{"id":214,"date":"2023-07-11T19:16:54","date_gmt":"2023-07-11T19:16:54","guid":{"rendered":"https:\/\/mathority.org\/ru\/%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b8%d1%82%d1%8c-%d1%81%d0%ba%d0%b0%d0%bb%d1%8f%d1%80%d0%bd%d0%be%d0%b5-%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%b5%d0%b4%d0%b5%d0%bd%d0%b8%d0%b5-%d0%b4%d0%b2\/"},"modified":"2023-07-11T19:16:54","modified_gmt":"2023-07-11T19:16:54","slug":"%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b8%d1%82%d1%8c-%d1%81%d0%ba%d0%b0%d0%bb%d1%8f%d1%80%d0%bd%d0%be%d0%b5-%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%b5%d0%b4%d0%b5%d0%bd%d0%b8%d0%b5-%d0%b4%d0%b2","status":"publish","type":"post","link":"https:\/\/mathority.org\/ru\/%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b8%d1%82%d1%8c-%d1%81%d0%ba%d0%b0%d0%bb%d1%8f%d1%80%d0%bd%d0%be%d0%b5-%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%b5%d0%b4%d0%b5%d0%bd%d0%b8%d0%b5-%d0%b4%d0%b2\/","title":{"rendered":"\u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432"},"content":{"rendered":"<p>\u041d\u0430 \u044d\u0442\u043e\u0439 \u0441\u0442\u0440\u0430\u043d\u0438\u0446\u0435 \u0432\u044b \u0443\u0432\u0438\u0434\u0438\u0442\u0435, \u0447\u0442\u043e \u044d\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0438 \u043a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432. \u0412\u044b \u0442\u0430\u043a\u0436\u0435 \u0443\u0437\u043d\u0430\u0435\u0442\u0435, \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, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0438, \u043a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0432\u0441\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f. \u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u0432\u044b \u0441\u043c\u043e\u0436\u0435\u0442\u0435 \u043f\u043e\u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0432\u0430\u0442\u044c\u0441\u044f \u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u0430\u0445 \u0438 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f\u0445, \u0440\u0435\u0448\u0430\u0435\u043c\u044b\u0445 \u0448\u0430\u0433 \u0437\u0430 \u0448\u0430\u0433\u043e\u043c. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-104\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-calcular-el-producto-escalar-entre-dos-vectores\"><\/span> \u041a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> \u0412 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u2014 \u044d\u0442\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u0430\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0443\u043c\u043d\u043e\u0436\u0430\u0435\u0442 \u0434\u0432\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u0438 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u0443\u0435\u0442 \u0438\u0445 \u0432 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u0418\u0442\u0430\u043a, \u0435\u0441\u0442\u044c \u0434\u0432\u0430 \u0441\u043f\u043e\u0441\u043e\u0431\u0430 \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432: <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> \u0415\u0441\u043b\u0438 \u043c\u044b \u0437\u043d\u0430\u0435\u043c \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043d\u0430\u0439\u0442\u0438 \u0438\u0445 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435, \u0443\u043c\u043d\u043e\u0436\u0438\u0432 \u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442\u044b X \u0438 Y, \u0430 \u0437\u0430\u0442\u0435\u043c \u0441\u043b\u043e\u0436\u0438\u0432 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b. \u0414\u0440\u0443\u0433\u0438\u043c\u0438 \u0441\u043b\u043e\u0432\u0430\u043c\u0438, \u0435\u0441\u043b\u0438 \u0443 \u043d\u0430\u0441 \u0435\u0441\u0442\u044c \u0434\u0432\u0430 \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-c06f6c1238ab7803750f830e16891f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (\\text{u}_x,\\text{u}_y) \\qquad \\vv{\\text{v}} = (\\text{v}_x,\\text{v}_y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"216\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> \u0421\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u043d\u0438\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-c469402aae308ebcf911d56f71dadef2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\vv{\\text{u}} \\cdot \\vv{\\text{v}} = \\text{u}_x\\cdot \\text{v}_x + \\text{u}_y\\cdot \\text{v}_y\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"174\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<\/div>\n<p> \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0445 \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-78adae8ec882bbd01c5bb2704f3ffa0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (1,2) \\qquad \\vv{\\text{v}} = (-1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" 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-756e86b9b5b7a9f5df2cacf728d82855_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}}&amp;=(1,2)\\cdot (-1,3) \\\\[1.5ex]&amp;=1\\cdot (-1) + 2 \\cdot 3 \\\\[1.5ex] &amp; = -1+6  \\\\[1.5ex] &amp; =\\bm{5} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"166\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> \u042d\u0442\u043e \u0441\u043f\u043e\u0441\u043e\u0431 \u043d\u0430\u0439\u0442\u0438 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438. \u041e\u0434\u043d\u0430\u043a\u043e \u0435\u0441\u0442\u044c \u0438 \u0434\u0440\u0443\u0433\u043e\u0439 \u043c\u0435\u0442\u043e\u0434: <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> \u0421 \u0434\u0440\u0443\u0433\u043e\u0439 \u0441\u0442\u043e\u0440\u043e\u043d\u044b, \u0435\u0441\u043b\u0438 \u043c\u044b \u0437\u043d\u0430\u0435\u043c \u043c\u043e\u0434\u0443\u043b\u044c \u0438 \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\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 \u043c\u043e\u0436\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c, \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0438\u0445 \u043c\u043e\u0434\u0443\u043b\u0435\u0439 \u043d\u0430 \u043a\u043e\u0441\u0438\u043d\u0443\u0441 \u0443\u0433\u043b\u0430, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043e\u043d\u0438 \u043e\u0431\u0440\u0430\u0437\u0443\u044e\u0442:<\/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-46f454c7efa9e990e47b8fd3858fbade_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 style=\"text-align:left\"> \u0417\u043e\u043b\u043e\u0442\u043e<\/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-4501274336c637b37c6332eae5c6c229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -5px;\"><\/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-9a59cd4f2581db3318d38a2a77340a64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{v}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"15\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u043c\u043e\u0434\u0443\u043b\u044f\u043c\u0438 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" 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-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0435\u043d\u043d\u043e \u0438<\/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> \u0443\u0433\u043e\u043b, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043e\u043d\u0438 \u043e\u0431\u0440\u0430\u0437\u0443\u044e\u0442. <\/p>\n<\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<p> \u041d\u0430\u043f\u043e\u043c\u043d\u0438\u043c, \u0447\u0442\u043e \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u0440\u0430\u0432\u043d\u0430 \u043a\u043e\u0440\u043d\u044e \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043e\u0432 \u0435\u0433\u043e \u0441\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u044e\u0449\u0438\u0445:<\/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-f63fa0a6f4110553705d4e3d6cf23692_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ \\text{u}_x^2+\\text{u}_y^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"117\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p> \u0412 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u043f\u0440\u0438\u043c\u0435\u0440\u0430 \u043c\u044b \u0440\u0435\u0448\u0438\u043c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432, \u043c\u043e\u0434\u0443\u043b\u0438 \u0438 \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u043a\u043e\u0442\u043e\u0440\u044b\u043c\u0438 \u0440\u0430\u0432\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-46c26555360d66fd213087ee2432e68e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert =3 \\qquad \\lvert \\vv{\\text{v}} \\rvert = 4 \\qquad \\alpha=60\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"226\" 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-a363625d93f01339437d5ad065050025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(\\alpha ) \\\\[1.5ex] &amp;= 3 \\cdot 4 \\cdot \\cos(60\u00ba)\\\\[1.5ex] &amp; = 3 \\cdot 4 \\cdot 0,5 \\\\[1.5ex] &amp;= \\bm{6} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"168\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> \u0421 \u0434\u0440\u0443\u0433\u043e\u0439 \u0441\u0442\u043e\u0440\u043e\u043d\u044b, \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0442\u0430\u043a\u0436\u0435 \u043d\u0430\u0437\u044b\u0432\u0430\u044e\u0442 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u044b\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c, \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u044b\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c \u0438\u043b\u0438 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u044b\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c.<\/p>\n<p> <strong>\u041f\u0440\u0438\u043c\u0435\u0447\u0430\u043d\u0438\u0435.<\/strong> \u041d\u0435 \u043f\u0443\u0442\u0430\u0439\u0442\u0435 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0441 \u043f\u0435\u0440\u0435\u043a\u0440\u0435\u0441\u0442\u043d\u044b\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443, \u0445\u043e\u0442\u044f \u043e\u043d\u0438 \u0438 \u0438\u043c\u0435\u044e\u0442 \u0441\u0445\u043e\u0436\u0438\u0435 \u043d\u0430\u0437\u0432\u0430\u043d\u0438\u044f, \u044d\u0442\u043e \u0441\u043e\u0432\u0435\u0440\u0448\u0435\u043d\u043d\u043e \u0440\u0430\u0437\u043d\u044b\u0435 \u043a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u0438. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"encontrar-el-angulo-entre-dos-vectores-utilizando-el-producto-escalar\"><\/span> \u041d\u0430\u0439\u0434\u0438\u0442\u0435 \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><\/div>\n<\/div>\n<p> \u041a\u0430\u043a \u0442\u043e\u043b\u044c\u043a\u043e \u043c\u044b \u0443\u0432\u0438\u0434\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, \u0443 \u0432\u0430\u0441 \u043c\u043e\u0436\u0435\u0442 \u0432\u043e\u0437\u043d\u0438\u043a\u043d\u0443\u0442\u044c \u0432\u043e\u043f\u0440\u043e\u0441, \u043a\u0430\u043a\u043e\u0432\u0430 \u0446\u0435\u043b\u044c \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u044f \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432? \u041d\u0443, \u043e\u0434\u043d\u043e \u0438\u0437 \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0439 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u2014 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \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. <\/p>\n<figure class=\"wp-block-image 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=\"172\" height=\"175\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> \u0420\u0435\u0448\u0430\u044f \u043a\u043e\u0441\u0438\u043d\u0443\u0441 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/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-0534ed8d223cf93e44493b80ebfa83d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}\\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*} \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0414\u0430\u0432\u0430\u0439\u0442\u0435 \u043f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u043c, \u043a\u0430\u043a \u044d\u0442\u043e \u0434\u0435\u043b\u0430\u0435\u0442\u0441\u044f, \u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u0435:<\/p>\n<ul>\n<li> \u041d\u0430\u0439\u0434\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:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d65d095d633d49431fcf3ee16757025e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,2) \\qquad \\vv{\\text{v}} = (-1,5)\" 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 \u043d\u0430\u043c \u043d\u0443\u0436\u043d\u043e \u043d\u0430\u0439\u0442\u0438 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b \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-b8435f99e0a124c838bc163cb3ac1c67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ 4^2+2^2}= \\sqrt{20}\" 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-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> \u0422\u0435\u043f\u0435\u0440\u044c \u0432\u043e\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c\u0441\u044f \u0444\u043e\u0440\u043c\u0443\u043b\u043e\u0439 \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-808047a2b37f1b5534038383a4d4b111_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 (-1) + 2\\cdot 5}{\\sqrt{20}\\cdot \\sqrt{26}} = \\cfrac{6}{\\sqrt{520}} = 0,26\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"387\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> \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-36498fae23e4068e3d1c3735dafd2e64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos^{-1}(0,26) = \\bm{74,93\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"158\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043e\u0431\u0440\u0430\u0437\u0443\u044e\u0442 \u0443\u0433\u043e\u043b 74,93 \u0433\u0440\u0430\u0434\u0443\u0441\u0430. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-del-producto-escalar-de-dos-vectores\"><\/span> \u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \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<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> \u0421\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0438\u043c\u0435\u0435\u0442 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u043a\u0438:<\/p>\n<ul>\n<li> <strong>\u041a\u043e\u043c\u043c\u0443\u0442\u0430\u0442\u0438\u0432\u043d\u043e\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e<\/strong> : \u043f\u043e\u0440\u044f\u0434\u043e\u043a \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 \u043d\u0435 \u0438\u043c\u0435\u0435\u0442 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fcc9e9f7ec3121774701301a70313b86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}} \\cdot \\vv{\\text{v}} =\\vv{\\text{v}} \\cdot \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"88\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>\u0420\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e<\/strong> : \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u043c \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u0438 \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0853c6a7d518e218e60c0d047bde152e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}( \\vv{\\text{v}}+ \\vv{\\text{w}} )=\\vv{\\text{u}} \\cdot \\vv{\\text{v}}+ \\vv{\\text{u}} \\cdot \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"180\" 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-585d05c6a06b6c57b9ab0e640a371a9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}( \\vv{\\text{v}}- \\vv{\\text{w}} )=\\vv{\\text{u}} \\cdot \\vv{\\text{v}}- \\vv{\\text{u}} \\cdot \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"180\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>\u0410\u0441\u0441\u043e\u0446\u0438\u0430\u0442\u0438\u0432\u043d\u043e\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e<\/strong> : \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0443\u043c\u043d\u043e\u0436\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043d\u0430 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0443 \u0434\u043e \u0438\u043b\u0438 \u043f\u043e\u0441\u043b\u0435 \u0432\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u044b:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d8c1312b87c767bb4439f6c6c693dad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle k\\cdot (\\vv{\\text{u}} \\cdot \\vv{\\text{v}}) = (k\\cdot\\vv{\\text{u}}) \\cdot \\vv{\\text{v}} =\\vv{\\text{u}} \\cdot (k\\cdot\\vv{\\text{v}})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"252\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> \u0415\u0441\u043b\u0438 \u0434\u0432\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 <strong>\u043e\u0440\u0442\u043e\u0433\u043e\u043d\u0430\u043b\u044c\u043d\u044b<\/strong> (\u0438\u043b\u0438 \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b), \u0442\u043e \u0438\u0445 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0440\u0430\u0432\u043d\u043e \u043d\u0443\u043b\u044e. \u042d\u0442\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u043b\u0435\u0433\u043a\u043e \u043f\u0440\u043e\u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0434\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 \u0441\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u044e\u0442 \u0443\u0433\u043e\u043b 90\u00b0, \u0430 \u043a\u043e\u0441\u0438\u043d\u0443\u0441 90\u00b0 \u0440\u0430\u0432\u0435\u043d 0:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-77e3b216b12f49a9dde99c0fbf626658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(90\u00ba ) \\\\[1.5ex] &amp;=\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot 0 \\\\[1.5ex] &amp;= 0 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"91\" width=\"175\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> \u041d\u0430\u043f\u0440\u043e\u0442\u0438\u0432, \u0435\u0441\u043b\u0438 \u0434\u0432\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 <strong>\u043f\u0430\u0440\u0430\u043b\u043b\u0435\u043b\u044c\u043d\u044b<\/strong> , \u0442\u043e \u0438\u0445 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0440\u0430\u0432\u043d\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044e \u0438\u0445 \u043c\u043e\u0434\u0443\u043b\u0435\u0439. \u042d\u0442\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u0442\u0430\u043a\u0436\u0435 \u043b\u0435\u0433\u043a\u043e \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u0442\u044c, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0434\u0432\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u043e\u0434\u043d\u043e\u0433\u043e \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043e\u0431\u0440\u0430\u0437\u0443\u044e\u0442 \u0443\u0433\u043e\u043b 0\u00b0, \u043a\u043e\u0441\u0438\u043d\u0443\u0441 \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u0440\u0430\u0432\u0435\u043d 1:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-777ec2544cac64aa065df21872989b4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(0\u00ba) \\\\[1.5ex] &amp;=\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot 1 \\\\[1.5ex] &amp;= \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"96\" width=\"166\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> \u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u0441\u0430\u043c\u043e \u043f\u043e \u0441\u0435\u0431\u0435 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u043e \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0443 \u0435\u0433\u043e \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b: <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0161de621030dcfe2a1d5fccc94048bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\vv{\\text{u}} \\cdot \\vv{\\text{u}} &amp; = \\lvert \\vv{\\text{u}} \\rvert ^2\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-productos-escalares-entre-dos-vectores\"><\/span> \u0420\u0435\u0448\u0435\u043d\u044b \u0437\u0430\u0434\u0430\u0447\u0438 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \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<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 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432 \u043f\u043b\u043e\u0441\u043a\u043e\u0441\u0442\u0438 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0445 \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-8c09eac43618120b3a2365fcd22278ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,-3) \\qquad \\vv{\\text{v}} = (5,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" 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>\u043f\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\"> \u0427\u0442\u043e\u0431\u044b \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432, \u043d\u0430\u043c \u043d\u0443\u0436\u043d\u043e \u043f\u0435\u0440\u0435\u043c\u043d\u043e\u0436\u0438\u0442\u044c \u0438\u0445 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b X \u0438 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b Y, \u0430 \u0437\u0430\u0442\u0435\u043c \u0441\u043b\u043e\u0436\u0438\u0442\u044c \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\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-f9cb0f372eee67ad149d5b2cff8d2f99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned}\\vv{\\text{u}} \\cdot \\vv{\\text{v}}  &amp; = (4,-3)\\cdot (5,2)  \\\\[1.5ex] &amp; = 4\\cdot 5 + (-3) \\cdot 2 \\\\[1.5ex] &amp; = 20-6\\\\[1.5ex] &amp; =\\bm{14} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"165\" 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 2<\/h3>\n<p> \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u0435 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432, \u043c\u043e\u0434\u0443\u043b\u0438 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0438 \u043e\u0431\u0440\u0430\u0437\u0443\u0435\u043c\u044b\u0439 \u0438\u043c\u0438 \u0443\u0433\u043e\u043b \u0440\u0430\u0432\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-55cc75d8aa625082e47ed6396449f550_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert =6 \\qquad \\lvert \\vv{\\text{v}} \\rvert = 3 \\qquad \\alpha=45\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"225\" 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>\u043f\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\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043c\u044b \u0437\u043d\u0430\u0435\u043c \u0438\u0445 \u043c\u043e\u0434\u0443\u043b\u0438 \u0438 \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u043d\u0438\u043c\u0438, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043d\u0430\u043f\u0440\u044f\u043c\u0443\u044e \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u0442\u044c \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-38999386ccbd92758c9968f025beff72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(\\alpha ) \\\\[1.5ex] &amp;= 6 \\cdot 3 \\cdot \\cos(45\u00ba)\\\\[1.5ex] &amp; = 6 \\cdot 3 \\cdot 0,71 \\\\[1.5ex] &amp;= \\bm{12,73} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"133\" width=\"168\" 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 3<\/h3>\n<p> \u041a\u0430\u043a\u043e\u0432 \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-256770a8f550eb60828f1027831ec423_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(3,8) \\qquad  \\vv{\\text{v}} =(-4,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" 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>\u043f\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 \u043d\u0430\u043c \u043d\u0443\u0436\u043d\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0443 \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-3a43677b59edab4031eae3cf64775e63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ 3^2+8^2}= \\sqrt{73}\" 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-34ffe2db4622d64f22c5ec525dbc01ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{v}} \\rvert = \\sqrt{ (-4)^2+1^2}= \\sqrt{17}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"197\" 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-eecb349d0dcd0362cfd6a67b639e4edb_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{ 3\\cdot (-4) + 8\\cdot 1}{\\sqrt{73}\\cdot \\sqrt{17}} = \\cfrac{-4}{\\sqrt{1241}} = -0,11\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"409\" 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-b3e2f0498a5af3652b484888d36d1f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos^{-1}(-0,11) = \\bm{96,52\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"170\" 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 4<\/h3>\n<p> \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0434\u0432\u0430 \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-ae35d4962f496060eec23769832a6649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(5,2) \\qquad \\vv{\\text{v}} =(-1,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u0435 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0443\u044e \u043e\u043f\u0435\u0440\u0430\u0446\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-256d0ec6c6c20b5914f3ab0716119a97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(\\vv{\\text{u}} \\cdot\\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -7px;\"><\/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>\u043f\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 \u043d\u0430\u043c \u043d\u0443\u0436\u043d\u043e \u043d\u0430\u0439\u0442\u0438 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432\u043d\u0443\u0442\u0440\u0438 \u043a\u0440\u0443\u0433\u043b\u044b\u0445 \u0441\u043a\u043e\u0431\u043e\u043a, \u0430 \u0437\u0430\u0442\u0435\u043c \u0432\u044b\u043f\u043e\u043b\u043d\u0438\u0442\u044c \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0441\u043d\u0430\u0440\u0443\u0436\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-256d0ec6c6c20b5914f3ab0716119a97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(\\vv{\\text{u}} \\cdot\\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -7px;\"><\/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-8652a87d0713db4774dae56ca4328e49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl((5,2) \\cdot (-1,6) \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"129\" style=\"vertical-align: -7px;\"><\/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-c6f9d3b4f48b49bdfdfc69a19dd13903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(5 \\cdot (-1) + 2 \\cdot 6 \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"133\" style=\"vertical-align: -7px;\"><\/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-afe85bb11fcdcc61ae43a2d7f6b7603c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(-5 + 12 \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -7px;\"><\/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-64de0e6750bcabc711c6d73ec0d0d869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\cdot 7\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" 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-32d67549905fec64c67a14e7a8b694e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{28}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" 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> \u0414\u0430\u043d\u044b \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0442\u0440\u0438 \u0434\u0432\u0443\u043c\u0435\u0440\u043d\u044b\u0445 \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-202792a7dea6027ec366cffa5ff30e22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(-2,6) \\qquad \\vv{\\text{v}} =(4,-3)\\qquad \\vv{\\text{w}} =(-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"333\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u0435 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0443\u044e \u043e\u043f\u0435\u0440\u0430\u0446\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-8413dfd8859a9b5a549dbfd002683439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} \\cdot \\bigl( 5 \\vv{\\text{u}}- 2 \\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -7px;\"><\/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>\u043f\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 \u0443\u043c\u043d\u043e\u0436\u0438\u043c \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043d\u0430 \u0441\u043a\u0430\u043b\u044f\u0440\u044b \u0432 \u0441\u043a\u043e\u0431\u043a\u0430\u0445: <\/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-8413dfd8859a9b5a549dbfd002683439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} \\cdot \\bigl( 5 \\vv{\\text{u}}- 2 \\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -7px;\"><\/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-dc363692e0558fc697dee94018108f2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot \\bigl( 5 (-2,6)- 2(4,-3)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"225\" style=\"vertical-align: -7px;\"><\/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-c51b85108b7dc83bc81a75702db939bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot \\bigl( (-10,30)- (8,-6)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"225\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0422\u0435\u043f\u0435\u0440\u044c \u0434\u0435\u043b\u0430\u0435\u043c \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u0435 \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-ca704e156d875e9ca62b62452c77cc3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot  (-10 -8,30-(-6))\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"224\" 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-28e80b50176016afcc5901693ef5426d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot  (-18,36)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0418, \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \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-498a8066459db0849a08866445623dbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1)\\cdot (-18) + 2 \\cdot 36\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"155\" 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-f3c885a9e81d347e3914a736b32bb382_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18 + 72\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"55\" 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-56a66a32d168d33967dda62795778226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{90}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" 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 6<\/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-1b1c6281d6e6570fd94edcd837293e2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(-2,-3) \\qquad  \\vv{\\text{v}} =(k,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"208\" 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>\u043f\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. \u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043a\u043e\u0441\u0438\u043d\u0443\u0441 \u0443\u0433\u043b\u0430 \u0434\u043e\u043b\u0436\u0435\u043d \u0431\u044b\u0442\u044c \u0440\u0430\u0432\u0435\u043d \u043d\u0443\u043b\u044e, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 cos(90\u00ba)=0. \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 \u043f\u0435\u0440\u0435\u0434\u0430\u0442\u044c \u0435\u0433\u043e, \u0443\u043c\u043d\u043e\u0436\u0438\u0432 \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-67bb2b17e1eb4d090327e96f5f3a8bcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =(-2,-3) \\cdot (k,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" 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-77ccd428939d3270f6feeef3ca9681e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-2 \\cdot k + (-3)\\cdot 6\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" 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-4dacdcd81cad7dcb737b48de38e3b4a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-2 k -18\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"105\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0418, \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u043f\u0440\u043e\u044f\u0441\u043d\u044f\u0435\u043c \u043d\u0435\u0438\u0437\u0432\u0435\u0441\u0442\u043d\u043e\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-35a0330d3e83b419577d9448ff01008f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2k =-18\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"74\" 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-6abbb3c976dc4f8d66093a3fb3a40cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle k =\\cfrac{-18}{2}\" 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-67556bf37fde34c4177ff3f3c037f95c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{k =-9}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" 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 7<\/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<figure class=\"wp-block-image 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, \"><\/figure>\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>\u043f\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\u0432\u0438\u0434\u0438\u0442\u0435, \u0447\u0442\u043e \u044d\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0438 \u043a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432. \u0412\u044b \u0442\u0430\u043a\u0436\u0435 \u0443\u0437\u043d\u0430\u0435\u0442\u0435, \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, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0438, \u043a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0432\u0441\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f. \u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u0432\u044b \u0441\u043c\u043e\u0436\u0435\u0442\u0435 \u043f\u043e\u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0432\u0430\u0442\u044c\u0441\u044f \u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u0430\u0445 \u0438 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f\u0445, \u0440\u0435\u0448\u0430\u0435\u043c\u044b\u0445 \u0448\u0430\u0433 \u0437\u0430 \u0448\u0430\u0433\u043e\u043c. \u041a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 \u0412 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/ru\/%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b8%d1%82%d1%8c-%d1%81%d0%ba%d0%b0%d0%bb%d1%8f%d1%80%d0%bd%d0%be%d0%b5-%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%b5%d0%b4%d0%b5%d0%bd%d0%b8%d0%b5-%d0%b4%d0%b2\/\"> <span class=\"screen-reader-text\">\u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432<\/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":[24],"tags":[],"class_list":["post-214","post","type-post","status-publish","format-standard","hentry","category-24"],"yoast_head":"<!-- This site is optimized 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\u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 \u0412 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 &hellip; \u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 \u0427\u0438\u0442\u0430\u0442\u044c \u0434\u0430\u043b\u0435\u0435 &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/ru\/\u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c-\u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435-\u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435-\u0434\u0432\/\" \/>\n<meta property=\"article:published_time\" 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