{"id":34290,"date":"2022-03-29T13:00:37","date_gmt":"2022-03-29T13:00:37","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34290"},"modified":"2025-09-07T00:56:07","modified_gmt":"2025-09-07T00:56:07","slug":"%d8%a7%d9%84%d8%ac%d8%a8%d8%b1-%d9%88%d8%a7%d9%84%d8%a5%d8%b3%d9%82%d8%a7%d8%b7%d8%a7%d8%aa-%d9%81%d9%8a-rn%d8%8c-%d9%88%d8%a7%d9%84%d8%ad%d8%a7%d8%b5%d9%84-%d8%a7%d9%84%d8%a7%d8%aa%d8%ac%d8%a7%d9%87","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ar\/%d8%a7%d9%84%d8%ac%d8%a8%d8%b1-%d9%88%d8%a7%d9%84%d8%a5%d8%b3%d9%82%d8%a7%d8%b7%d8%a7%d8%aa-%d9%81%d9%8a-rn%d8%8c-%d9%88%d8%a7%d9%84%d8%ad%d8%a7%d8%b5%d9%84-%d8%a7%d9%84%d8%a7%d8%aa%d8%ac%d8%a7%d9%87\/","title":{"rendered":"\u0627\u0644\u062c\u0628\u0631 \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637\u0627\u062a \u0641\u064a Rn\u060c \u0648\u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0641\u064a R3"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>\u0627\u0644\u062c\u0628\u0631 \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637\u0627\u062a \u0641\u064a Rn\u060c \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{\\mathbb{R}^3}<\/span><\/span><\/h1>\n<p style=\"text-align:center;\"><em><strong>\u0645\u0644\u062e\u0635:<\/strong><\/br>\u062a\u0634\u0643\u0644 \u0647\u0630\u0647 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0645\u062a\u0627\u0628\u0639\u0629 \u0645\u0628\u0627\u0634\u0631\u0629 \u0644\u0644\u0633\u0644\u0633\u0644\u0629 \u062d\u0648\u0644 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u0630\u064a \u0627\u0644\u0623\u0628\u0639\u0627\u062f n. \u0647\u0646\u0627 \u0633\u0646\u0631\u0627\u062c\u0639 \u0628\u0639\u0636 \u0645\u0641\u0627\u0647\u064a\u0645 \u0627\u0644\u062c\u0628\u0631 \u0627\u0644\u062e\u0637\u064a \u0627\u0644\u062a\u064a \u062a\u0633\u0627\u0639\u062f \u0639\u0644\u0649 \u0641\u0647\u0645 \u0623\u0641\u0636\u0644 \u0644\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a n-\u0627\u0644\u0623\u0628\u0639\u0627\u062f\u060c \u0648\u0633\u0646\u0631\u0627\u062c\u0639 \u0645\u0641\u0627\u0647\u064a\u0645 \u0625\u0633\u0642\u0627\u0637 \u0645\u062a\u062c\u0647 \u0639\u0644\u0649 \u0622\u062e\u0631\u060c \u0648\u0646\u0628\u0631\u0647\u0646 \u0639\u0644\u0649 \u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633\u060c \u0648\u0646\u062e\u062a\u062a\u0645 \u0628\u0645\u0631\u0627\u062c\u0639\u0629 \u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/span> \u0648\u0639\u0644\u0627\u0642\u062a\u0647 \u0645\u0639 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u0623\u062e\u0631\u0649 \u0641\u064a \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u062b\u0644\u0627\u062b\u064a \u0627\u0644\u0623\u0628\u0639\u0627\u062f. <\/p>\n<p style=\"text-align:center;\"><strong>\u0627\u0644\u0641\u0647\u0631\u0633<\/strong><br \/>\n<a href=\"#Independencia-Lineal-Ortogonalidad-y-Proyecciones\">\u0627\u0644\u0627\u0633\u062a\u0642\u0644\u0627\u0644\u064a\u0629 \u0627\u0644\u062e\u0637\u064a\u0629\u060c \u0627\u0644\u062a\u0639\u0627\u0645\u062f \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637\u0627\u062a<\/a><br \/>\n<a href=\"#El-Teorema-de-Pitagoras-y-la-Proyecci\u00f3n-sobre-un-Subespacio\">\u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633 \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637 \u0639\u0644\u0649 \u0641\u0636\u0627\u0621 \u062c\u0632\u0626\u064a<\/a><br \/>\n<a href=\"#El-Producto-Escalar-y-Vectorial-en-R3\">\u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0646\u0642\u0637\u064a \u0648\u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/span><\/a>\n<\/p>\n<p><a name=\"Independencia-Lineal-Ortogonalidad-y-Proyecciones\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/vtNHkaHD3aA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h2>\u0627\u0644\u0627\u0633\u062a\u0642\u0644\u0627\u0644\u064a\u0629 \u0627\u0644\u062e\u0637\u064a\u0629\u060c \u0627\u0644\u062a\u0639\u0627\u0645\u062f \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637\u0627\u062a<\/h2>\n<h3>\u0627\u0644\u0645\u0632\u062c \u0627\u0644\u062e\u0637\u064a \u0648\u0627\u0644\u0627\u0633\u062a\u0642\u0644\u0627\u0644\u064a\u0629 \u0627\u0644\u062e\u0637\u064a\u0629<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=138s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0645\u062a\u062c\u0647 \u063a\u064a\u0631 \u0635\u0641\u0631\u064a<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span><\/span> \u064a\u0645\u0643\u0646 \u0628\u0646\u0627\u0624\u0647 \u0643\u0640 <strong>\u0645\u0632\u062c \u062e\u0637\u064a<\/strong> \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0645\u062a\u062c\u0647\u0627\u062a \u063a\u064a\u0631 \u0635\u0641\u0631\u064a\u0629 \u0623\u062e\u0631\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u0625\u0630\u0627 \u0648\u064f\u062c\u062f \u0632\u0648\u062c \u0645\u0646 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u062d\u0642\u064a\u0642\u064a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span>\u060c \u0644\u064a\u0633 \u0643\u0644\u0627\u0647\u0645\u0627 \u0635\u0641\u0631\u064a\u0627\u064b \u0641\u064a \u0622\u0646 \u0648\u0627\u062d\u062f\u060c \u0628\u062d\u064a\u062b:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z} = \\alpha \\vec{x} + \\beta\\vec{y}<\/span>\n<p>\u0623\u064a \u0623\u0646 \u0627\u0644\u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span><\/span> \u064a\u0645\u0643\u0646 \u0628\u0646\u0627\u0624\u0647 \u0643\u0645\u062c\u0645\u0648\u0639 \u0645\u0648\u0632\u0648\u0646 \u0644\u0644\u0645\u062a\u062c\u0647\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}.<\/span><\/span><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=609s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0648\u0628\u0627\u0644\u0645\u062b\u0644 \u064a\u0642\u0627\u0644<\/span><\/strong><\/a> \u0625\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u0647\u0645\u0627 <strong>\u0645\u0633\u062a\u0642\u0644\u0627\u0646 \u062e\u0637\u064a\u0627\u064b<\/strong> \u0625\u0630\u0627 <\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vec{x} + \\beta\\vec{y} = \\vec{0} ) \\longleftrightarrow (\\alpha=0 \\wedge \\beta=0 )<\/span>\n<p>\u0625\u0646 \u0627\u0644\u0627\u0633\u062a\u0642\u0644\u0627\u0644\u064a\u0629 \u0627\u0644\u062e\u0637\u064a\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u062a\u0639\u0646\u064a \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u0644\u0627 \u064a\u0645\u0643\u0646 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u064a\u0647 \u0643\u0645\u0636\u0627\u0639\u0641 \u0639\u062f\u062f\u064a (\u063a\u064a\u0631 \u0635\u0641\u0631\u064a) \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648\u0644\u0627 \u0627\u0644\u0639\u0643\u0633.<\/p>\n<p>\u064a\u0645\u0643\u0646 \u062a\u0648\u0633\u064a\u0639 \u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u0627\u0633\u062a\u0642\u0644\u0627\u0644\u064a\u0629 \u0627\u0644\u062e\u0637\u064a\u0629 \u0627\u0644\u0630\u064a \u0631\u0627\u062c\u0639\u0646\u0627\u0647 \u0644\u0644\u062a\u0648 \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0627\u062a \u0623\u0643\u0628\u0631 \u0645\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a. \u0641\u064a\u0642\u0627\u0644 \u0625\u0646 \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u063a\u064a\u0631 \u0627\u0644\u0635\u0641\u0631\u064a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x}_1, \\cdots, \\vec{x}_n\\}<\/span><\/span> \u0645\u0633\u062a\u0642\u0644\u0629 \u062e\u0637\u064a\u0627\u064b \u0639\u0646\u062f\u0645\u0627<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\displaystyle \\left[\\left(\\sum_{i=1}^n \\alpha_i \\vec{x}_i \\right) = \\vec{0} \\right] \\longleftrightarrow \\left[\\bigwedge_{i=1}^n (\\alpha_i = 0) \\right]<\/span>\n<h3>\u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u0645\u062a\u0634\u0643\u0644\u0629 \u0628\u064a\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0648\u0627\u0644\u062a\u0639\u0627\u0645\u062f<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=1289s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0625\u0630\u0627 \u062a\u0630\u0643\u0631\u0646\u0627 \u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0643\u0648\u0634\u064a-\u0634\u0641\u0627\u0631\u062a\u0633\u060c<\/span><\/strong><\/a> \u0641\u0625\u0646\u0647\u0627 \u062a\u062e\u0628\u0631\u0646\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\vec{x},\\vec{y}\\in\\mathbb{R}^n)(|\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\| \\|\\vec{y}\\|).<\/span><\/span> \u0648\u0628\u0623\u062e\u0630 \u0647\u0630\u0627 \u0628\u0639\u064a\u0646 \u0627\u0644\u0627\u0639\u062a\u0628\u0627\u0631\u060c \u0645\u0646 \u0627\u0644\u0633\u0647\u0644 \u0627\u0644\u062a\u0623\u0643\u062f \u0645\u0646 \u0623\u0646\u0647 \u0644\u0623\u064a \u0632\u0648\u062c \u0645\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span><\/span> \u062a\u062a\u062d\u0642\u0642 \u0627\u0644\u0639\u0644\u0627\u0642\u0629:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle -1 \\leq \\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{x}\\|\\|\\vec{y}\\|}\\leq 1<\/span>\n<p>\u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0622\u0646 \u0627\u0633\u062a\u0646\u062a\u0627\u062c \u0648\u062c\u0648\u062f \u0639\u0644\u0627\u0642\u0629 \u0628\u064a\u0646 \u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0646\u0642\u0637\u064a \u0648\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 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span>\u060c \u0644\u0623\u0646\u0647\u0645\u0627 \u064a\u0648\u0644\u062f\u0627\u0646 \u0645\u0633\u062a\u0648\u0649 \u0645\u062a\u0637\u0627\u0628\u0642 \u0627\u0644\u0634\u0643\u0644 \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span><\/span>. \u0648\u0644\u0647\u0630\u0627\u060c \u0648\u0645\u0646 \u062f\u0648\u0646 \u0641\u0642\u062f\u0627\u0646 \u0627\u0644\u0639\u0645\u0648\u0645\u064a\u0629\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u062e\u064a\u0644\u0647\u0645\u0627 \u0643\u0639\u0646\u0627\u0635\u0631 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span><\/span> \u0645\u0639 \u0632\u0648\u0627\u064a\u0627 \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0644\u0645\u062d\u0648\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/span> \u0642\u062f\u0631\u0647\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_y,<\/span><\/span> \u0639\u0644\u0649 \u0627\u0644\u062a\u0648\u0627\u0644\u064a\u060c \u0628\u062d\u064a\u062b \u062a\u064f\u0643\u062a\u0628 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0628\u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0642\u0637\u0628\u064a \u0643\u0645\u0627 \u064a\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\vec{x} &amp;= \\|\\vec{x}\\|(\\cos(\\theta_x) , \\sin(\\theta_x)) \\\\ \\\\ \\vec{y} &amp;= \\|\\vec{y}\\|(\\cos(\\theta_y) , \\sin(\\theta_y))\n\n\\end{array}<\/span>\n<p>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a \u064a\u0645\u0643\u0646\u0646\u0627 \u0623\u0646 \u0646\u0641\u062a\u0631\u0636 (\u062f\u0648\u0646 \u0641\u0642\u062f\u0627\u0646 \u0627\u0644\u0639\u0645\u0648\u0645\u064a\u0629\u060c \u0645\u0631\u0629 \u0623\u062e\u0631\u0649) \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x \\lt \\theta_y,<\/span><\/span> \u0644\u0646\u062d\u0633\u0628 \u0628\u0639\u062f \u0630\u0644\u0643 \u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0646\u0642\u0637\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}.<\/span><\/span> \u0639\u0646\u062f \u0627\u0644\u0642\u064a\u0627\u0645 \u0628\u0630\u0644\u0643 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\\vec{x}\\cdot \\vec{y} &amp;=  \\|\\vec{x}\\|  \\|\\vec{y}\\| (\\cos(\\theta_x)\\cos(\\theta_y) + \\sin(\\theta_x)\\sin(\\theta_y)) \\\\ \\\\ &amp;=  \\|\\vec{x}\\|  \\|\\vec{y}\\| \\cos(\\theta_y-\\theta_x)\n\n\\end{array}<\/span>\n<p>\u0627\u0644\u0622\u0646\u060c \u0648\u0628\u0623\u062e\u0630 \u0627\u0644\u0641\u0631\u0642 \u0628\u064a\u0646 \u0627\u0644\u0645\u0648\u0636\u0639 \u0627\u0644\u0632\u0627\u0648\u064a \u0627\u0644\u0623\u0643\u0628\u0631 \u0648\u0627\u0644\u0623\u0635\u063a\u0631 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 \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\u060c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})=\\theta_y - \\theta_x.<\/span><\/span> \u0648\u0645\u0646 \u0647\u0646\u0627 \u064a\u0645\u0643\u0646\u0646\u0627 \u0643\u062a\u0627\u0628\u0629:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\cos\\left(\\angle(\\vec{x},\\vec{y}) \\right) = \\frac{\\vec{x} \\cdot \\vec{y}}{\\|\\vec{x}\\|\\|\\vec{y}\\|}\n\n<\/span>\n<p>\u0647\u0646\u0627 \u064a\u062c\u0628 \u0623\u0646 \u0646\u0624\u0643\u062f \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in [0, \\pi]<\/span><\/span><\/p>\n<p>\u0627\u0646\u0637\u0644\u0627\u0642\u0627\u064b \u0645\u0646 \u0647\u0630\u0627 \u064a\u0645\u0643\u0646\u0646\u0627 \u0631\u0628\u0637 \u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0643\u0648\u0634\u064a-\u0634\u0641\u0627\u0631\u062a\u0633 \u0645\u0639 \u0647\u0646\u062f\u0633\u0629 \u0627\u0644\u0632\u0648\u0627\u064a\u0627\u060c \u0643\u0645\u0627 \u064a\u0633\u0645\u062d \u0644\u0646\u0627 \u0630\u0644\u0643 \u0628\u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0645\u0641\u0647\u0648\u0645 \u0635\u0627\u0631\u0645 \u0644\u0644\u062a\u0639\u0627\u0645\u062f. \u064a\u0642\u0627\u0644 \u0639\u0646 \u0645\u062a\u062c\u0647\u064a\u0646 \u0623\u0646\u0647\u0645\u0627 <strong>\u0645\u062a\u0639\u0627\u0645\u062f\u0627\u0646<\/strong> \u0639\u0646\u062f\u0645\u0627 \u064a\u0634\u0643\u0644\u0627\u0646 \u0641\u064a\u0645\u0627 \u0628\u064a\u0646\u0647\u0645\u0627 \u0632\u0627\u0648\u064a\u0629 \u0645\u0642\u062f\u0627\u0631\u0647\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\pi\/2<\/span><\/span> \u0631\u0627\u062f\u064a\u0627\u0646\u060c \u0628\u0627\u0644\u0645\u0639\u0646\u0649 \u0627\u0644\u0645\u0648\u0636\u062d \u0641\u064a \u0627\u0644\u0641\u0642\u0631\u0629 \u0627\u0644\u0633\u0627\u0628\u0642\u0629. \u0648\u0647\u0630\u0627 \u0645\u0643\u0627\u0641\u0626 \u0644\u0644\u0642\u0648\u0644 \u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\cos\\left(\\angle(\\vec{x},\\vec{y})\\right) = 0,<\/span><\/span> \u0648\u0647\u0648 \u0628\u062f\u0648\u0631\u0647 \u0645\u0643\u0627\u0641\u0626 \u0644\u0644\u0642\u0648\u0644 \u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = 0.<\/span><\/span> \u0648\u0644\u0647\u0630\u0627 \u0627\u0644\u0633\u0628\u0628 \u0641\u0625\u0646 \u062a\u0623\u0643\u064a\u062f \u062a\u0639\u0627\u0645\u062f \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u064a\u0639\u0627\u062f\u0644 \u0627\u0644\u0642\u0648\u0644 \u0628\u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0.<\/span><\/span><\/p>\n<h4>\u0625\u0630\u0627 \u0643\u0627\u0646 \u0645\u062a\u062c\u0647\u0627\u0646 \u063a\u064a\u0631 \u0635\u0641\u0631\u064a\u064a\u0646 \u0645\u062a\u0639\u0627\u0645\u062f\u064a\u0646\u060c \u0641\u0625\u0646\u0647\u0645\u0627 \u0645\u0633\u062a\u0642\u0644\u0627\u0646 \u062e\u0637\u064a\u0627\u064b<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=2365s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0647\u0630\u0647 \u062e\u0627\u0635\u064a\u0629 \u064a\u0645\u0643\u0646 \u0627\u0639\u062a\u0628\u0627\u0631\u0647\u0627 \u0628\u062f\u064a\u0647\u064a\u0629 \u0646\u0648\u0639\u0627\u064b \u0645\u0627 \u0644\u0644\u0645\u062a\u062c\u0647\u0627\u062a<\/span><\/strong><\/a> \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>\u060c \u0625\u0644\u0627 \u0623\u0646 \u0628\u0631\u0647\u0627\u0646\u0647\u0627 \u0627\u0644\u0631\u0633\u0645\u064a \u0644\u064a\u0633 \u0645\u0628\u0627\u0634\u0631\u0627\u064b\u060c \u0648\u0647\u064a \u0623\u064a\u0636\u0627\u064b \u062e\u0627\u0635\u064a\u0629 \u0642\u062f \u062a\u0633\u0628\u0628 \u0623\u062d\u064a\u0627\u0646\u0627\u064b \u0628\u0639\u0636 \u0627\u0644\u0627\u0644\u062a\u0628\u0627\u0633: \u0625\u0646 \u062a\u0639\u0627\u0645\u062f \u0645\u062a\u062c\u0647\u064a\u0646 \u064a\u0639\u0646\u064a \u0627\u0633\u062a\u0642\u0644\u0627\u0644\u0647\u0645\u0627 \u0627\u0644\u062e\u0637\u064a\u060c \u0648\u0644\u0643\u0646 \u0627\u0633\u062a\u0642\u0644\u0627\u0644\u0647\u0645\u0627 \u0627\u0644\u062e\u0637\u064a \u0644\u0627 \u064a\u0639\u0646\u064a \u0628\u0627\u0644\u0636\u0631\u0648\u0631\u0629 \u0623\u0646\u0647\u0645\u0627 \u0645\u062a\u0639\u0627\u0645\u062f\u0627\u0646. \u0648\u0644\u0631\u0624\u064a\u0629 \u0647\u0630\u0627 \u064a\u0643\u0641\u064a \u0645\u062b\u0627\u0644 \u0645\u0639\u0627\u0643\u0633 \u0628\u0633\u064a\u0637:<\/p>\n<p>\u0625\u0630\u0627 \u0623\u062e\u0630\u0646\u0627 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}=(1,0)<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}=(1,1),<\/span><\/span> \u0627\u0644\u0644\u0630\u064a\u0646 \u0645\u0646 \u0627\u0644\u0648\u0627\u0636\u062d \u0623\u0646\u0647\u0645\u0627 \u063a\u064a\u0631 \u0645\u062a\u0639\u0627\u0645\u062f\u064a\u0646 \u0644\u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}\\cdot\\vec{B}=1,<\/span><\/span> \u0633\u0646\u0631\u0649 \u0623\u0646\u0647 \u0625\u0630\u0627 \u0642\u0645\u0646\u0627 \u0628\u0627\u0644\u0639\u0645\u0644\u064a\u0629<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\alpha\\vec{A} + \\beta\\vec{B} = \\vec{0}\n\n<\/span>\n<p>\u0625\u0630\u0646 \u0644\u062f\u064a\u0646\u0627<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\alpha + \\beta &amp;= 0 \\\\ \\beta &amp;= 0\n\n\\end{array}<\/span>\n<p>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = 0  \\wedge \\beta=0.<\/span><\/span> \u0648\u0628\u0647\u0630\u0627 \u0646\u0633\u062a\u0646\u062a\u062c \u0623\u0646:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\alpha\\vec{A} + \\beta\\vec{B} = \\vec{0} \\longleftrightarrow  \\alpha = 0  \\wedge \\beta=0\n\n<\/span>\n<p>\u0648\u0647\u0630\u0627 \u0645\u0643\u0627\u0641\u0626 \u0644\u0644\u0642\u0648\u0644 \u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}<\/span><\/span> \u0645\u0633\u062a\u0642\u0644\u0627\u0646 \u062e\u0637\u064a\u0627\u064b. \u0648\u0645\u0646 \u0647\u0646\u0627 \u064a\u062a\u0636\u062d \u0628\u0634\u0643\u0644 \u0635\u0631\u064a\u062d \u062c\u062f\u0627\u064b \u0623\u0646 \u0627\u0644\u0627\u0633\u062a\u0642\u0644\u0627\u0644\u064a\u0629 \u0627\u0644\u062e\u0637\u064a\u0629 \u0644\u0627 \u062a\u0639\u0646\u064a \u0628\u0627\u0644\u0636\u0631\u0648\u0631\u0629 \u0627\u0644\u062a\u0639\u0627\u0645\u062f. \u0648\u0645\u0639 \u0630\u0644\u0643 \u0641\u0625\u0646 \u0627\u0644\u062a\u0639\u0627\u0645\u062f \u064a\u0639\u0646\u064a \u0627\u0644\u0627\u0633\u062a\u0642\u0644\u0627\u0644\u064a\u0629 \u0627\u0644\u062e\u0637\u064a\u0629\u060c \u0648\u0647\u0648 \u0645\u0627 \u0633\u0623\u0628\u0631\u0647\u0646\u0647 \u0631\u0633\u0645\u064a\u0627\u064b \u0641\u064a\u0645\u0627 \u064a\u0644\u064a\u060c \u0648\u0644\u0623\u062c\u0644 \u0630\u0644\u0643 \u0644\u0646\u0646\u0638\u0631 \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u0641\u0631\u0636\u064a\u0627\u062a \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\mathcal{H}= \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\cdot\\vec{y}=0, \\alpha\\vec{x}+\\beta\\vec{y} = \\vec{0}\\}<\/span>\n<p>\u0627\u0646\u0637\u0644\u0627\u0642\u0627\u064b \u0645\u0646 \u0647\u0630\u0627 \u064a\u0645\u0643\u0646\u0646\u0627 \u0628\u0646\u0627\u0621 \u0627\u0644\u0627\u0633\u062a\u062f\u0644\u0627\u0644 \u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp;\\mathcal{H}\\vdash \\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\} &amp;{;\\;\u0627\u0641\u062a\u0631\u0627\u0636}\\\\ \\\\\n\n(2) &amp;\\mathcal{H}\\vdash \\vec{x}\\cdot\\vec{y}=0 &amp;{\\;\u0627\u0641\u062a\u0631\u0627\u0636} \\\\ \\\\\n\n(3) &amp;\\mathcal{H}\\vdash \\alpha\\vec{x} + \\beta\\vec{y} = \\vec{0} &amp;{\\;\u0627\u0641\u062a\u0631\u0627\u0636} \\\\ \\\\\n\n(4) &amp;\\mathcal{H}\\vdash (\\alpha\\vec{x} + \\beta\\vec{y})\\cdot\\vec{x} = \\alpha\\|\\vec{x}\\|^2 + \\beta(\\vec{x}\\cdot\\vec{y}) &amp;{;\\; \u062b\u0646\u0627\u0626\u064a\u0629 \u0627\u0644\u062e\u0637\u064a\u0629} \\\\ \\\\\n\n(5) &amp;\\mathcal{H}\\vdash  \\alpha\\|\\vec{x}\\|^2 = 0 &amp; {;\\; \u0645\u0646 (2,3,4)} \\\\ \\\\\n\n(6) &amp;\\mathcal{H}\\vdash  \\alpha  = 0 &amp; {;\\; \u0645\u0646 (1,5)} \\\\ \\\\\n\n(7) &amp;\\mathcal{H}\\vdash (\\alpha\\vec{x} + \\beta\\vec{y})\\cdot\\vec{y} = \\alpha(\\vec{x}\\cdot\\vec{y}) + \\beta\\|\\vec{y}\\|^2 &amp; {;\\;\u062b\u0646\u0627\u0626\u064a\u0629 \u0627\u0644\u062e\u0637\u064a\u0629} \\\\ \\\\\n\n(8) &amp;\\mathcal{H}\\vdash \\beta\\|\\vec{y}\\|^2 = 0 &amp;{;\\;\u0645\u0646 (2,3,7)} \\\\ \\\\\n\n(9) &amp;\\mathcal{H}\\vdash \\beta = 0 &amp;{;\\;\u0645\u0646 (1,8)} \\\\ \\\\\n\n(10) &amp;\\mathcal{H}\\vdash \\alpha= 0 \\wedge \\beta = 0 &amp;{;\\;\\wedge-int(6,9)}\n\n\\end{array}<\/span>\n<p>\u0648\u0628\u0630\u0644\u0643 \u0646\u0633\u062a\u0646\u062a\u062c \u0623\u0646<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\cdot\\vec{y}=0, \\alpha\\vec{x}+\\beta\\vec{y} = \\vec{0}\\} \\vdash \\alpha= 0 \\wedge \\beta = 0\n\n<\/span>\n<p>\u0648\u0623\u062e\u064a\u0631\u0627\u064b\u060c \u0628\u062a\u0637\u0628\u064a\u0642 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u0627\u0633\u062a\u0646\u062a\u0627\u062c \u0639\u0644\u0649 \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u0627\u063a\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\cdot\\vec{y}=0\\} \\vdash (\\alpha\\vec{x}+\\beta\\vec{y} = \\vec{0}) \\rightarrow (\\alpha= 0 \\wedge \\beta = 0)<\/span>\n<p>\u0627\u0644\u0628\u0631\u0647\u0627\u0646 \u0627\u0644\u0630\u064a \u064a\u0639\u0637\u064a \u0627\u0644\u0633\u0647\u0645 \u0641\u064a \u0627\u0644\u0627\u062a\u062c\u0627\u0647 \u0627\u0644\u0645\u0639\u0627\u0643\u0633 \u0628\u062f\u064a\u0647\u064a.<\/p>\n<p>\u0623\u064a \u0623\u0646\u0647 \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u0645\u062a\u062c\u0647\u064a\u0646 \u063a\u064a\u0631 \u0635\u0641\u0631\u064a\u064a\u0646 \u0648\u0645\u062a\u0639\u0627\u0645\u062f\u064a\u0646\u060c \u0641\u0625\u0646\u0647\u0645\u0627 \u0645\u0633\u062a\u0642\u0644\u0627\u0646 \u062e\u0637\u064a\u0627\u064b.<\/p>\n<h3>\u0625\u0633\u0642\u0627\u0637 \u0645\u062a\u062c\u0647 \u0639\u0644\u0649 \u0622\u062e\u0631<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=3055s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u0646\u0641\u062a\u0631\u0636 \u0623\u0646 \u0644\u062f\u064a\u0646\u0627 \u0645\u062a\u062c\u0647\u064a\u0646 \u063a\u064a\u0631 \u0635\u0641\u0631\u064a\u064a\u0646<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u064a\u0634\u0643\u0644\u0627\u0646 \u0641\u064a\u0645\u0627 \u0628\u064a\u0646\u0647\u0645\u0627 \u0632\u0627\u0648\u064a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})<\/span><\/span> \u0648\u0646\u062a\u0633\u0627\u0621\u0644: \u00ab\u0628\u0623\u064a \u0645\u0642\u062f\u0627\u0631 \u064a\u0648\u062c\u062f \u0627\u0644\u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0639\u0644\u0649 \u0627\u0644\u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span>\u061f\u00bb \u0623\u0648 \u00ab\u0645\u0627 \u062d\u062c\u0645 \u0638\u0644 \u0627\u0644\u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0639\u0646\u062f \u0625\u0633\u0642\u0627\u0637\u0647 \u0639\u0644\u0649 \u0627\u062a\u062c\u0627\u0647 \u0627\u0644\u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span>\u061f\u00bb. \u064a\u0645\u0643\u0646\u0646\u0627 \u062d\u0644 \u0647\u0630\u0627 \u0627\u0644\u0633\u0624\u0627\u0644 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0639\u0644\u0645 \u0627\u0644\u0645\u062b\u0644\u062b\u0627\u062a\u060c \u0648\u0628\u0630\u0644\u0643 \u0646\u0639\u0631\u0651\u0641 \u0625\u0633\u0642\u0627\u0637 \u0627\u0644\u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0639\u0644\u0649 \u0645\u062a\u062c\u0647 \u0622\u062e\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y},<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Proy_{\\vec{y}}(\\vec{x}),<\/span><\/span> \u0645\u0646 \u062e\u0644\u0627\u0644 \u0627\u0644\u062a\u0639\u0628\u064a\u0631:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Proy_{\\vec{y}}(\\vec{x}) = \\| \\vec{x}\\| \\cos(\\angle(\\vec{x},\\vec{y})) \\hat{y}<\/span>\n<p>\u0648\u0625\u0630\u0627 \u062c\u0645\u0639\u0646\u0627 \u0647\u0630\u0627 \u0645\u0639 \u0645\u0627 \u0631\u0623\u064a\u0646\u0627\u0647 \u0641\u064a \u0627\u0644\u0641\u0642\u0631\u0627\u062a \u0627\u0644\u0633\u0627\u0628\u0642\u0629 \u064a\u0645\u0643\u0646\u0646\u0627 \u0643\u062a\u0627\u0628\u0629:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle Proy_{\\vec{y}}(\\vec{x}) = {\\| \\vec{x}\\|} \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{{\\|\\vec{x}\\|} \\|\\vec{y}\\|}\\right)\\color{red}{\\hat{y}} =  \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{y}\\|} \\right)\\color{red}{\\frac{\\vec{y}}{\\|\\vec{y}\\|}} = \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{y}\\|^2}\\right)\\vec{y} = \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\vec{y}\\cdot\\vec{y}}\\right)\\vec{y}<\/span>\n<p>\u0644\u0623\u0646\u0646\u0627 \u0646\u062a\u0630\u0643\u0631<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\cos(\\angle(\\vec{x},\\vec{y}))  = \\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{x}\\| \\|\\vec{y}\\|}<\/span>\n<p>\u062a\u064f\u0639\u062a\u0628\u0631 \u0627\u0644\u0625\u0633\u0642\u0627\u0637\u0627\u062a \u0645\u0647\u0645\u0629 \u0644\u0623\u0646\u0647\u0627 \u062a\u0633\u0645\u062d \u0644\u0646\u0627 \u0628\u062a\u0645\u062b\u064a\u0644 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0628\u062f\u0644\u0627\u0644\u0629 \u0623\u064a \u0642\u0627\u0639\u062f\u0629 \u0639\u0644\u0649 \u0623\u0646\u0647\u0627 \u0645\u062c\u0645\u0648\u0639 \u0625\u0633\u0642\u0627\u0637\u0627\u062a\u0647\u0627:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} = \\displaystyle \\sum_{i=1}^n \\alpha_i \\hat{u}_i<\/span>\n<p>\u062d\u064a\u062b \u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{u}_i\\}_{i=1,\\cdots, n}<\/span><\/span> \u0647\u064a \u0642\u0627\u0639\u062f\u0629 \u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0645\u0633\u062a\u0642\u0644\u0629 \u062e\u0637\u064a\u0627\u064b \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>\u060c \u0648\u0627\u0644\u0645\u0639\u0627\u0645\u0644\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_i = (\\vec{x}\\cdot\\vec{u}_i)\/\\|\\vec{u}_i\\|<\/span><\/span> \u0647\u064a \u0628\u0627\u0644\u0636\u0628\u0637 \u0627\u0644\u0625\u0633\u0642\u0627\u0637\u0627\u062a \u0639\u0644\u0649 \u0643\u0644 \u0639\u0646\u0635\u0631 \u0645\u0646 \u0639\u0646\u0627\u0635\u0631 \u0627\u0644\u0642\u0627\u0639\u062f\u0629 \u0648\u062a\u0634\u0643\u0644 \u0625\u062d\u062f\u0627\u062b\u064a\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0644\u0642\u0627\u0639\u062f\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{u}_i\\}_{i=1,\\cdots, n}<\/span><\/span> \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span><\/p>\n<p><a name=\"El-Teorema-de-Pitagoras-y-la-Proyecci\u00f3n-sobre-un-Subespacio\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/CGrr6IDnvjs\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h2>\u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633 \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637 \u0639\u0644\u0649 \u0641\u0636\u0627\u0621 \u062c\u0632\u0626\u064a<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=254s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633 \u0647\u064a \u0646\u062a\u064a\u062c\u0629<\/span><\/strong><\/a> \u0645\u0639\u0631\u0648\u0641\u0629 \u0644\u062f\u0649 \u0627\u0644\u062c\u0645\u064a\u0639 \u0648\u0644\u0647\u0627 \u0639\u062f\u062f \u0644\u0627 \u064a\u062d\u0635\u0649 \u0645\u0646 \u0627\u0644\u0628\u0631\u0627\u0647\u064a\u0646. \u0623\u062d\u062f \u0627\u0644\u0628\u0631\u0627\u0647\u064a\u0646 \u0627\u0644\u0645\u0645\u0643\u0646\u0629 \u0644\u0647\u0630\u0647 \u0627\u0644\u0646\u0638\u0631\u064a\u0629 \u064a\u0638\u0647\u0631 \u0628\u0627\u0644\u0636\u0628\u0637 \u0645\u0646 \u0627\u0644\u0645\u0648\u0636\u0648\u0639\u0627\u062a \u0627\u0644\u062a\u064a \u0637\u0648\u0631\u0646\u0627\u0647\u0627 \u0644\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u0645\u0639 \u0645\u064a\u0632\u0629 \u0625\u0636\u0627\u0641\u064a\u0629 \u0647\u064a \u0635\u0644\u0627\u062d\u064a\u062a\u0647 \u0644\u0623\u064a \u0639\u062f\u062f \u0645\u0646 \u0627\u0644\u0623\u0628\u0639\u0627\u062f.<\/p>\n<h3>\u0628\u0631\u0647\u0646\u0629 \u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=533s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0625\u0630\u0627 \u0643\u0627\u0646 \u0644\u062f\u064a\u0646\u0627 \u0645\u062b\u0644\u062b \u0642\u0627\u0626\u0645 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u0623\u0636\u0644\u0627\u0639<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b,<\/span><\/span> \u0648\u0627\u0644\u0648\u062a\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c,<\/span><\/span> \u0641\u0625\u0646 \u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633 \u062a\u062e\u0628\u0631\u0646\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a^2+b^2=c^2.<\/span><\/span>\u0648\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643 \u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u0645\u062b\u064a\u0644 \u0643\u0644 \u0636\u0644\u0639 \u0628\u0648\u0627\u0633\u0637\u0629 \u0632\u0648\u062c \u0645\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0627\u0644\u0645\u062a\u0639\u0627\u0645\u062f\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> \u0648\u0643\u062a\u0627\u0628\u0629 \u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633 \u0628\u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}\\} \\vdash\n\n \\vec{x}\\bot\\vec{y} \\leftrightarrow (\\|\\vec{x} + \\vec{y}\\|^2 = \\|\\vec{x}\\|^2 + \\|\\vec{y}\\|^2)<\/span>\n<p>\u062d\u064a\u062b \u0625\u0646 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\bot\\vec{y}<\/span><\/span> \u064a\u0634\u064a\u0631 \u0625\u0644\u0649 \u0623\u0646 \u0643\u0644\u0627 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 \u0645\u062a\u0639\u0627\u0645\u062f\u0627\u0646\u060c \u0623\u064a \u063a\u064a\u0631 \u0635\u0641\u0631\u064a\u064a\u0646 \u0648\u0645\u062b\u0644 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0.<\/span><\/span> \u0648\u0628\u0647\u0630\u0627 \u064a\u062a\u0645 \u062a\u0623\u0633\u064a\u0633 \u0639\u0644\u0627\u0642\u0629 \u062b\u0646\u0627\u0626\u064a\u0629 \u0627\u0644\u0634\u0631\u0637 \u0628\u064a\u0646 \u0627\u0644\u062a\u0639\u0627\u0645\u062f \u0648\u062c\u0645\u0639 \u0645\u0631\u0628\u0639\u0627\u062a \u0627\u0644\u0642\u064a\u0645 \u0627\u0644\u0639\u062f\u062f\u064a\u0629 \u0644\u0645\u062a\u062c\u0647\u064a\u0646.<\/p>\n<p>\u064a\u0645\u0643\u0646 \u0628\u0631\u0647\u0646\u0629 \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u0627\u063a\u0629 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0629 \u0644\u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633 \u0645\u0646 \u062e\u0644\u0627\u0644 \u0627\u0644\u0627\u0633\u062a\u062f\u0644\u0627\u0644\u064a\u0646 \u0627\u0644\u062a\u0627\u0644\u064a\u064a\u0646:<\/p>\n<p><strong>\u0623\u0648\u0644\u0627\u064b \u0641\u064a \u0627\u062a\u062c\u0627\u0647 \u0627\u0644\u0630\u0647\u0627\u0628:<\/strong><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\bot\\vec{y}\\} \\vdash \\vec{x}\\bot\\vec{y} &amp; {;\\;\u0627\u0641\u062a\u0631\u0627\u0636} \\\\ \\\\\n\n(2) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\bot\\vec{y}\\} \\vdash \\vec{x}\\cdot\\vec{y}= 0 &amp; {;\\;\u0645\u0646 (1)} \\\\ \\\\\n\n(3) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\bot\\vec{y}\\} \\vdash \\|\\vec{x} + \\vec{y}\\|^2 = (\\vec{x} + \\vec{y})\\cdot(\\vec{x} + \\vec{y}) = \\|\\vec{x}\\|^2 + 2(\\vec{x}\\cdot\\vec{y}) + \\|\\vec{y}\\|^2 &amp; \\\\\n\n&amp;;\\; \u062e\u0627\u0635\u064a\u0629\\;\u0627\u0644\u0645\u0639\u064a\u0627\u0631\\;\u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\\;\u0648\u0627\u0644\u062d\u0627\u0635\u0644\\;\u0627\u0644\u0646\u0642\u0637\u064a &amp; \\\\ \\\\\n\n(4) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\bot\\vec{y}\\} \\vdash \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2 &amp; {;\\;\u0645\u0646 (2,3)} \\\\ \\\\\n\n(5) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}\\} \\vdash \\vec{x}\\bot\\vec{y} \\rightarrow ( \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2) &amp; {;\\;TD(4)} \\end{array}<\/span>\n<p><strong>\u0648\u0627\u0644\u0622\u0646 \u0641\u064a \u0627\u0644\u0627\u062a\u062c\u0627\u0647 \u0627\u0644\u0639\u0643\u0633\u064a:<\/strong><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2\\} \\vdash \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2 &amp; {;\\;\u0627\u0641\u062a\u0631\u0627\u0636} \\\\ \\\\\n\n(2) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2\\} \\vdash \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2 +2(\\vec{x}\\cdot\\vec{y}) + \\|\\vec{y}\\|^2 &amp;  \\\\\n\n&amp;;\\; \u062e\u0627\u0635\u064a\u0629\\;\u0627\u0644\u0645\u0639\u064a\u0627\u0631\\;\u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\\;\u0648\u0627\u0644\u062d\u0627\u0635\u0644\\;\u0627\u0644\u0646\u0642\u0637\u064a &amp;\\\\ \\\\\n\n(3) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2\\} \\vdash  \\vec{x}\\cdot\\vec{y}=0 &amp; {;\\;\u0645\u0646 (1,2)} \\\\ \\\\\n\n(4) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2\\} \\vdash  \\vec{x}\\bot\\vec{y} &amp; {;\\;\u0645\u0646 (3)} \\\\ \\\\\n\n(5) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}\\} \\vdash (\\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2) \\rightarrow  \\vec{x}\\bot\\vec{y} &amp; {;\\;TD(4)} \\end{array}<\/span>\n<p><strong>\u0648\u0623\u062e\u064a\u0631\u0627\u064b\u060c \u0628\u0636\u0645 \u0627\u0644\u0627\u0633\u062a\u062f\u0644\u0627\u0644\u064a\u0646 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 \u0645\u0627 \u0643\u0646\u0627 \u0646\u0631\u064a\u062f \u0628\u0631\u0647\u0646\u062a\u0647:<\/strong><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}\\} \\vdash   \\vec{x}\\bot\\vec{y} \\leftrightarrow (\\|\\vec{x} + \\vec{y}\\|^2 = \\|\\vec{x}\\|^2 + \\|\\vec{y}\\|^2)<\/span>\n<h3>\u0625\u0633\u0642\u0627\u0637 \u0645\u062a\u062c\u0647 \u0639\u0644\u0649 \u0641\u0636\u0627\u0621 \u062c\u0632\u0626\u064a \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=1545s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u0646\u0639\u062a\u0628\u0631 \u0641\u0636\u0627\u0621\u064b \u062c\u0632\u0626\u064a\u0627\u064b<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u064a\u062a\u0643\u0648\u0646 \u0645\u0646 \u0642\u0627\u0639\u062f\u0629 \u0645\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0627\u0644\u0648\u062d\u062f\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}.<\/span><\/span> \u0625\u0630\u0627 \u0623\u062e\u0630\u0646\u0627 \u0645\u062a\u062c\u0647\u0627\u064b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\},<\/span><\/span> \u0641\u0625\u0646 \u0625\u0633\u0642\u0627\u0637 \u0627\u0644\u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0639\u0644\u0649 \u0627\u0644\u0641\u0636\u0627\u0621 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u064a\u064f\u0639\u0631\u0651\u0641 \u0628\u0648\u0627\u0633\u0637\u0629 \u0627\u0644\u062a\u0639\u0628\u064a\u0631:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Proy_{H}(\\vec{x}) = \\displaystyle \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j<\/span>\n<p>\u0625\u0646 \u0643\u0648\u0646 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0627 \u0645\u062a\u0639\u0627\u0645\u062f\u0629-\u0645\u0642\u0646\u0646\u0629 \u064a\u0639\u0646\u064a \u0623\u0646 \u062c\u0645\u064a\u0639 \u0639\u0646\u0627\u0635\u0631\u0647\u0627 \u0645\u062a\u0639\u0627\u0645\u062f\u0629 \u0641\u064a\u0645\u0627 \u0628\u064a\u0646\u0647\u0627 \u0648\u0643\u0644 \u0645\u0646\u0647\u0627 \u0644\u0647 \u0645\u0639\u064a\u0627\u0631 \u064a\u0633\u0627\u0648\u064a \u0627\u0644\u0648\u0627\u062d\u062f.<\/p>\n<p>\u0623\u064a \u0623\u0646 \u0647\u0630\u0627\u060c \u0625\u0646 \u0635\u062d \u0627\u0644\u062a\u0639\u0628\u064a\u0631\u060c \u0647\u0648 \u0627\u0644\u0638\u0644 \u0627\u0644\u0630\u064a \u064a\u0644\u0642\u064a\u0647 \u0645\u062a\u062c\u0647 \u0639\u0644\u0649 \u0643\u0644 \u0645\u0646 \u0645\u0643\u0648\u0646\u0627\u062a \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u062c\u0632\u0626\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> <\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEga986LBrInk-B_9gUKPe01TF10dNECXU54KK1bSf3mAPakWE-FqdqyPbb0TVy88OfGxQmJRd-yW4dwAfcC21i2dM0KZqQjPe_Qx0M5OUz4f_P6IipJQ6PcxtkOmcO7-GqRiGZ-3StQpzy8FMIfPYE89Wae6JZIC2Jk9dSTPFTK1L4TsnpkcdpV1Dbr\" width=\"578\" height=\"591\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEga986LBrInk-B_9gUKPe01TF10dNECXU54KK1bSf3mAPakWE-FqdqyPbb0TVy88OfGxQmJRd-yW4dwAfcC21i2dM0KZqQjPe_Qx0M5OUz4f_P6IipJQ6PcxtkOmcO7-GqRiGZ-3StQpzy8FMIfPYE89Wae6JZIC2Jk9dSTPFTK1L4TsnpkcdpV1Dbr\" width=\"578\" height=\"591\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<h3>\u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0628\u064a\u0646 \u0646\u0642\u0637\u0629 \u0623\u0648 \u0645\u062a\u062c\u0647 \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0648\u0641\u0636\u0627\u0621 \u062c\u0632\u0626\u064a \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=1974s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0627\u0646\u0637\u0644\u0627\u0642\u0627\u064b \u0645\u0646 \u0625\u0633\u0642\u0627\u0637 \u0645\u062a\u062c\u0647<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span><\/span> \u0639\u0644\u0649 \u0641\u0636\u0627\u0621 \u062c\u0632\u0626\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u064a\u0645\u0643\u0646\u0646\u0627 \u0628\u0646\u0627\u0621 \u0645\u062a\u062c\u0647 \u0639\u0644\u0649 \u0627\u0644\u0646\u062d\u0648 \u0627\u0644\u062a\u0627\u0644\u064a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} - Proy_{H}(\\vec{x})<\/span>\n<p>\u0627\u0644\u0645\u062a\u062c\u0647 \u0627\u0644\u0645\u0634\u0643\u0644 \u0628\u0647\u0630\u0647 \u0627\u0644\u0637\u0631\u064a\u0642\u0629 \u0633\u064a\u0643\u0648\u0646 \u0645\u062a\u062c\u0647\u0627\u064b \u064a\u0635\u0644 \u0628\u064a\u0646 \u0646\u0642\u0637\u0629 \u0645\u0646 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u062c\u0632\u0626\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u0648\u0627\u0644\u0646\u0642\u0637\u0629 \u0630\u0627\u062a \u0627\u0644\u0625\u062d\u062f\u0627\u062b\u064a\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},<\/span><\/span> \u0628\u062d\u064a\u062b \u064a\u062e\u0631\u062c \u0639\u0645\u0648\u062f\u064a\u0627\u064b \u0639\u0644\u0649 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u062c\u0632\u0626\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H.<\/span><\/span> \u0648\u0647\u0630\u0627 \u0644\u064a\u0633 \u0635\u0639\u0628 \u0627\u0644\u0628\u0631\u0647\u0627\u0646\u060c \u0641\u0625\u0630\u0627 \u0623\u062e\u0630\u0646\u0627 \u0645\u062a\u062c\u0647\u0627\u064b \u0643\u064a\u0641\u064a\u0627\u064b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span><\/span> \u0648\u062d\u0633\u0628\u0646\u0627 \u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0646\u0642\u0637\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z},<\/span><\/span> \u064a\u0643\u0641\u064a \u0623\u0646 \u0646\u0631\u0649 \u0623\u0646 \u0646\u062a\u064a\u062c\u0629 \u0647\u0630\u0647 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0647\u064a \u0627\u0644\u0635\u0641\u0631. \u0644\u0646\u0642\u0645 \u0628\u0627\u0644\u062d\u0633\u0627\u0628 \u0644\u0644\u062a\u0623\u0643\u062f \u0645\u0646 \u0630\u0644\u0643:<\/p>\n<p>\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H,<\/span><\/span> \u0641\u0633\u064a\u0643\u0648\u0646 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}=\\displaystyle \\sum_{j=1}^k \\beta_j\\hat{v}_j<\/span>\n<p>\u062d\u064a\u062b \u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_j\\}_{j=1}^k<\/span><\/span> \u0647\u064a \u0642\u0627\u0639\u062f\u0629 \u0645\u062a\u0639\u0627\u0645\u062f\u0629-\u0645\u0642\u0646\u0646\u0629 \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta_j \\in\\mathbb{R}<\/span><\/span> \u0647\u064a \u0645\u0639\u0627\u0645\u0644\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span><\/span> \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H.<\/span><\/span> \u0648\u0645\u0639 \u0623\u062e\u0630 \u0647\u0630\u0627 \u0628\u0639\u064a\u0646 \u0627\u0644\u0627\u0639\u062a\u0628\u0627\u0631 \u0641\u0625\u0646 \u062d\u0633\u0627\u0628 \u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0646\u0642\u0637\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z},<\/span><\/span> \u0633\u064a\u0639\u0637\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} (\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z} &amp;= \\left(\\vec{x} - \\displaystyle \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\right) \\cdot \\displaystyle \\sum_{j=1}^k \\beta_j\\hat{v}_j \\\\ \\\\ &amp;= \\vec{x} \\cdot \\displaystyle \\sum_{j=1}^k \\beta_j\\hat{v}_j - \\displaystyle \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\cdot \\displaystyle \\sum_{j=1}^k \\beta_j\\hat{v}_j \\end{array}<\/span>\n<p>\u0648\u0644\u0643\u0646 \u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0645\u062a\u062c\u0647 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u0641\u0636\u0627\u0621 \u062c\u0632\u0626\u064a \u0645\u0646\u0647\u060c \u0641\u0645\u0646 \u0627\u0644\u0645\u0645\u0643\u0646 \u0625\u064a\u062c\u0627\u062f \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n-k<\/span><\/span> \u0645\u062a\u062c\u0647\u0627\u062a \u0645\u062a\u0639\u0627\u0645\u062f\u0629-\u0645\u0642\u0646\u0646\u0629 \u0641\u064a\u0645\u0627 \u0628\u064a\u0646\u0647\u0627 \u0648\u0641\u064a \u0627\u0644\u0648\u0642\u062a \u0646\u0641\u0633\u0647 \u0645\u062a\u0639\u0627\u0645\u062f\u0629-\u0645\u0642\u0646\u0646\u0629 \u0645\u0639 \u062c\u0645\u064a\u0639 \u0645\u062a\u062c\u0647\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H,<\/span><\/span> \u0644\u0646\u0642\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_{k+1}, \\cdots, \\hat{v}_n\\},<\/span><\/span> \u0628\u062d\u064a\u062b \u062a\u0634\u0643\u0644 \u0645\u0639 \u0642\u0627\u0639\u062f\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u0642\u0627\u0639\u062f\u0629 \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0648\u064a\u0645\u0643\u0646 \u0643\u062a\u0627\u0628\u0629<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} = \\displaystyle  \\sum_{j=1}^k (\\vec{x}\\cdot\\hat{v}_j )\\hat{v}_j + \\sum_{j=k+1}^n \\alpha_j \\hat{v}_j <\/span>\n<p>\u0648\u0628\u0630\u0644\u0643 \u064a\u062a\u0627\u0628\u0639 \u0627\u0644\u062a\u0637\u0648\u064a\u0631 \u0623\u0639\u0644\u0627\u0647 \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z} &amp;= \\displaystyle \\left( \\sum_{j=1}^k (\\vec{x}\\cdot\\hat{v}_j )\\hat{v}_j + \\sum_{j=k+1}^n \\alpha_j \\hat{v}_j\\right) \\cdot  \\sum_{j=1}^k \\beta_j\\hat{v}_j -  \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\cdot  \\sum_{j=1}^k \\beta_j\\hat{v}_j \\\\ \\\\\n\n&amp;=  \\displaystyle \\sum_{j=1}^k (\\vec{x}\\cdot\\hat{v}_j )\\hat{v}_j \\cdot \\sum_{j=1}^k \\beta_j\\hat{v}_j + \\underbrace{\\color{red}{\\sum_{j=k+1}^n \\alpha_j \\hat{v}_j \\cdot \\sum_{j=1}^k \\beta_j\\hat{v}_j}}_{(*)} - \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\cdot  \\sum_{j=1}^k \\beta_j\\hat{v}_j \\\\ \\\\\n\n&amp;=  \\displaystyle \\sum_{j=1}^k (\\vec{x}\\cdot\\hat{v}_j )\\hat{v}_j \\cdot \\sum_{j=1}^k \\beta_j\\hat{v}_j  - \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\cdot  \\sum_{j=1}^k \\beta_j\\hat{v}_j \\\\ \\\\\n\n&amp;= 0  \\end{array}<\/span>\n<p>(*) \u0645\u062c\u0645\u0648\u0639 \u064a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u0627\u064b \u0644\u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{v_j\\}_{j=1}^n<\/span><\/span> \u0642\u0627\u0639\u062f\u0629 \u0645\u062a\u0639\u0627\u0645\u062f\u0629-\u0645\u0642\u0646\u0646\u0629 \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span><\/p>\n<p>\u0627\u0646\u0637\u0644\u0627\u0642\u0627\u064b \u0645\u0646 \u0647\u0630\u0627 \u064a\u0645\u0643\u0646\u0646\u0627 \u0623\u0646 \u0646\u0628\u0631\u0647\u0646 \u0623\u0646 \u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0628\u064a\u0646 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u062c\u0632\u0626\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u0648\u0627\u0644\u0645\u062a\u062c\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u062a\u064f\u0639\u0637\u0649 \u0628\u0640:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\|<\/span>\n<h4>\u0628\u0631\u0647\u0627\u0646<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=2995s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u0628\u0631\u0647\u0646\u0629 \u0647\u0630\u0647 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0633\u0646\u0628\u064a\u0646<\/span><\/strong><\/a> \u0623\u0646\u0647 \u0644\u0623\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span><\/span> \u0633\u064a\u062a\u062d\u0642\u0642 \u062f\u0627\u0626\u0645\u0627\u064b \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\| \\leq \\|\\vec{x} - \\vec{z}\\|,<\/span><\/span> \u0648\u0644\u0623\u062c\u0644 \u0630\u0644\u0643 \u0633\u0646\u0633\u062a\u062e\u062f\u0645 \u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633 \u0639\u0644\u0649 \u0627\u0644\u0646\u062d\u0648 \u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\|\\vec{x} - \\vec{z}\\|^2 &amp;= \\| \\left(\\vec{x} -Proy_{H}(\\vec{x}) \\right) + \\left(Proy_{H}(\\vec{x}) - \\vec{z}\\right)\\|^2 \\\\ \\\\ &amp;= \\| \\vec{x} -Proy_{H}(\\vec{x}) \\|^2 + \\|Proy_{H}(\\vec{x}) - \\vec{z}\\|^2 \\\\ \\\\ \\end{array}<\/span>\n<p>\u062a\u064f\u0633\u062a\u0646\u062a\u062c \u0647\u0630\u0647 \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u0644\u0623\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} -Proy_{H}(\\vec{x})<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Proy_{H}(\\vec{x}) - \\vec{z}<\/span><\/span> \u0645\u062a\u0639\u0627\u0645\u062f\u0627\u0646. \u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\|^2 \\leq \\|\\vec{x} - \\vec{z}\\|^2<\/span>\n<p>\u0648\u0647\u0648 \u0645\u0627 \u0643\u0627\u0646 \u0645\u0637\u0644\u0648\u0628\u0627\u064b \u0628\u0631\u0647\u0646\u062a\u0647.<\/p>\n<p>\u0648\u0628\u0647\u0630\u0627 \u062a\u0643\u0648\u0646 \u0644\u062f\u064a\u0646\u0627 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u062a\u064a \u062a\u0645\u0643\u0651\u0646\u0646\u0627 \u0645\u0646 \u0627\u0644\u0642\u0648\u0644 \u0625\u0646 \u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0628\u064a\u0646 \u0646\u0642\u0637\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span> \u0648\u0641\u0636\u0627\u0621 \u062c\u0632\u0626\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/span> \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0645\u0648\u0644\u0651\u064e\u062f \u0628\u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0627\u0644\u0645\u062a\u0639\u0627\u0645\u062f\u0629-\u0627\u0644\u0645\u0642\u0646\u0646\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}<\/span><\/span> \u062a\u064f\u0639\u0637\u0649 \u0628\u0640:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist(\\vec{x},H) =\\left\\|\\vec{x} - Proy_{H}(\\vec{x})\\right\\|= \\left\\|\\vec{x} - \\displaystyle \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j\\right\\|<\/span>\n<p><a name=\"El-Producto-Escalar-y-Vectorial-en-R3\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/uei6y2tniOc\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h2>\u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0646\u0642\u0637\u064a \u0648\u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/span><\/h2>\n<p><strong><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=242s\" rel=\"noopener\" target=\"_blank\"><span style=\"color: #ff0000;\">\u0633\u0646\u063a\u064a\u0631 \u0627\u0644\u0622\u0646 \u062a\u0631\u0643\u064a\u0632\u0646\u0627 \u0642\u0644\u064a\u0644\u0627\u064b<\/span><\/a><\/strong> \u0644\u0646\u062a\u0648\u062c\u0647 \u0646\u062d\u0648 \u0645\u062a\u062c\u0647\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3.<\/span><\/span> \u0647\u0646\u0627\u060c \u0628\u0627\u0644\u0625\u0636\u0627\u0641\u0629 \u0625\u0644\u0649 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u062a\u064a \u0631\u0627\u062c\u0639\u0646\u0627\u0647\u0627 \u0633\u0627\u0628\u0642\u0627\u064b \u0639\u0645\u0648\u0645\u0627\u064b \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n,<\/span><\/span> \u064a\u0645\u0643\u0646 \u0623\u064a\u0636\u0627\u064b \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0627\u0644\u0630\u064a \u064a\u0639\u0637\u064a \u0643\u0646\u062a\u064a\u062c\u0629 \u0645\u062a\u062c\u0647\u0627\u064b \u0622\u062e\u0631 \u0627\u0646\u0637\u0644\u0627\u0642\u0627\u064b \u0645\u0646 \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0645\u062a\u062c\u0647\u064a\u0646. \u0647\u0630\u0627 \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u062e\u0627\u0635 \u0628\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/span> (\u0648\u0631\u0628\u0645\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^7<\/span><\/span>\u060c \u0648\u0647\u0648 \u0645\u0627 \u0644\u0646 \u0646\u0646\u0627\u0642\u0634\u0647 \u0647\u0646\u0627). \u0648\u063a\u0627\u0644\u0628\u0627\u064b \u0645\u0627 \u064a\u064f\u0645\u062b\u0651\u064e\u0644 \u0645\u062a\u062c\u0647\u0648 \u0627\u0644\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0642\u064a\u0627\u0633\u064a\u0629 \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/span> \u0628\u0648\u0627\u0633\u0637\u0629 \u0627\u0644\u0631\u0645\u0648\u0632 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}, \\hat{y}, \\hat{z}<\/span><\/span> \u0623\u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{\\imath}, \\hat{\\jmath}, \\hat{k}<\/span><\/span>. \u0648\u062a\u0628\u0642\u0649 \u0627\u0644\u0623\u0641\u0636\u0644\u064a\u0629 \u0628\u064a\u0646\u0647\u0645\u0627 \u0634\u062e\u0635\u064a\u0629.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\hat{\\imath} = \\hat{x}&amp;=(1,0,0)\\\\ \\hat{\\jmath} =\\hat{y}&amp;=(0,1,0)\\\\ \\hat{k} =\\hat{z}&amp;=(0,0,1)\\\\ \\end{array}<\/span>\n<p>\u0648\u0647\u0643\u0630\u0627\u060c \u0625\u0630\u0627 \u0643\u0627\u0646 \u0644\u062f\u064a\u0646\u0627 \u0645\u062a\u062c\u0647 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c),<\/span><\/span> \u0641\u064a\u0645\u0643\u0646 \u0643\u062a\u0627\u0628\u062a\u0647 \u0641\u064a \u0627\u0644\u0635\u0648\u0631\u0629 \u0627\u0644\u062c\u0628\u0631\u064a\u0629 \u0643\u0645\u0627 \u064a\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c) = a\\hat{x} + b\\hat{y} + c\\hat{z}<\/span>\n<h3>\u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=330s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u062a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span><\/span> \u0645\u062a\u062c\u0647\u064a\u0646 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3.<\/span><\/span><\/span><\/strong><\/a> \u064a\u064f\u0639\u0631\u0651\u064e\u0641 \u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y},<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y}<\/span><\/span> \u0639\u0644\u0649 \u0627\u0644\u0646\u062d\u0648 \u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl} \\vec{x}\\times\\vec{y} &amp;= \\left|\\begin{array}{ccc} \\hat{x} &amp; \\hat{y} &amp; \\hat{z} \\\\ x_1 &amp; x_2 &amp; x_3 \\\\ y_1 &amp; y_2 &amp; y_3 \\end{array}\\right| \\\\ \\\\ &amp;=\\hat{x}x_2y_3 + \\hat{y}x_3y_1 + \\hat{z} x_1y_2 - \\left( \\hat{z} x_2 y_1 + \\hat{y} x_1 y_3 + \\hat{x}x_3y_2\\right) \\\\ \\\\ &amp;=\\hat{x}(x_2y_3 - x_3y_2) + \\hat{y}(x_3y_1 - x_1y_3) + \\hat{z}(x_1y_2 - x_2y_1) \\end{array}<\/span>\n<h3>\u0647\u0648\u064a\u0629 \u0644\u0627\u063a\u0631\u0627\u0646\u062c<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=1399s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0641\u064a \u062d\u0627\u0644\u0629 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/span><\/span><\/strong><\/a> \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u062a\u0639\u0631\u0641 \u0639\u0644\u0649 \u062b\u0644\u0627\u062b\u0629 \u0623\u0646\u0648\u0627\u0639 \u0645\u0646 \u00ab\u0627\u0644\u0636\u0631\u0628\u00bb: \u0627\u0644\u0646\u0642\u0637\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y},<\/span><\/span> \u0648\u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y},<\/span><\/span> \u0648\u0636\u0631\u0628 \u0627\u0644\u0645\u0639\u0627\u064a\u064a\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|\\|\\vec{y}\\|.<\/span><\/span> \u0647\u0630\u0647 \u0627\u0644\u0623\u0646\u0648\u0627\u0639 \u0627\u0644\u062b\u0644\u0627\u062b\u0629 \u0645\u0646 \u0627\u0644\u0636\u0631\u0628 \u0645\u0631\u062a\u0628\u0637\u0629 \u0641\u064a\u0645\u0627 \u0628\u064a\u0646\u0647\u0627 \u0645\u0646 \u062e\u0644\u0627\u0644 \u0647\u0648\u064a\u0629 \u0644\u0627\u063a\u0631\u0627\u0646\u062c<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\|^2  = \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2- (\\vec{x}\\cdot\\vec{y})^2 <\/span>\n<h4>\u0628\u0631\u0647\u0627\u0646 \u0647\u0648\u064a\u0629 \u0644\u0627\u063a\u0631\u0627\u0646\u062c<\/h4>\n<p>\u0644\u062a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span><\/span> \u0645\u062a\u062c\u0647\u064a\u0646 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3,<\/span><\/span> \u0625\u0630\u0646 \u0644\u062f\u064a\u0646\u0627:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl} \\vec{x}\\times\\vec{y} &amp;=(x_2y_3 - x_3y_2) \\hat{x} + (x_3y_1 - x_1y_3)\\hat{y} + (x_1y_2 - x_2y_1)\\hat{z} \\end{array}<\/span>\n<p>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl}\n\n\\|\\vec{x}\\times\\vec{y}\\|^2 &amp;=(x_2y_3 - x_3y_2)^2 + (x_3y_1 - x_1y_3)^2 + (x_1y_2 - x_2y_1)^2 \\\\ \\\\\n\n&amp;= \\color{green}{x_2^2y_3^2 - 2x_2x_3y_3y_2 + x_3^2y_2^2} + \\cdots\\\\ \\\\\n\n&amp;\\cdots + \\color{blue}{x_3^2y_1^2 - 2x_3x_1y_1y_3 + x_1^2y_3^2} + \\cdots \\\\ \\\\\n\n&amp;\\cdots + \\color{red}{x_1^2y_2^2 - 2x_1x_2y_2y_1 + x_2^2y_1^2} \\end{array}<\/span>\n<p>\u0645\u0646 \u0646\u0627\u062d\u064a\u0629 \u0623\u062e\u0631\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl}\n\n\\|\\vec{x}\\|^2 \\|\\vec{y}\\|^2 - (\\vec{x}\\cdot\\vec{y})^2 &amp;= (x_1^2 + x_2^2 + x_3^2)(y_1^2+y_2^2 + y_3^2) - (x_1y_1 + x_2y_2 + x_3 y_3)^2 \\\\ \\\\ \\\\\n\n&amp;=  {x_1^2y_1^2} + \\color{red}{x_1^2y_2^2} + \\color{blue}{x_1^2y_3^2} + \\cdots \\\\ \\\\\n\n&amp;\\cdots + \\color{red}{x_2^2y_1^2} +  {x_2^2y_2^2} + \\color{green}{x_2^2y_3^2} + \\cdots \\\\ \\\\\n\n&amp;\\cdots + \\color{blue}{x_3^2y_1^2} + \\color{green}{x_3^2y_2^2} +  {x_3^2y_3^2} + \\cdots \\\\ \\\\\n\n&amp;\\cdots - \\left[ {x_1^2y_1^2} +  {x_2^2y_2^2} +  {x_3^2y_3^2} + \\right. \\cdots \\\\ \\\\\n\n&amp;\\cdots + 2\\left(\\color{red}{x_1x_2y_1y_2} + \\color{blue}{x_1x_3y_1y_3} + \\color{green}{x_2x_3y_2y_3} \\right)\\left.\\right] \\\\ \\\\ \\\\\n\n&amp;= \\color{red}{x_1^2y_2^2 - 2x_1x_2y_2y_1 + x_2^2y_1^2} + \\cdots \\\\ \\\\\n\n&amp; \\cdots + \\color{blue}{x_1^2y_3^2 - 2x_1x_3y_3y_1 + x_3^2y_1^2} + \\cdots \\\\ \\\\\n\n&amp; \\cdots + \\color{green}{x_2^2y_3^2 - 2x_2x_3y_3y_2 + x_3^2y_2^2}\n\n\\end{array}<\/span>\n<p>\u0648\u0623\u062e\u064a\u0631\u0627\u064b\u060c \u0628\u0645\u0642\u0627\u0631\u0646\u0629 \u0627\u0644\u062a\u0639\u0627\u0628\u064a\u0631 \u0627\u0644\u0645\u0644\u0648\u0646\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 \u0645\u0627 \u0643\u0627\u0646 \u0645\u0637\u0644\u0648\u0628\u0627\u064b \u0628\u0631\u0647\u0646\u062a\u0647.<\/p>\n<h3>\u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0648\u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u0627\u062a<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=1954s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0631\u0623\u064a\u0646\u0627 \u0633\u0627\u0628\u0642\u0627\u064b \u0623\u0646 \u0647\u0646\u0627\u0643 \u0639\u0644\u0627\u0642\u0629 \u0648\u062b\u064a\u0642\u0629<\/span><\/strong><\/a> \u0628\u064a\u0646 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 \u0627\u0644\u062a\u064a \u064a\u0634\u0643\u0644\u0647\u0627 \u0645\u062a\u062c\u0647\u0627\u0646 \u0648\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0646\u0642\u0637\u064a\u060c \u0648\u0647\u0630\u0627 \u064a\u064f\u0639\u0637\u0649 \u0628\u0627\u0644\u0639\u0644\u0627\u0642\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = \\|\\vec{x}\\|\\|\\vec{y}\\|\\cos(\\angle(\\vec{x},\\vec{y})).<\/span><\/span> \u0648\u064a\u062a\u0636\u062d \u0623\u0646 \u0634\u064a\u0626\u0627\u064b \u0645\u0634\u0627\u0628\u0647\u0627\u064b \u064a\u062d\u062f\u062b \u0645\u0639 \u0627\u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0648\u064a\u064f\u0639\u0637\u0649 \u0628\u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\| = \\|\\vec{x}\\|\\|\\vec{y}\\| \\sin(\\angle(\\vec{x},\\vec{y}))<\/span>\n<p>\u0648\u0647\u0630\u0627 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u0647\u0648 \u0646\u062a\u064a\u062c\u0629 \u0645\u0628\u0627\u0634\u0631\u0629 \u0644\u0647\u0648\u064a\u0629 \u0644\u0627\u063a\u0631\u0627\u0646\u062c \u0627\u0644\u062a\u064a \u0628\u0631\u0647\u0646\u0627\u0647\u0627 \u0623\u0639\u0644\u0627\u0647\u060c \u0648\u064a\u0643\u0648\u0646 \u0627\u0644\u0628\u0631\u0647\u0627\u0646 \u062a\u0642\u0631\u064a\u0628\u0627\u064b \u0639\u0644\u0649 \u0627\u0644\u0646\u062d\u0648 \u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\|\\vec{x}\\times\\vec{y}\\|^2 &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 - (\\vec{x}\\cdot\\vec{y})^2 \\\\ \\\\ &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 - (\\|\\vec{x}\\|\\|\\vec{y}\\|\\cos(\\angle(\\vec{x},\\vec{y})))^2 \\\\ \\\\ &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 - \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2\\cos^2(\\angle(\\vec{x},\\vec{y})) \\\\ \\\\ &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 (1 - \\cos^2(\\angle(\\vec{x},\\vec{y}))) \\\\ \\\\ &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 \\sin^2(\\angle(\\vec{x},\\vec{y})) \\end{array}<\/span>\n<p>\u0648\u0623\u062e\u064a\u0631\u0627\u064b\u060c \u0628\u0623\u062e\u0630 \u0627\u0644\u062c\u0630\u0648\u0631 \u0646\u0635\u0644 \u0625\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\| = \\|\\vec{x}\\|\\|\\vec{y}\\|\\; |\\sin(\\angle(\\vec{x},\\vec{y}))|<\/span>\n<p>\u0644\u0643\u0646 \u0644\u0646\u062a\u0630\u0643\u0631 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in[0,\\pi],<\/span><\/span> \u0648\u0641\u064a \u0647\u0630\u0627 \u0627\u0644\u0646\u0637\u0627\u0642 \u062a\u0643\u0648\u0646 \u062f\u0627\u0644\u0629 \u0627\u0644\u062c\u064a\u0628 \u062f\u0627\u0626\u0645\u0627\u064b \u063a\u064a\u0631 \u0633\u0627\u0644\u0628\u0629\u060c \u0648\u0628\u0630\u0644\u0643 \u064a\u0645\u0643\u0646\u0646\u0627 \u0625\u0632\u0627\u0644\u0629 \u0627\u0644\u0642\u064a\u0645\u0629 \u0627\u0644\u0645\u0637\u0644\u0642\u0629 \u0648\u0646\u0635\u0644 \u0625\u0644\u0649 \u0645\u0627 \u0643\u0627\u0646 \u0645\u0637\u0644\u0648\u0628\u0627\u064b \u0628\u0631\u0647\u0646\u062a\u0647.<\/p>\n<p>\u0648\u0627\u0646\u0637\u0644\u0627\u0642\u0627\u064b \u0645\u0646 \u0647\u0630\u0627 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u064a\u0645\u0643\u0646\u0646\u0627 \u0623\u0646 \u0646\u0633\u062a\u0646\u062a\u062c \u0623\u0646 \u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\|<\/span><\/span> \u062a\u0639\u0637\u064a\u0646\u0627 \u0645\u0633\u0627\u062d\u0629 \u0627\u0644\u0645\u062a\u0648\u0627\u0632\u064a \u0627\u0644\u0646\u0627\u062a\u062c \u0639\u0646 \u0627\u0644\u0645\u062a\u062c\u0647\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}.<\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0627\u0644\u062c\u0628\u0631 \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637\u0627\u062a \u0641\u064a Rn\u060c \u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0641\u064a \u0645\u0644\u062e\u0635:\u062a\u0634\u0643\u0644 \u0647\u0630\u0647 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0645\u062a\u0627\u0628\u0639\u0629 \u0645\u0628\u0627\u0634\u0631\u0629 \u0644\u0644\u0633\u0644\u0633\u0644\u0629 \u062d\u0648\u0644 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u0630\u064a \u0627\u0644\u0623\u0628\u0639\u0627\u062f n. \u0647\u0646\u0627 \u0633\u0646\u0631\u0627\u062c\u0639 \u0628\u0639\u0636 \u0645\u0641\u0627\u0647\u064a\u0645 \u0627\u0644\u062c\u0628\u0631 \u0627\u0644\u062e\u0637\u064a \u0627\u0644\u062a\u064a \u062a\u0633\u0627\u0639\u062f \u0639\u0644\u0649 \u0641\u0647\u0645 \u0623\u0641\u0636\u0644 \u0644\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a n-\u0627\u0644\u0623\u0628\u0639\u0627\u062f\u060c \u0648\u0633\u0646\u0631\u0627\u062c\u0639 \u0645\u0641\u0627\u0647\u064a\u0645 \u0625\u0633\u0642\u0627\u0637 \u0645\u062a\u062c\u0647 \u0639\u0644\u0649 \u0622\u062e\u0631\u060c \u0648\u0646\u0628\u0631\u0647\u0646 \u0639\u0644\u0649 \u0646\u0638\u0631\u064a\u0629 \u0641\u064a\u062b\u0627\u063a\u0648\u0631\u0633\u060c \u0648\u0646\u062e\u062a\u062a\u0645 \u0628\u0645\u0631\u0627\u062c\u0639\u0629 \u0644\u062d\u0627\u0635\u0644 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0627\u062a\u062c\u0627\u0647\u064a \u0641\u064a \u0648\u0639\u0644\u0627\u0642\u062a\u0647 \u0645\u0639 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u0623\u062e\u0631\u0649 \u0641\u064a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":34241,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1122,565],"tags":[],"class_list":["post-34290","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-1122","category-565"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u0627\u0644\u062c\u0628\u0631 \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637\u0627\u062a \u0641\u064a Rn\u060c 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