 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0442\u044c \u044d\u0442\u0443 \u0444\u043e\u0440\u043c\u0443\u043b\u0443, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 \u043c\u043e\u0436\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c:<\/p>\n<ul>\n<li> \u0423\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f <strong>\u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b\u043c\u0438<\/strong> \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 (\u0438\u043c\u0435\u044e\u0449\u0438\u043c\u0438 \u043e\u0434\u0438\u043d\u0430\u043a\u043e\u0432\u043e\u0435 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435) \u0440\u0430\u0432\u0435\u043d 0\u00b0.<\/li>\n<li> \u0423\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f <strong>\u043e\u0440\u0442\u043e\u0433\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c\u0438<\/strong> (\u0438\u043b\u0438 \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b\u043c\u0438) \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 \u0440\u0430\u0432\u0435\u043d 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> \u041f\u0440\u0438\u043c\u0435\u0440 \u0442\u043e\u0433\u043e, \u043a\u0430\u043a \u043d\u0430\u0439\u0442\u0438 \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> \u0412 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u043f\u0440\u0438\u043c\u0435\u0440\u0430 \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u043c \u0443\u0433\u043e\u043b, \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u043d\u044b\u0439 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c\u0438 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438:<\/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> \u0421\u043d\u0430\u0447\u0430\u043b\u0430 \u043c\u044b \u0434\u043e\u043b\u0436\u043d\u044b \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u043c\u043e\u0434\u0443\u043b\u044c \u043a\u0430\u0436\u0434\u043e\u0433\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u0430:<\/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> \u0422\u0435\u043f\u0435\u0440\u044c \u043c\u044b \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u0434\u043b\u044f \u0440\u0430\u0441\u0447\u0435\u0442\u0430 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430 \u0443\u0433\u043b\u0430 \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438:<\/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> \u0418, \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u043d\u0430\u0445\u043e\u0434\u0438\u043c \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044e\u0449\u0438\u0439 \u0443\u0433\u043e\u043b, \u0432\u044b\u043f\u043e\u043b\u043d\u0438\u0432 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0439 \u043a\u043e\u0441\u0438\u043d\u0443\u0441 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043a\u0430\u043b\u044c\u043a\u0443\u043b\u044f\u0442\u043e\u0440\u0430:<\/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> \u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u0434\u0432\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u043e\u0431\u0440\u0430\u0437\u0443\u044e\u0442 \u0443\u0433\u043e\u043b 81,95\u00b0. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-angulos-entre-vectores\"><\/span> \u0420\u0435\u0448\u0435\u043d\u043d\u044b\u0435 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u043d\u0430 \u0443\u0433\u043b\u044b \u043c\u0435\u0436\u0434\u0443 \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> \u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 1<\/h3>\n<p> \u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u0435 \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c\u0438 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438: <\/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>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u041f\u0440\u0435\u0436\u0434\u0435 \u0432\u0441\u0435\u0433\u043e, \u043c\u044b \u0434\u043e\u043b\u0436\u043d\u044b \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u043c\u043e\u0434\u0443\u043b\u044c \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432: <\/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\"> \u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u0434\u043b\u044f \u0440\u0430\u0441\u0447\u0435\u0442\u0430 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430 \u0443\u0433\u043b\u0430, \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u043d\u043e\u0433\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438:<\/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\"> \u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u043c\u044b \u043d\u0430\u0445\u043e\u0434\u0438\u043c \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044e\u0449\u0438\u0439 \u0443\u0433\u043e\u043b, \u0432\u044b\u043f\u043e\u043b\u043d\u0438\u0432 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0439 \u043a\u043e\u0441\u0438\u043d\u0443\u0441 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043a\u0430\u043b\u044c\u043a\u0443\u043b\u044f\u0442\u043e\u0440\u0430: <\/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\">\u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 2<\/h3>\n<p> \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u0435 \u0443\u0433\u043e\u043b, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043c\u0435\u0436\u0434\u0443 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c\u0438 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438: <\/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>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u041f\u0440\u0435\u0436\u0434\u0435 \u0432\u0441\u0435\u0433\u043e \u043d\u0430\u043c \u043d\u0443\u0436\u043d\u043e \u043d\u0430\u0439\u0442\u0438 \u043c\u043e\u0434\u0443\u043b\u0438 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432: <\/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\"> \u041c\u044b \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u043a\u043e\u0441\u0438\u043d\u0443\u0441 \u0443\u0433\u043b\u0430, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u0438\u043c\u0435\u044e\u0442 \u0432\u0435\u043a\u0442\u043e\u0440\u044b:<\/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\"> \u0418, \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u043d\u0430\u0445\u043e\u0434\u0438\u043c \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044e\u0449\u0438\u0439 \u0443\u0433\u043e\u043b, \u0432\u044b\u043f\u043e\u043b\u043d\u0438\u0432 \u043d\u0430 \u043a\u0430\u043b\u044c\u043a\u0443\u043b\u044f\u0442\u043e\u0440\u0435 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0439 \u043a\u043e\u0441\u0438\u043d\u0443\u0441: <\/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\">\u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 3<\/h3>\n<p> \u0420\u0430\u0441\u0441\u0447\u0438\u0442\u0430\u0439\u0442\u0435 \u0441\u0442\u043e\u0438\u043c\u043e\u0441\u0442\u044c<\/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> \u0442\u0430\u043a \u0447\u0442\u043e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b: <\/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>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u0414\u0432\u0430 \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u043e\u0431\u0440\u0430\u0437\u0443\u044e\u0442 \u0443\u0433\u043e\u043b 90\u00b0. \u0415\u0449\u0435: <\/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\"> \u0417\u043d\u0430\u043c\u0435\u043d\u0430\u0442\u0435\u043b\u044c \u0434\u0440\u043e\u0431\u0438 \u0434\u0435\u043b\u0438\u0442 \u0432\u0441\u044e \u043f\u0440\u0430\u0432\u0443\u044e \u0447\u0430\u0441\u0442\u044c \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0443\u043c\u043d\u043e\u0436\u0438\u0442\u044c \u0435\u0435 \u043d\u0430 \u0434\u0440\u0443\u0433\u0443\u044e \u0447\u0430\u0441\u0442\u044c: <\/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\"> \u0422\u0435\u043f\u0435\u0440\u044c \u0440\u0435\u0448\u0430\u0435\u043c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435: <\/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\"> \u0418 \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u043c\u044b \u043f\u0440\u043e\u044f\u0441\u043d\u044f\u0435\u043c \u0442\u0430\u0439\u043d\u0443: <\/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\">\u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 4<\/h3>\n<p> \u041d\u0430\u0439\u0434\u0438\u0442\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0434\u043e\u043b\u0436\u043d\u044b \u0438\u043c\u0435\u0442\u044c \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u044b<\/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> \u0418<\/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> \u0442\u0430\u043a \u0447\u0442\u043e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b \u0438, \u043a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0432\u0435\u0440\u043d\u043e <\/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>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u0421\u043d\u0430\u0447\u0430\u043b\u0430 \u043c\u044b \u0432\u043e\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c\u0441\u044f \u0443\u0441\u043b\u043e\u0432\u0438\u0435\u043c \u043c\u043e\u0434\u0443\u043b\u044f, \u0447\u0442\u043e\u0431\u044b \u043d\u0430\u0439\u0442\u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 <\/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\"> \u041f\u043e\u0434\u043d\u0438\u043c\u0435\u043c \u043e\u0431\u0435 \u0447\u0430\u0441\u0442\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f, \u0447\u0442\u043e\u0431\u044b \u0443\u0434\u0430\u043b\u0438\u0442\u044c \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u044b\u0439 \u043a\u043e\u0440\u0435\u043d\u044c: <\/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\"> \u0418 \u043c\u044b \u043f\u0440\u043e\u044f\u0441\u043d\u044f\u0435\u043c \u0442\u0430\u0439\u043d\u0443: <\/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\"> \u041a\u0430\u043a \u0442\u043e\u043b\u044c\u043a\u043e \u043c\u044b \u0443\u0437\u043d\u0430\u0435\u043c \u0446\u0435\u043d\u043d\u043e\u0441\u0442\u044c<\/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> , \u043d\u0430\u0439\u0434\u0438\u0442\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435<\/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> \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u0432 \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u0443\u0433\u043b\u0430 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0432 \u044d\u0442\u043e\u043c \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u0438 \u0433\u043e\u0432\u043e\u0440\u0438\u0442\u0441\u044f, \u0447\u0442\u043e \u043e\u043d\u0438 \u0434\u043e\u043b\u0436\u043d\u044b \u0431\u044b\u0442\u044c \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b \u0438\u043b\u0438, \u0447\u0442\u043e \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u043e, \u043e\u043d\u0438 \u0434\u043e\u043b\u0436\u043d\u044b \u0441\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u0442\u044c 90\u00b0. <\/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\"> \u0417\u043d\u0430\u043c\u0435\u043d\u0430\u0442\u0435\u043b\u044c \u0434\u0440\u043e\u0431\u0438 \u0434\u0435\u043b\u0438\u0442 \u0432\u0441\u044e \u043f\u0440\u0430\u0432\u0443\u044e \u0447\u0430\u0441\u0442\u044c \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0443\u043c\u043d\u043e\u0436\u0438\u0442\u044c \u0435\u0435 \u043d\u0430 \u0434\u0440\u0443\u0433\u0443\u044e \u0447\u0430\u0441\u0442\u044c: <\/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\"> \u0422\u0435\u043f\u0435\u0440\u044c \u043f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u0440\u0435\u0448\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435: <\/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\"> \u0418 \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u043c\u044b \u043f\u0440\u043e\u044f\u0441\u043d\u044f\u0435\u043c \u0442\u0430\u0439\u043d\u0443: <\/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\">\u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 5<\/h3>\n<p> \u0420\u0430\u0441\u0441\u0447\u0438\u0442\u0430\u0442\u044c \u0443\u0433\u043b\u044b<\/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> \u0418<\/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> \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043e\u0431\u0440\u0430\u0437\u0443\u044e\u0442 \u0441\u0442\u043e\u0440\u043e\u043d\u044b \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0433\u043e \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430: <\/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=\"\u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u0438 \u0437\u0430\u0434\u0430\u0447\u0438, \u0440\u0435\u0448\u0430\u0435\u043c\u044b\u0435 \u0448\u0430\u0433 \u0437\u0430 \u0448\u0430\u0433\u043e\u043c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432\" 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>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u0412\u0435\u0440\u0448\u0438\u043d\u0430\u043c\u0438, \u0441\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u044e\u0449\u0438\u043c\u0438 \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a, \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0442\u043e\u0447\u043a\u0438:<\/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\"> \u0427\u0442\u043e\u0431\u044b \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0432\u043d\u0443\u0442\u0440\u0435\u043d\u043d\u0438\u0435 \u0443\u0433\u043b\u044b \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043a\u0430\u0436\u0434\u043e\u0439 \u0438\u0437 \u0435\u0433\u043e \u0441\u0442\u043e\u0440\u043e\u043d, \u0430 \u0437\u0430\u0442\u0435\u043c \u043d\u0430\u0439\u0442\u0438 \u0443\u0433\u043e\u043b, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043e\u043d\u0438 \u043e\u0431\u0440\u0430\u0437\u0443\u044e\u0442, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f.<\/p>\n<p class=\"has-text-align-left\"> \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0447\u0442\u043e\u0431\u044b \u043d\u0430\u0439\u0442\u0438 \u0443\u0433\u043e\u043b<\/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> \u0412\u044b\u0447\u0438\u0441\u043b\u044f\u0435\u043c \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u0435\u0433\u043e \u0441\u0442\u043e\u0440\u043e\u043d: <\/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\"> \u0418 \u043c\u044b \u043d\u0430\u0445\u043e\u0434\u0438\u043c \u0443\u0433\u043e\u043b, \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u043d\u044b\u0439 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f: <\/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\"> \u0422\u0435\u043f\u0435\u0440\u044c \u043f\u043e\u0432\u0442\u043e\u0440\u044f\u0435\u043c \u0442\u0443 \u0436\u0435 \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440\u0443 \u0434\u043b\u044f \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0443\u0433\u043b\u0430 <\/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\"> \u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u0447\u0442\u043e\u0431\u044b \u043d\u0430\u0439\u0442\u0438 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0439 \u0443\u0433\u043e\u043b, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043f\u043e\u0432\u0442\u043e\u0440\u0438\u0442\u044c \u0442\u0443 \u0436\u0435 \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440\u0443. \u041e\u0434\u043d\u0430\u043a\u043e \u0441\u0443\u043c\u043c\u0430 \u0432\u0441\u0435\u0445 \u0443\u0433\u043b\u043e\u0432 \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 \u0434\u043e\u043b\u0436\u043d\u0430 \u0441\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u0442\u044c 180 \u0433\u0440\u0430\u0434\u0443\u0441\u043e\u0432, \u043f\u043e\u044d\u0442\u043e\u043c\u0443: <\/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>\u041d\u0430 \u044d\u0442\u043e\u0439 \u0441\u0442\u0440\u0430\u043d\u0438\u0446\u0435 \u0432\u044b \u0443\u0437\u043d\u0430\u0435\u0442\u0435, \u043a\u0430\u043a \u0440\u0430\u0441\u0441\u0447\u0438\u0442\u0430\u0442\u044c \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0432\u044b \u0442\u0430\u043a\u0436\u0435 \u0443\u0432\u0438\u0434\u0438\u0442\u0435 \u043f\u0440\u0438\u043c\u0435\u0440\u044b \u0438 \u0441\u043c\u043e\u0436\u0435\u0442\u0435 \u043f\u043e\u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0432\u0430\u0442\u044c\u0441\u044f, \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u044f \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u0438 \u0437\u0430\u0434\u0430\u0447\u0438, \u0440\u0435\u0448\u0430\u0435\u043c\u044b\u0435 \u0448\u0430\u0433 \u0437\u0430 \u0448\u0430\u0433\u043e\u043c. \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u0443\u0433\u043b\u0430 \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 \u0415\u0441\u043b\u0438 \u043c\u044b \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f , \u0435\u0433\u043e \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u0447\u0438\u0442\u0430\u0442\u044c \u043f\u043e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u043c\u0443 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044e: \u0418\u0437 \u044d\u0442\u043e\u0433\u043e \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430 \u043c\u043e\u0436\u043d\u043e \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u0444\u043e\u0440\u043c\u0443\u043b\u0443, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u043f\u043e\u043c\u043e\u0436\u0435\u0442 \u043d\u0430\u043c &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/ru\/%d0%ba%d0%b0%d0%ba-%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b8%d1%82%d1%8c-%d1%83%d0%b3%d0%be%d0%bb-%d0%bc%d0%b5%d0%b6%d0%b4%d1%83-%d0%b4%d0%b2%d1%83%d0%bc%d1%8f-%d0%b2%d0%b5%d0%ba%d1%82%d0%be%d1%80\/\"> <span class=\"screen-reader-text\">\u041a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438<\/span> \u0427\u0438\u0442\u0430\u0442\u044c \u0434\u0430\u043b\u0435\u0435 &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-22"],"yoast_head":"<!-- This site is optimized 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