{"id":215,"date":"2023-07-11T19:16:54","date_gmt":"2023-07-11T19:16:54","guid":{"rendered":"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/"},"modified":"2023-07-11T19:16:54","modified_gmt":"2023-07-11T19:16:54","slug":"%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac","status":"publish","type":"post","link":"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/","title":{"rendered":"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646"},"content":{"rendered":"<p style=\";text-align:right;direction:rtl\">\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u0641\u062d\u0629 \u0633\u062a\u0631\u0649 \u0645\u0627 \u0647\u0648 \u0648\u0643\u064a\u0641\u064a\u0629 \u062d\u0633\u0627\u0628 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646. \u0633\u062a\u062a\u0639\u0644\u0645 \u0623\u064a\u0636\u064b\u0627 \u0643\u064a\u0641\u064a\u0629 \u0625\u064a\u062c\u0627\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a\u060c \u0628\u0627\u0644\u0625\u0636\u0627\u0641\u0629 \u0625\u0644\u0649 \u062c\u0645\u064a\u0639 \u062e\u0635\u0627\u0626\u0635 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a. \u0648\u0623\u062e\u064a\u0631\u0627\u060c \u0633\u0648\u0641 \u062a\u0643\u0648\u0646 \u0642\u0627\u062f\u0631\u0627 \u0639\u0644\u0649 \u0627\u0644\u062a\u062f\u0631\u0628 \u0645\u0639 \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0648\u0627\u0644\u062a\u0645\u0627\u0631\u064a\u0646 \u0627\u0644\u062a\u064a \u062a\u0645 \u062d\u0644\u0647\u0627 \u062e\u0637\u0648\u0629 \u0628\u062e\u0637\u0648\u0629. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center;text-align:right;direction:rtl\">\n<div id=\"ezoic-pub-ad-placeholder-104\" style=\";text-align:right;direction:rtl\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\"><span class=\"ez-toc-section\" id=\"como-calcular-el-producto-escalar-entre-dos-vectores\"><\/span> \u0643\u064a\u0641\u064a\u0629 \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\";text-align:right;direction:rtl\"> \u0641\u064a \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0627\u062a\u060c \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0647\u0648 \u0639\u0645\u0644\u064a\u0629 \u0645\u062a\u062c\u0647\u0629 \u062a\u0642\u0648\u0645 \u0628\u0636\u0631\u0628 \u0645\u062a\u062c\u0647\u064a\u0646 \u0648\u062a\u062d\u0648\u064a\u0644\u0647\u0645\u0627 \u0625\u0644\u0649 \u0639\u062f\u062f \u062d\u0642\u064a\u0642\u064a. \u0644\u0630\u0644\u0643\u060c \u0647\u0646\u0627\u0643 \u0637\u0631\u064a\u0642\u062a\u0627\u0646 \u0644\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646: <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center;text-align:right;direction:rtl\">\n<div id=\"ezoic-pub-ad-placeholder-105\" style=\";text-align:right;direction:rtl\"><\/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;;text-align:right;direction:rtl\">\n<p style=\"text-align:left;text-align:right;direction:rtl\"> \u0625\u0630\u0627 \u0639\u0631\u0641\u0646\u0627 \u0625\u062d\u062f\u0627\u062b\u064a\u0627\u062a \u0645\u062a\u062c\u0647\u064a\u0646\u060c \u0641\u064a\u0645\u0643\u0646\u0646\u0627 \u0625\u064a\u062c\u0627\u062f \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0647\u0645\u0627 \u0639\u0646 \u0637\u0631\u064a\u0642 \u0636\u0631\u0628 \u0645\u0643\u0648\u0646\u0627\u062a X \u0648Y \u0645\u0639\u064b\u0627 \u062b\u0645 \u0625\u0636\u0627\u0641\u0629 \u0627\u0644\u0646\u062a\u0627\u0626\u062c. \u0628\u0645\u0639\u0646\u0649 \u0622\u062e\u0631\u060c \u0625\u0630\u0627 \u0643\u0627\u0646 \u0644\u062f\u064a\u0646\u0627 \u0645\u062a\u062c\u0647\u0627\u0646:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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;text-align:right;direction:rtl\"> \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0639\u062f\u062f\u064a \u0628\u064a\u0646\u0647\u0645\u0627 \u0647\u0648:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0627\u0644\u062a\u0627\u0644\u064a\u064a\u0646 \u0647\u0648:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0625\u0646\u0647\u0627 \u0637\u0631\u064a\u0642\u0629 \u0644\u0644\u0639\u062b\u0648\u0631 \u0639\u0644\u0649 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u0647\u0646\u0627\u0643 \u0623\u064a\u0636\u064b\u0627 \u0637\u0631\u064a\u0642\u0629 \u0623\u062e\u0631\u0649: <\/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;;text-align:right;direction:rtl\">\n<p style=\"text-align:left;text-align:right;direction:rtl\"> \u0645\u0646 \u0646\u0627\u062d\u064a\u0629 \u0623\u062e\u0631\u0649\u060c \u0625\u0630\u0627 \u0639\u0631\u0641\u0646\u0627 \u0627\u0644\u0648\u062d\u062f\u0629 \u0648\u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646\u060c \u0641\u064a\u0645\u0643\u0646 \u062a\u062d\u062f\u064a\u062f \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0639\u0646 \u0637\u0631\u064a\u0642 \u062d\u0633\u0627\u0628 \u0645\u0646\u062a\u062c \u0648\u062d\u062f\u0627\u062a\u0647\u0645\u0627 \u0628\u0648\u0627\u0633\u0637\u0629 \u062c\u064a\u0628 \u062a\u0645\u0627\u0645 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u062a\u064a \u064a\u0634\u0643\u0644\u0627\u0646\u0647\u0627:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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;text-align:right;direction:rtl\"> \u0630\u0647\u0628<\/p>\n<p style=\";text-align:right;direction:rtl\">\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> \u0648<\/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> \u0647\u064a \u0648\u062d\u062f\u0627\u062a \u0645\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a<\/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> \u0648<\/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> \u0639\u0644\u0649 \u0627\u0644\u062a\u0648\u0627\u0644\u064a \u0648<\/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> \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u062a\u064a \u064a\u0635\u0646\u0639\u0648\u0646\u0647\u0627. <\/p>\n<\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center;text-align:right;direction:rtl\">\n<div id=\"ezoic-pub-ad-placeholder-106\" style=\";text-align:right;direction:rtl\"><\/div>\n<\/div>\n<p style=\";text-align:right;direction:rtl\"> \u062a\u0630\u0643\u0631 \u0623\u0646 \u0645\u0642\u062f\u0627\u0631 \u0627\u0644\u0645\u062a\u062c\u0647 \u0647\u0648 \u062c\u0630\u0631 \u0645\u0631\u0628\u0639\u0627\u062a \u0645\u0643\u0648\u0646\u0627\u062a\u0647:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0633\u0646\u062d\u0644 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0648\u062d\u062f\u062a\u064a\u0647\u0645\u0627 \u0648\u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646\u0647\u0645\u0627 \u0647\u064a:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0645\u0646 \u0646\u0627\u062d\u064a\u0629 \u0623\u062e\u0631\u0649\u060c \u064a\u064f\u0637\u0644\u0642 \u0639\u0644\u0649 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0623\u064a\u0636\u064b\u0627 \u0627\u0633\u0645 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0623\u0648 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0639\u062f\u062f\u064a \u0623\u0648 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a.<\/p>\n<p style=\";text-align:right;direction:rtl\"> <strong>\u0645\u0644\u0627\u062d\u0638\u0629:<\/strong> \u0644\u0627 \u062a\u062e\u0644\u0637 \u0628\u064a\u0646 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0648\u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0645\u062a\u0642\u0627\u0637\u0639\u060c \u0644\u0623\u0646\u0647 \u0639\u0644\u0649 \u0627\u0644\u0631\u063a\u0645 \u0645\u0646 \u0623\u0646 \u0644\u0647\u0645\u0627 \u0623\u0633\u0645\u0627\u0621 \u0645\u062a\u0634\u0627\u0628\u0647\u0629\u060c \u0625\u0644\u0627 \u0623\u0646\u0647\u0645\u0627 \u0645\u0641\u0647\u0648\u0645\u0627\u0646 \u0645\u062e\u062a\u0644\u0641\u0627\u0646 \u062a\u0645\u0627\u0645\u064b\u0627. <\/p>\n<h2 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\"><span class=\"ez-toc-section\" id=\"encontrar-el-angulo-entre-dos-vectores-utilizando-el-producto-escalar\"><\/span> \u0623\u0648\u062c\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center;text-align:right;direction:rtl\">\n<div id=\"ezoic-pub-ad-placeholder-109\" style=\";text-align:right;direction:rtl\"><\/div>\n<\/div>\n<p style=\";text-align:right;direction:rtl\"> \u0628\u0645\u062c\u0631\u062f \u0623\u0646 \u0646\u0631\u0649 \u062a\u0639\u0631\u064a\u0641 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a\u060c \u0631\u0628\u0645\u0627 \u062a\u062a\u0633\u0627\u0621\u0644 \u0645\u0627 \u0647\u0648 \u0627\u0644\u063a\u0631\u0636 \u0645\u0646 \u0636\u0631\u0628 \u0645\u062a\u062c\u0647\u064a\u0646\u061f \u062d\u0633\u0646\u064b\u0627\u060c \u0623\u062d\u062f \u062a\u0637\u0628\u064a\u0642\u0627\u062a \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0647\u0648 \u062d\u0633\u0627\u0628 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u062a\u064a \u064a\u0634\u0643\u0644\u0647\u0627 \u0645\u062a\u062c\u0647\u0627\u0646. <\/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=\"\u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0644\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a\" class=\"wp-image-583\" width=\"172\" height=\"175\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p style=\";text-align:right;direction:rtl\"> \u0648\u0628\u062d\u0644 \u062c\u064a\u0628 \u062a\u0645\u0627\u0645 \u0635\u064a\u063a\u0629 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a\u060c \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u062f\u0639\u0648\u0646\u0627 \u0646\u0631\u0649 \u0643\u064a\u0641 \u064a\u062a\u0645 \u0630\u0644\u0643 \u0645\u0646 \u062e\u0644\u0627\u0644 \u0645\u062b\u0627\u0644:<\/p>\n<ul style=\";text-align:right;direction:rtl\">\n<li style=\";text-align:right;direction:rtl\"> \u0623\u0648\u062c\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u0645\u062d\u0635\u0648\u0631\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0627\u0644\u062a\u0627\u0644\u064a\u064a\u0646:<\/li>\n<\/ul>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0646\u062d\u062a\u0627\u062c \u0623\u0648\u0644\u0627\u064b \u0625\u0644\u0649 \u0625\u064a\u062c\u0627\u062f \u0645\u0642\u062f\u0627\u0631 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0646\u0633\u062a\u062e\u062f\u0645 \u0627\u0644\u0622\u0646 \u0627\u0644\u0635\u064a\u063a\u0629 \u0644\u062d\u0633\u0627\u0628 \u062c\u064a\u0628 \u062a\u0645\u0627\u0645 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0625\u064a\u062c\u0627\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u0645\u0642\u0627\u0628\u0644\u0629 \u0639\u0646 \u0637\u0631\u064a\u0642 \u0625\u062c\u0631\u0627\u0621 \u0645\u0639\u0643\u0648\u0633 \u062c\u064a\u0628 \u0627\u0644\u062a\u0645\u0627\u0645 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0622\u0644\u0629 \u0627\u0644\u062d\u0627\u0633\u0628\u0629:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0648\u0644\u0630\u0644\u0643\u060c \u062a\u0634\u0643\u0644 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0632\u0627\u0648\u064a\u0629 \u0642\u062f\u0631\u0647\u0627 74.93 \u062f\u0631\u062c\u0629. <\/p>\n<h2 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\"><span class=\"ez-toc-section\" id=\"propiedades-del-producto-escalar-de-dos-vectores\"><\/span> \u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\";text-align:right;direction:rtl\"> \u064a\u062a\u0645\u064a\u0632 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0628\u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<ul style=\";text-align:right;direction:rtl\">\n<li style=\";text-align:right;direction:rtl\"> <strong>\u0627\u0644\u062e\u0627\u0635\u064a\u0629 \u0627\u0644\u062a\u0628\u0627\u062f\u0644\u064a\u0629<\/strong> : \u0644\u0627 \u064a\u0647\u0645 \u062a\u0631\u062a\u064a\u0628 \u0636\u0631\u0628 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a.<\/li>\n<\/ul>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\n<li style=\";text-align:right;direction:rtl\"> <strong>\u062e\u0627\u0635\u064a\u0629 \u0627\u0644\u062a\u0648\u0632\u064a\u0639<\/strong> : \u064a\u0643\u0648\u0646 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u062a\u0648\u0632\u064a\u0639\u064a\u064b\u0627 \u0641\u064a\u0645\u0627 \u064a\u062a\u0639\u0644\u0642 \u0628\u062c\u0645\u0639 \u0648\u0637\u0631\u062d \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a:<\/li>\n<\/ul>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\n<li style=\";text-align:right;direction:rtl\"> <strong>\u062e\u0627\u0635\u064a\u0629 \u0627\u0644\u062f\u0645\u062c<\/strong> : \u064a\u0645\u0643\u0646\u0646\u0627 \u0636\u0631\u0628 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0628\u062b\u0627\u0628\u062a \u0642\u0628\u0644 \u0623\u0648 \u0628\u0639\u062f \u0625\u062c\u0631\u0627\u0621 \u0627\u0644\u0639\u0645\u0644\u064a\u0629\u060c \u062d\u064a\u062b \u0623\u0646 \u0627\u0644\u0646\u062a\u0627\u0626\u062c \u0645\u062a\u0643\u0627\u0641\u0626\u0629:<\/li>\n<\/ul>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\n<li style=\";text-align:right;direction:rtl\"> \u0625\u0630\u0627 \u0643\u0627\u0646 \u0645\u062a\u062c\u0647\u0627\u0646 <strong>\u0645\u062a\u0639\u0627\u0645\u062f\u064a\u0646<\/strong> (\u0623\u0648 \u0645\u062a\u0639\u0627\u0645\u062f\u064a\u0646)\u060c \u0641\u0625\u0646 \u062d\u0627\u0635\u0644 \u0636\u0631\u0628\u0647\u0645\u0627 \u0627\u0644\u0646\u0642\u0637\u064a \u064a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u064b\u0627. \u064a\u0645\u0643\u0646 \u0625\u062b\u0628\u0627\u062a \u0647\u0630\u0647 \u0627\u0644\u062e\u0627\u0635\u064a\u0629 \u0628\u0633\u0647\u0648\u0644\u0629 \u0644\u0623\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0645\u062a\u0639\u0627\u0645\u062f\u064a\u0646 \u064a\u0634\u0643\u0644\u0627\u0646 \u0632\u0627\u0648\u064a\u0629 \u0642\u062f\u0631\u0647\u0627 90 \u062f\u0631\u062c\u0629\u060c \u0648\u062c\u064a\u0628 \u0627\u0644\u062a\u0645\u0627\u0645 90 \u062f\u0631\u062c\u0629 \u064a\u0633\u0627\u0648\u064a 0:<\/li>\n<\/ul>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\n<li style=\";text-align:right;direction:rtl\"> \u0639\u0644\u0649 \u0627\u0644\u0639\u0643\u0633 \u0645\u0646 \u0630\u0644\u0643\u060c \u0625\u0630\u0627 \u0643\u0627\u0646 \u0645\u062a\u062c\u0647\u0627\u0646 <strong>\u0645\u062a\u0648\u0627\u0632\u064a\u064a\u0646<\/strong> \u060c \u0641\u0625\u0646 \u0645\u0646\u062a\u062c\u0647\u0645\u0627 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u064a\u0643\u0648\u0646 \u0647\u0648 \u0646\u0641\u0633\u0647 \u0645\u0646\u062a\u062c \u0648\u062d\u062f\u0627\u062a\u0647\u0645\u0627. \u064a\u0645\u0643\u0646 \u0623\u064a\u0636\u064b\u0627 \u0627\u0644\u062a\u062d\u0642\u0642 \u0645\u0646 \u0647\u0630\u0647 \u0627\u0644\u062e\u0627\u0635\u064a\u0629 \u0628\u0633\u0647\u0648\u0644\u0629 \u0646\u0638\u0631\u064b\u0627 \u0644\u0623\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0645\u0646 \u0646\u0641\u0633 \u0627\u0644\u0627\u062a\u062c\u0627\u0647 \u064a\u0634\u0643\u0644\u0627\u0646 \u0632\u0627\u0648\u064a\u0629 \u0642\u062f\u0631\u0647\u0627 0 \u062f\u0631\u062c\u0629\u060c \u0648\u062c\u064a\u0628 \u062a\u0645\u0627\u0645\u0647\u0627 \u064a\u0633\u0627\u0648\u064a 1:<\/li>\n<\/ul>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\n<li style=\";text-align:right;direction:rtl\"> \u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0641\u0625\u0646 \u0627\u0644\u0646\u0627\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0644\u0645\u062a\u062c\u0647 \u0641\u064a \u062d\u062f \u0630\u0627\u062a\u0647 \u064a\u0639\u0627\u062f\u0644 \u0645\u0631\u0628\u0639 \u062d\u062c\u0645\u0647: <\/li>\n<\/ul>\n<p style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-productos-escalares-entre-dos-vectores\"><\/span> \u062d\u0644 \u0645\u0633\u0627\u0626\u0644 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0639\u062f\u062f\u064a \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\"> \u0627\u0644\u062a\u0645\u0631\u064a\u0646 1<\/h3>\n<p style=\";text-align:right;direction:rtl\"> \u0627\u062d\u0633\u0628 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0641\u064a \u0645\u0633\u062a\u0648\u0649 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0627\u0644\u062a\u0627\u0644\u064a\u064a\u0646: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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;text-align:right;direction:rtl\">\n<div class=\"otfm-sp__title\" style=\";text-align:right;direction:rtl\"> <strong>\u0627\u0646\u0638\u0631 \u0627\u0644\u062d\u0644<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\" style=\";text-align:right;direction:rtl\"> \u0644\u062d\u0633\u0627\u0628 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646\u060c \u0646\u062d\u062a\u0627\u062c \u0625\u0644\u0649 \u0636\u0631\u0628 \u0625\u062d\u062f\u0627\u062b\u064a\u0627\u062a X \u0648\u0625\u062d\u062f\u0627\u062b\u064a\u0627\u062a Y \u0645\u0639\u064b\u0627\u060c \u062b\u0645 \u0625\u0636\u0627\u0641\u0629 \u0627\u0644\u0646\u062a\u0627\u0626\u062c: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"><\/div>\n<h3 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\">\u062a\u0645\u0631\u064a\u0646 2<\/h3>\n<p style=\";text-align:right;direction:rtl\"> \u062a\u062d\u062f\u064a\u062f \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u062a\u0643\u0648\u0646 \u0648\u062d\u062f\u0627\u062a\u0647\u0645\u0627 \u0648\u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u062a\u064a \u064a\u0634\u0643\u0644\u0627\u0646\u0647\u0627 \u0647\u064a: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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;text-align:right;direction:rtl\">\n<div class=\"otfm-sp__title\" style=\";text-align:right;direction:rtl\"> <strong>\u0627\u0646\u0638\u0631 \u0627\u0644\u062d\u0644<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\" style=\";text-align:right;direction:rtl\"> \u0648\u0628\u0645\u0627 \u0623\u0646\u0646\u0627 \u0646\u0639\u0631\u0641 \u0648\u062d\u062f\u0627\u062a\u0647\u0627 \u0648\u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646\u0647\u0627\u060c \u0641\u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u0637\u0628\u064a\u0642 \u0635\u064a\u063a\u0629 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0645\u0628\u0627\u0634\u0631\u0629: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"><\/div>\n<h3 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\">\u0627\u0644\u062a\u0645\u0631\u064a\u0646 3<\/h3>\n<p style=\";text-align:right;direction:rtl\"> \u0645\u0627 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u0645\u062d\u0635\u0648\u0631\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0627\u0644\u062a\u0627\u0644\u064a\u064a\u0646\u061f <\/p>\n<p style=\";text-align:right;direction:rtl\">\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;text-align:right;direction:rtl\">\n<div class=\"otfm-sp__title\" style=\";text-align:right;direction:rtl\"> <strong>\u0627\u0646\u0638\u0631 \u0627\u0644\u062d\u0644<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\" style=\";text-align:right;direction:rtl\"> \u0623\u0648\u0644\u0627\u064b\u060c \u0639\u0644\u064a\u0646\u0627 \u062d\u0633\u0627\u0628 \u0645\u0642\u062f\u0627\u0631 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0646\u0633\u062a\u062e\u062f\u0645 \u0627\u0644\u0635\u064a\u063a\u0629 \u0644\u062d\u0633\u0627\u0628 \u062c\u064a\u0628 \u062a\u0645\u0627\u0645 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u062a\u064a \u062a\u0634\u0643\u0644\u0647\u0627 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0625\u064a\u062c\u0627\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u0645\u0642\u0627\u0628\u0644\u0629 \u0639\u0646 \u0637\u0631\u064a\u0642 \u0625\u062c\u0631\u0627\u0621 \u0645\u0639\u0643\u0648\u0633 \u062c\u064a\u0628 \u0627\u0644\u062a\u0645\u0627\u0645 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0622\u0644\u0629 \u0627\u0644\u062d\u0627\u0633\u0628\u0629: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"><\/div>\n<h3 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\">\u0627\u0644\u062a\u0645\u0631\u064a\u0646 4<\/h3>\n<p style=\";text-align:right;direction:rtl\"> \u062e\u0630 \u0628\u0639\u064a\u0646 \u0627\u0644\u0627\u0639\u062a\u0628\u0627\u0631 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0627\u0644\u062a\u0627\u0644\u064a\u064a\u0646:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0627\u062d\u0633\u0628 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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;text-align:right;direction:rtl\">\n<div class=\"otfm-sp__title\" style=\";text-align:right;direction:rtl\"> <strong>\u0627\u0646\u0638\u0631 \u0627\u0644\u062d\u0644<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\" style=\";text-align:right;direction:rtl\"> \u0639\u0644\u064a\u0646\u0627 \u0623\u0648\u0644\u064b\u0627 \u0625\u064a\u062c\u0627\u062f \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u062f\u0627\u062e\u0644 \u0627\u0644\u0642\u0648\u0633\u064a\u0646\u060c \u062b\u0645 \u0627\u0644\u0636\u0631\u0628 \u0641\u064a \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0648\u0633: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"><\/div>\n<h3 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\">\u0627\u0644\u062a\u0645\u0631\u064a\u0646 5<\/h3>\n<p style=\";text-align:right;direction:rtl\"> \u0628\u0627\u0644\u0646\u0638\u0631 \u0625\u0644\u0649 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0627\u0644\u062b\u0644\u0627\u062b\u0629 \u062b\u0646\u0627\u0626\u064a\u0629 \u0627\u0644\u0623\u0628\u0639\u0627\u062f \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\"> \u0627\u062d\u0633\u0628 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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;text-align:right;direction:rtl\">\n<div class=\"otfm-sp__title\" style=\";text-align:right;direction:rtl\"> <strong>\u0627\u0646\u0638\u0631 \u0627\u0644\u062d\u0644<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\" style=\";text-align:right;direction:rtl\"> \u0623\u0648\u0644\u0627\u064b\u060c \u0646\u0636\u0631\u0628 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0641\u064a \u0627\u0644\u0643\u0645\u064a\u0627\u062a \u0627\u0644\u0642\u064a\u0627\u0633\u064a\u0629 \u0627\u0644\u0645\u0648\u062c\u0648\u062f\u0629 \u0628\u064a\u0646 \u0642\u0648\u0633\u064a\u0646: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0627\u0644\u0622\u0646 \u0646\u0642\u0648\u0645 \u0628\u0637\u0631\u062d \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0646\u062d\u0644 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0642\u064a\u0627\u0633\u064a: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"><\/div>\n<h3 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\">\u0627\u0644\u062a\u0645\u0631\u064a\u0646 6<\/h3>\n<p style=\";text-align:right;direction:rtl\"> \u0627\u062d\u0633\u0628 \u0642\u064a\u0645\u0629<\/p>\n<p style=\";text-align:right;direction:rtl\">\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> \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0627\u0644\u062a\u0627\u0644\u064a\u0629 \u0645\u062a\u0639\u0627\u0645\u062f\u0629: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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;text-align:right;direction:rtl\">\n<div class=\"otfm-sp__title\" style=\";text-align:right;direction:rtl\"> <strong>\u0627\u0646\u0638\u0631 \u0627\u0644\u062d\u0644<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\" style=\";text-align:right;direction:rtl\"> \u064a\u0634\u0643\u0644 \u0645\u062a\u062c\u0647\u0627\u0646 \u0645\u062a\u0639\u0627\u0645\u062f\u0627\u0646 \u0632\u0627\u0648\u064a\u0629 \u0642\u062f\u0631\u0647\u0627 90 \u062f\u0631\u062c\u0629. \u0644\u0630\u0627 \u064a\u062c\u0628 \u0623\u0646 \u064a\u0643\u0648\u0646 \u062c\u064a\u0628 \u062a\u0645\u0627\u0645 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0635\u0641\u0631\u064b\u0627\u060c \u062d\u064a\u062b \u0623\u0646 cos(90\u00b0)=0. \u062d\u062a\u0649 \u0627\u0644\u0622\u0646: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0645\u0642\u0627\u0645 \u0627\u0644\u0643\u0633\u0631 \u064a\u0642\u0633\u0645 \u0627\u0644\u062c\u0627\u0646\u0628 \u0627\u0644\u0623\u064a\u0645\u0646 \u0628\u0627\u0644\u0643\u0627\u0645\u0644 \u0645\u0646 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629\u060c \u062d\u062a\u0649 \u0646\u062a\u0645\u0643\u0646 \u0645\u0646 \u062a\u0645\u0631\u064a\u0631\u0647 \u0639\u0646 \u0637\u0631\u064a\u0642 \u0627\u0644\u0636\u0631\u0628 \u0641\u064a \u0627\u0644\u062c\u0627\u0646\u0628 \u0627\u0644\u0622\u062e\u0631: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0646\u062d\u0646 \u0627\u0644\u0622\u0646 \u0646\u062d\u0644 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0639\u062f\u062f\u064a: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0648\u0623\u062e\u064a\u0631\u0627 \u0646\u0648\u0636\u062d \u0627\u0644\u0645\u062c\u0647\u0648\u0644: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"><\/div>\n<h3 class=\"wp-block-heading\" style=\";text-align:right;direction:rtl\">\u0627\u0644\u062a\u0645\u0631\u064a\u0646 7<\/h3>\n<p style=\";text-align:right;direction:rtl\"> \u062d\u0633\u0627\u0628 \u0627\u0644\u0632\u0648\u0627\u064a\u0627<\/p>\n<p style=\";text-align:right;direction:rtl\">\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> \u0648<\/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> \u0648\u0627\u0644\u062a\u064a \u062a\u0634\u0643\u0644 \u0623\u0636\u0644\u0627\u0639 \u0627\u0644\u0645\u062b\u0644\u062b \u0627\u0644\u062a\u0627\u0644\u064a: <\/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=\"\u062a\u0645\u0627\u0631\u064a\u0646 \u0648\u0645\u0633\u0627\u0626\u0644 \u062a\u0645 \u062d\u0644\u0647\u0627 \u062e\u0637\u0648\u0629 \u0628\u062e\u0637\u0648\u0629 \u0644\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0642\u064a\u0627\u0633\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646\" 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;text-align:right;direction:rtl\">\n<div class=\"otfm-sp__title\" style=\";text-align:right;direction:rtl\"> <strong>\u0627\u0646\u0638\u0631 \u0627\u0644\u062d\u0644<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\" style=\";text-align:right;direction:rtl\"> \u0627\u0644\u0631\u0621\u0648\u0633 \u0627\u0644\u062a\u064a \u064a\u062a\u0643\u0648\u0646 \u0645\u0646\u0647\u0627 \u0627\u0644\u0645\u062b\u0644\u062b \u0647\u064a \u0627\u0644\u0646\u0642\u0627\u0637 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<p style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0644\u062d\u0633\u0627\u0628 \u0627\u0644\u0632\u0648\u0627\u064a\u0627 \u0627\u0644\u062f\u0627\u062e\u0644\u064a\u0629 \u0644\u0644\u0645\u062b\u0644\u062b\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u062d\u0633\u0627\u0628 \u0645\u062a\u062c\u0647\u0627\u062a \u0643\u0644 \u0636\u0644\u0639 \u0645\u0646 \u0623\u0636\u0644\u0627\u0639\u0647\u060c \u062b\u0645 \u0625\u064a\u062c\u0627\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u062a\u064a \u062a\u0634\u0643\u0644\u0647\u0627 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0635\u064a\u063a\u0629 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a.<\/p>\n<p class=\"has-text-align-left\" style=\";text-align:right;direction:rtl\"> \u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0644\u0644\u0639\u062b\u0648\u0631 \u0639\u0644\u0649 \u0627\u0644\u0632\u0627\u0648\u064a\u0629<\/p>\n<p style=\";text-align:right;direction:rtl\">\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> \u0646\u062d\u0633\u0628 \u0645\u062a\u062c\u0647\u0627\u062a \u062c\u0648\u0627\u0646\u0628\u0647: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0648\u0646\u0648\u062c\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u062a\u064a \u064a\u0634\u0643\u0644\u0647\u0627 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u0646 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0635\u064a\u063a\u0629 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0627\u0644\u0622\u0646 \u0646\u0643\u0631\u0631 \u0646\u0641\u0633 \u0627\u0644\u0625\u062c\u0631\u0627\u0621 \u0644\u062a\u062d\u062f\u064a\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 <\/p>\n<p style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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 style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"> \u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0644\u0644\u0639\u062b\u0648\u0631 \u0639\u0644\u0649 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u0643\u0631\u0627\u0631 \u0646\u0641\u0633 \u0627\u0644\u0625\u062c\u0631\u0627\u0621. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u062c\u0645\u064a\u0639 \u0627\u0644\u0632\u0648\u0627\u064a\u0627 \u0641\u064a \u0627\u0644\u0645\u062b\u0644\u062b \u064a\u062c\u0628 \u0623\u0646 \u064a\u0643\u0648\u0646 \u0645\u062c\u0645\u0648\u0639\u0647\u0627 180 \u062f\u0631\u062c\u0629\u060c \u0644\u0630\u0644\u0643: <\/p>\n<p style=\";text-align:right;direction:rtl\">\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\" style=\";text-align:right;direction:rtl\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u0641\u062d\u0629 \u0633\u062a\u0631\u0649 \u0645\u0627 \u0647\u0648 \u0648\u0643\u064a\u0641\u064a\u0629 \u062d\u0633\u0627\u0628 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646. \u0633\u062a\u062a\u0639\u0644\u0645 \u0623\u064a\u0636\u064b\u0627 \u0643\u064a\u0641\u064a\u0629 \u0625\u064a\u062c\u0627\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a\u060c \u0628\u0627\u0644\u0625\u0636\u0627\u0641\u0629 \u0625\u0644\u0649 \u062c\u0645\u064a\u0639 \u062e\u0635\u0627\u0626\u0635 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a. \u0648\u0623\u062e\u064a\u0631\u0627\u060c \u0633\u0648\u0641 \u062a\u0643\u0648\u0646 \u0642\u0627\u062f\u0631\u0627 \u0639\u0644\u0649 \u0627\u0644\u062a\u062f\u0631\u0628 \u0645\u0639 \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0648\u0627\u0644\u062a\u0645\u0627\u0631\u064a\u0646 \u0627\u0644\u062a\u064a \u062a\u0645 \u062d\u0644\u0647\u0627 \u062e\u0637\u0648\u0629 \u0628\u062e\u0637\u0648\u0629. \u0643\u064a\u0641\u064a\u0629 \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0641\u064a \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0627\u062a\u060c \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0647\u0648 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/\"> <span class=\"screen-reader-text\">\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646<\/span> \u0642\u0631\u0627\u0621\u0629 \u0627\u0644\u0645\u0632\u064a\u062f &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":[23],"tags":[],"class_list":["post-215","post","type-post","status-publish","format-standard","hentry","category-23"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/ar\/\u062d\u0633\u0627\u0628-\u0627\u0644\u0645\u0646\u062a\u062c-\u0627\u0644\u0642\u064a\u0627\u0633\u064a-\u0628\u064a\u0646-\u0627\u062b\u0646\u064a\u0646-\u0645\u0646-\u0627\u0644\u0645\u062a\u062c\/\" \/>\n<meta property=\"og:locale\" content=\"ar_AR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 -\" \/>\n<meta property=\"og:description\" content=\"\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u0641\u062d\u0629 \u0633\u062a\u0631\u0649 \u0645\u0627 \u0647\u0648 \u0648\u0643\u064a\u0641\u064a\u0629 \u062d\u0633\u0627\u0628 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646. \u0633\u062a\u062a\u0639\u0644\u0645 \u0623\u064a\u0636\u064b\u0627 \u0643\u064a\u0641\u064a\u0629 \u0625\u064a\u062c\u0627\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a\u060c \u0628\u0627\u0644\u0625\u0636\u0627\u0641\u0629 \u0625\u0644\u0649 \u062c\u0645\u064a\u0639 \u062e\u0635\u0627\u0626\u0635 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a. \u0648\u0623\u062e\u064a\u0631\u0627\u060c \u0633\u0648\u0641 \u062a\u0643\u0648\u0646 \u0642\u0627\u062f\u0631\u0627 \u0639\u0644\u0649 \u0627\u0644\u062a\u062f\u0631\u0628 \u0645\u0639 \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0648\u0627\u0644\u062a\u0645\u0627\u0631\u064a\u0646 \u0627\u0644\u062a\u064a \u062a\u0645 \u062d\u0644\u0647\u0627 \u062e\u0637\u0648\u0629 \u0628\u062e\u0637\u0648\u0629. \u0643\u064a\u0641\u064a\u0629 \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0641\u064a \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0627\u062a\u060c \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0647\u0648 &hellip; \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0642\u0631\u0627\u0621\u0629 \u0627\u0644\u0645\u0632\u064a\u062f &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/ar\/\u062d\u0633\u0627\u0628-\u0627\u0644\u0645\u0646\u062a\u062c-\u0627\u0644\u0642\u064a\u0627\u0633\u064a-\u0628\u064a\u0646-\u0627\u062b\u0646\u064a\u0646-\u0645\u0646-\u0627\u0644\u0645\u062a\u062c\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-11T19:16:54+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c06f6c1238ab7803750f830e16891f1f_l3.png\" \/>\n<meta name=\"author\" content=\"\u0641\u0631\u064a\u0642 Mathority\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u0643\u064f\u062a\u0628 \u0628\u0648\u0627\u0633\u0637\u0629\" \/>\n\t<meta name=\"twitter:data1\" content=\"\u0641\u0631\u064a\u0642 Mathority\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/\"},\"author\":{\"name\":\"\u0641\u0631\u064a\u0642 Mathority\",\"@id\":\"https:\/\/mathority.org\/ar\/#\/schema\/person\/b6c8cd3c82fb4228ce41b667b17bd613\"},\"headline\":\"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646\",\"datePublished\":\"2023-07-11T19:16:54+00:00\",\"dateModified\":\"2023-07-11T19:16:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/\"},\"wordCount\":5,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/ar\/#organization\"},\"articleSection\":[\"\u0627\u0644\u0622\u0644\u0627\u062a \u0627\u0644\u062d\u0627\u0633\u0628\u0629 \u0627\u0644\u0639\u0644\u0645\u064a\u0629\"],\"inLanguage\":\"ar\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/\",\"url\":\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/\",\"name\":\"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 -\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/ar\/#website\"},\"datePublished\":\"2023-07-11T19:16:54+00:00\",\"dateModified\":\"2023-07-11T19:16:54+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/#breadcrumb\"},\"inLanguage\":\"ar\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/ar\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/ar\/#website\",\"url\":\"https:\/\/mathority.org\/ar\/\",\"name\":\"Mathority\",\"description\":\"\u062d\u064a\u062b \u064a\u0644\u062a\u0642\u064a \u0627\u0644\u0641\u0636\u0648\u0644 \u0628\u0627\u0644\u062d\u0633\u0627\u0628!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/ar\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/ar\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"ar\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/ar\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/ar\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"ar\",\"@id\":\"https:\/\/mathority.org\/ar\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/ar\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/ar\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/ar\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/ar\/#\/schema\/person\/b6c8cd3c82fb4228ce41b667b17bd613\",\"name\":\"\u0641\u0631\u064a\u0642 Mathority\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"ar\",\"@id\":\"https:\/\/mathority.org\/ar\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/dd3f98feaeb54f342e72193f2059ae90dc3178f59adb7c82b33fc7d71c6ee19d?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/dd3f98feaeb54f342e72193f2059ae90dc3178f59adb7c82b33fc7d71c6ee19d?s=96&d=mm&r=g\",\"caption\":\"\u0641\u0631\u064a\u0642 Mathority\"},\"sameAs\":[\"http:\/\/mathority.org\/ar\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 -","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/ar\/\u062d\u0633\u0627\u0628-\u0627\u0644\u0645\u0646\u062a\u062c-\u0627\u0644\u0642\u064a\u0627\u0633\u064a-\u0628\u064a\u0646-\u0627\u062b\u0646\u064a\u0646-\u0645\u0646-\u0627\u0644\u0645\u062a\u062c\/","og_locale":"ar_AR","og_type":"article","og_title":"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 -","og_description":"\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u0641\u062d\u0629 \u0633\u062a\u0631\u0649 \u0645\u0627 \u0647\u0648 \u0648\u0643\u064a\u0641\u064a\u0629 \u062d\u0633\u0627\u0628 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646. \u0633\u062a\u062a\u0639\u0644\u0645 \u0623\u064a\u0636\u064b\u0627 \u0643\u064a\u0641\u064a\u0629 \u0625\u064a\u062c\u0627\u062f \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a\u060c \u0628\u0627\u0644\u0625\u0636\u0627\u0641\u0629 \u0625\u0644\u0649 \u062c\u0645\u064a\u0639 \u062e\u0635\u0627\u0626\u0635 \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0646\u0642\u0637\u064a. \u0648\u0623\u062e\u064a\u0631\u0627\u060c \u0633\u0648\u0641 \u062a\u0643\u0648\u0646 \u0642\u0627\u062f\u0631\u0627 \u0639\u0644\u0649 \u0627\u0644\u062a\u062f\u0631\u0628 \u0645\u0639 \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0648\u0627\u0644\u062a\u0645\u0627\u0631\u064a\u0646 \u0627\u0644\u062a\u064a \u062a\u0645 \u062d\u0644\u0647\u0627 \u062e\u0637\u0648\u0629 \u0628\u062e\u0637\u0648\u0629. \u0643\u064a\u0641\u064a\u0629 \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0641\u064a \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0627\u062a\u060c \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0647\u0648 &hellip; \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0642\u0631\u0627\u0621\u0629 \u0627\u0644\u0645\u0632\u064a\u062f &raquo;","og_url":"https:\/\/mathority.org\/ar\/\u062d\u0633\u0627\u0628-\u0627\u0644\u0645\u0646\u062a\u062c-\u0627\u0644\u0642\u064a\u0627\u0633\u064a-\u0628\u064a\u0646-\u0627\u062b\u0646\u064a\u0646-\u0645\u0646-\u0627\u0644\u0645\u062a\u062c\/","article_published_time":"2023-07-11T19:16:54+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c06f6c1238ab7803750f830e16891f1f_l3.png"}],"author":"\u0641\u0631\u064a\u0642 Mathority","twitter_card":"summary_large_image","twitter_misc":{"\u0643\u064f\u062a\u0628 \u0628\u0648\u0627\u0633\u0637\u0629":"\u0641\u0631\u064a\u0642 Mathority"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/"},"author":{"name":"\u0641\u0631\u064a\u0642 Mathority","@id":"https:\/\/mathority.org\/ar\/#\/schema\/person\/b6c8cd3c82fb4228ce41b667b17bd613"},"headline":"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646","datePublished":"2023-07-11T19:16:54+00:00","dateModified":"2023-07-11T19:16:54+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/"},"wordCount":5,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/ar\/#organization"},"articleSection":["\u0627\u0644\u0622\u0644\u0627\u062a \u0627\u0644\u062d\u0627\u0633\u0628\u0629 \u0627\u0644\u0639\u0644\u0645\u064a\u0629"],"inLanguage":"ar","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/","url":"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/","name":"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646 -","isPartOf":{"@id":"https:\/\/mathority.org\/ar\/#website"},"datePublished":"2023-07-11T19:16:54+00:00","dateModified":"2023-07-11T19:16:54+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/#breadcrumb"},"inLanguage":"ar","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/ar\/%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d9%85%d9%86%d8%aa%d8%ac-%d8%a7%d9%84%d9%82%d9%8a%d8%a7%d8%b3%d9%8a-%d8%a8%d9%8a%d9%86-%d8%a7%d8%ab%d9%86%d9%8a%d9%86-%d9%85%d9%86-%d8%a7%d9%84%d9%85%d8%aa%d8%ac\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/ar\/"},{"@type":"ListItem","position":2,"name":"\u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0646\u062a\u062c \u0627\u0644\u0646\u0642\u0637\u064a \u0644\u0645\u062a\u062c\u0647\u064a\u0646"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/ar\/#website","url":"https:\/\/mathority.org\/ar\/","name":"Mathority","description":"\u062d\u064a\u062b \u064a\u0644\u062a\u0642\u064a \u0627\u0644\u0641\u0636\u0648\u0644 \u0628\u0627\u0644\u062d\u0633\u0627\u0628!","publisher":{"@id":"https:\/\/mathority.org\/ar\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/ar\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"ar"},{"@type":"Organization","@id":"https:\/\/mathority.org\/ar\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/ar\/","logo":{"@type":"ImageObject","inLanguage":"ar","@id":"https:\/\/mathority.org\/ar\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/ar\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/ar\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/ar\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/ar\/#\/schema\/person\/b6c8cd3c82fb4228ce41b667b17bd613","name":"\u0641\u0631\u064a\u0642 Mathority","image":{"@type":"ImageObject","inLanguage":"ar","@id":"https:\/\/mathority.org\/ar\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/dd3f98feaeb54f342e72193f2059ae90dc3178f59adb7c82b33fc7d71c6ee19d?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/dd3f98feaeb54f342e72193f2059ae90dc3178f59adb7c82b33fc7d71c6ee19d?s=96&d=mm&r=g","caption":"\u0641\u0631\u064a\u0642 Mathority"},"sameAs":["http:\/\/mathority.org\/ar"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/ar\/wp-json\/wp\/v2\/posts\/215","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/ar\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/ar\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/ar\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/ar\/wp-json\/wp\/v2\/comments?post=215"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/ar\/wp-json\/wp\/v2\/posts\/215\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/ar\/wp-json\/wp\/v2\/media?parent=215"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/ar\/wp-json\/wp\/v2\/categories?post=215"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/ar\/wp-json\/wp\/v2\/tags?post=215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}