{"id":34257,"date":"2022-03-29T13:00:55","date_gmt":"2022-03-29T13:00:55","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34257"},"modified":"2025-08-27T21:39:37","modified_gmt":"2025-08-27T21:39:37","slug":"%e0%a4%86%e0%a4%b0%e2%81%bf-%e0%a4%ae%e0%a5%87%e0%a4%82-%e0%a4%ac%e0%a5%80%e0%a4%9c%e0%a4%97%e0%a4%a3%e0%a4%bf%e0%a4%a4-%e0%a4%94%e0%a4%b0-%e0%a4%aa%e0%a5%8d%e0%a4%b0%e0%a4%95%e0%a5%8d%e0%a4%b7","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/hi\/%e0%a4%86%e0%a4%b0%e2%81%bf-%e0%a4%ae%e0%a5%87%e0%a4%82-%e0%a4%ac%e0%a5%80%e0%a4%9c%e0%a4%97%e0%a4%a3%e0%a4%bf%e0%a4%a4-%e0%a4%94%e0%a4%b0-%e0%a4%aa%e0%a5%8d%e0%a4%b0%e0%a4%95%e0%a5%8d%e0%a4%b7\/","title":{"rendered":"\u0906\u0930\u207f \u092e\u0947\u0902 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0914\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923, \u211d\u00b3 \u092e\u0947\u0902 \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932"},"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>\u0906\u0930 \u092c\u0940 \u092e\u0947\u0902 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0914\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923, <span class=\"katex-eq\" data-katex-display=\"false\">{\\mathbb{R}^3}<\/span> \u092e\u0947\u0902 \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932<\/h1>\n<p style=\"text-align:center;\"><em><strong>\u0938\u093e\u0930\u093e\u0902\u0936:<\/strong><\/br>\u092f\u0939 \u0936\u094d\u0930\u0943\u0902\u0916\u0932\u093e n \u0906\u092f\u093e\u092e\u094b\u0902 \u0935\u093e\u0932\u0947 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u0940\u092f \u0938\u094d\u0925\u093e\u0928 \u092a\u0930 \u0936\u094d\u0930\u0943\u0902\u0916\u0932\u093e \u0915\u0940 \u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0928\u093f\u0930\u0902\u0924\u0930\u0924\u093e \u0939\u0948\u0964 \u092f\u0939\u093e\u0901 \u0939\u092e \u0930\u0948\u0916\u093f\u0915 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u0940 \u0915\u0941\u091b \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0947\u0902\u0917\u0947 \u091c\u094b n-\u0906\u092f\u093e\u092e\u0940 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u0940\u092f \u0938\u094d\u0925\u093e\u0928 \u0915\u094b \u092c\u0947\u0939\u0924\u0930 \u0938\u092e\u091d\u0928\u0947 \u092e\u0947\u0902 \u092e\u0926\u0926 \u0915\u0930\u0924\u0940 \u0939\u0948\u0902, \u0939\u092e \u090f\u0915 \u0938\u0926\u093f\u0936 \u0915\u093e \u0926\u0942\u0938\u0930\u0947 \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923 \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0947\u0902\u0917\u0947, \u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u094b \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u0930\u0947\u0902\u0917\u0947 \u0914\u0930 \u0905\u0902\u0924 \u092e\u0947\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u092e\u0947\u0902 \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932 \u0914\u0930 3-\u0906\u092f\u093e\u092e\u0940 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u0940\u092f \u0938\u094d\u0925\u093e\u0928 \u0915\u0947 \u0905\u0928\u094d\u092f \u0917\u0941\u0923\u0928\u092b\u0932\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0907\u0938\u0915\u0947 \u0938\u0902\u092c\u0902\u0927 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0947\u0902\u0917\u0947\u0964 <\/p>\n<p style=\"text-align:center;\"><strong>\u0938\u0942\u091a\u0940<\/strong><br \/>\n<a href=\"#Independencia-Lineal-Ortogonalidad-y-Proyecciones\">\u0930\u0947\u0916\u0940\u092f \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e, \u0932\u092e\u094d\u092c\u0935\u0924\u0924\u093e \u0914\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923<\/a><br \/>\n<a href=\"#El-Teorema-de-Pitagoras-y-la-Proyecci\u00f3n-sobre-un-Subespacio\">\u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0914\u0930 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923<\/a><br \/>\n<a href=\"#El-Producto-Escalar-y-Vectorial-en-R3\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u092e\u0947\u0902 \u0905\u0926\u093f\u0936 \u0914\u0930 \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932<\/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>\u0930\u0947\u0916\u0940\u092f \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e, \u0932\u092e\u094d\u092c\u0935\u0924 \u0914\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923<\/h2>\n<h3>\u0930\u0948\u0916\u093f\u0915 \u0938\u0902\u092f\u094b\u091c\u0928 \u0914\u0930 \u0930\u0947\u0916\u0940\u092f \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=138s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u090f\u0915 \u0917\u0948\u0930-\u0936\u0942\u0928\u094d\u092f \u0938\u0926\u093f\u0936<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> \u0915\u094b \u0905\u0928\u094d\u092f \u0917\u0948\u0930-\u0936\u0942\u0928\u094d\u092f \u0938\u0926\u093f\u0936\u094b\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0915\u0947 \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u090f\u0915 <strong>\u0930\u0948\u0916\u093f\u0915 \u0938\u0902\u092f\u094b\u091c\u0928<\/strong> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0928\u093f\u0930\u094d\u092e\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u092f\u0926\u093f \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u090f\u0915 \u092f\u0941\u0917\u094d\u092e <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> \u092e\u094c\u091c\u0942\u0926 \u0939\u0948, \u091c\u094b \u0926\u094b\u0928\u094b\u0902 \u090f\u0915 \u0938\u093e\u0925 \u0936\u0942\u0928\u094d\u092f \u0928 \u0939\u094b\u0902, \u0910\u0938\u093e \u0915\u093f:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z} = \\alpha \\vec{x} + \\beta\\vec{y}<\/span>\n<p>\u0905\u0930\u094d\u0925\u093e\u0924\u094d, \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> \u0915\u094b \u0938\u0926\u093f\u0936\u094b\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0915\u0947 \u090f\u0915 \u092d\u093e\u0930\u093f\u0924 \u092f\u094b\u0917 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0928\u093f\u0930\u094d\u092e\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=609s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947, \u092f\u0939 \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948<\/span><\/strong><\/a> \u0915\u093f \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> <strong>\u0930\u0947\u0916\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930<\/strong> \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u092f\u0926\u093f <\/p>\n<p style=\"text-align:center;\"><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>\u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0915\u0947 \u092c\u0940\u091a \u0930\u0948\u0916\u093f\u0915 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e \u0939\u092e\u0947\u0902 \u092c\u0924\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0915\u094b <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0915\u093e (\u0917\u0948\u0930-\u0936\u0942\u0928\u094d\u092f) \u0905\u0926\u093f\u0936 \u0917\u0941\u0923\u0915 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0928\u0939\u0940\u0902 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0914\u0930 \u0928 \u0939\u0940 \u0907\u0938\u0915\u0947 \u0935\u093f\u092a\u0930\u0940\u0924\u0964<\/p>\n<p>\u091c\u093f\u0938 \u0930\u0948\u0916\u093f\u0915 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0915\u0940 \u0939\u092e\u0928\u0947 \u0905\u092d\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0940 \u0939\u0948, \u0909\u0938\u0947 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0947 \u092c\u0921\u093c\u0947 \u0938\u092e\u0942\u0939\u094b\u0902 \u0924\u0915 \u0935\u093f\u0938\u094d\u0924\u093e\u0930\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0917\u0948\u0930-\u0936\u0942\u0928\u094d\u092f \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u093e \u0938\u092e\u0942\u0939 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x}_1, \\cdots, \\vec{x}_n\\}<\/span> \u0915\u094b \u0930\u0948\u0916\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u091c\u092c<\/p>\n<p style=\"text-align:center;\"><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>\u0926\u094b \u0938\u0926\u093f\u0936\u094b\u0902 \u0926\u094d\u0935\u093e\u0930\u093e \u0928\u093f\u0930\u094d\u092e\u093f\u0924 \u0915\u094b\u0923 \u0914\u0930 \u0932\u092e\u094d\u092c\u0935\u0924\u0924\u093e<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=1289s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092f\u0926\u093f \u0939\u092e \u0915\u0949\u0936\u0940-\u0936\u094d\u0935\u093e\u0930\u094d\u091c\u093c \u0905\u0938\u092e\u093e\u0928\u0924\u093e \u0915\u094b \u092f\u093e\u0926 \u0915\u0930\u0947\u0902,<\/span><\/strong><\/a> \u092f\u0939 \u0939\u092e\u0947\u0902 \u092c\u0924\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f <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> \u0907\u0938\u0947 \u0927\u094d\u092f\u093e\u0928 \u092e\u0947\u0902 \u0930\u0916\u0924\u0947 \u0939\u0941\u090f, \u092f\u0939 \u0906\u0938\u093e\u0928\u0940 \u0938\u0947 \u0938\u0924\u094d\u092f\u093e\u092a\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0915\u093f \u0915\u093f\u0938\u0940 \u092d\u0940 \u0938\u0926\u093f\u0936 \u092f\u0941\u0917\u094d\u092e <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span> \u0915\u0947 \u0932\u093f\u090f \u092f\u0939 \u0938\u0902\u092c\u0902\u0927 \u092a\u0942\u0930\u093e \u0939\u094b\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><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>\u0905\u092c \u0939\u092e \u0905\u0902\u0924\u0930\u094d\u091c\u094d\u091e\u093e\u0928 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0915\u0947 \u092c\u0940\u091a \u0921\u0949\u091f \u0917\u0941\u0923\u0928\u092b\u0932 \u0914\u0930 \u0909\u0928\u0915\u0947 \u0926\u094d\u0935\u093e\u0930\u093e \u092c\u0928\u093e\u090f \u0917\u090f \u0915\u094b\u0923 \u0915\u0947 \u092c\u0940\u091a \u090f\u0915 \u0938\u0902\u092c\u0902\u0927 \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0947 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span> \u0915\u0947 \u0938\u092e\u092e\u093f\u0924 \u0938\u092e\u0924\u0932 \u0915\u093e \u0928\u093f\u0930\u094d\u092e\u093e\u0923 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938\u0932\u093f\u090f, \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u093e \u0915\u0940 \u0939\u093e\u0928\u093f \u0915\u0947 \u092c\u093f\u0928\u093e, \u0939\u092e \u0909\u0928\u094d\u0939\u0947\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span> \u0915\u0947 \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0915\u0932\u094d\u092a\u0928\u093e \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0928\u0915\u0947 \u0915\u094b\u0923 \u0915\u094d\u0930\u092e\u0936\u0903 <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span> \u0905\u0915\u094d\u0937 \u0915\u0947 \u0938\u093e\u092a\u0947\u0915\u094d\u0937 <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_y<\/span> \u0939\u0948\u0902, \u0924\u093e\u0915\u093f \u0938\u0926\u093f\u0936 \u0927\u094d\u0930\u0941\u0935\u0940\u092f \u0930\u0942\u092a \u092e\u0947\u0902 \u0932\u093f\u0916\u0947 \u091c\u093e \u0938\u0915\u0947\u0902:<\/p>\n<p style=\"text-align:center;\"><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>\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u092e \u092e\u093e\u0928 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 (\u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u093e \u0915\u0940 \u0939\u093e\u0928\u093f \u0915\u0947 \u092c\u093f\u0928\u093e, \u092b\u093f\u0930 \u0938\u0947) \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x \\lt \\theta_y,<\/span> \u0914\u0930 \u092b\u093f\u0930 \u0921\u0949\u091f \u0917\u0941\u0923\u0928\u092b\u0932 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}<\/span> \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u0910\u0938\u093e \u0915\u0930\u0928\u0947 \u092a\u0930 \u0939\u092e\u0947\u0902 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092a\u0930\u093f\u0923\u093e\u092e \u092e\u093f\u0932\u0947\u0917\u093e:<\/p>\n<p style=\"text-align:center;\"><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>\u0905\u092c, \u092c\u0921\u093c\u093e \u0914\u0930 \u091b\u094b\u091f\u093e \u0915\u094b\u0923\u0940\u092f \u0938\u094d\u0925\u093e\u0928 \u0915\u0947 \u092c\u0940\u091a \u0915\u093e \u0905\u0902\u0924\u0930 \u0932\u0947\u0924\u0947 \u0939\u0941\u090f \u0939\u092e\u0947\u0902 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u092c\u0928\u093e \u0915\u094b\u0923 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948, <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})=\\theta_y - \\theta_x.<\/span> \u0914\u0930 \u0907\u0938\u0915\u0947 \u0938\u093e\u0925 \u0905\u092c \u0939\u092e \u0932\u093f\u0916 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align:center;\"><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>\u092f\u0939\u093e\u0901 \u0939\u092e\u0947\u0902 \u092f\u0939 \u0930\u0947\u0916\u093e\u0902\u0915\u093f\u0924 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u093f\u090f \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in [0, \\pi]<\/span>\n<p>\u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0939\u092e \u0915\u0949\u0936\u0940-\u0936\u094d\u0935\u093e\u0930\u094d\u091c\u093c \u0905\u0938\u092e\u093e\u0928\u0924\u093e \u0915\u094b \u0915\u094b\u0923\u094b\u0902 \u0915\u0940 \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u093f \u0938\u0947 \u091c\u094b\u0921\u093c \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u0914\u0930 \u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e \u092f\u0939 \u0939\u092e\u0947\u0902 \u0932\u092e\u094d\u092c\u0935\u0924\u0924\u093e \u0915\u0940 \u090f\u0915 \u0915\u0920\u094b\u0930 \u0927\u093e\u0930\u0923\u093e \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u093e \u0939\u0948\u0964 \u0926\u094b \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u094b <strong>\u0932\u092e\u094d\u092c\u0935\u0924<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u091c\u092c \u0935\u0947 \u0906\u092a\u0938 \u092e\u0947\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\pi\/2<\/span> \u0930\u0947\u0921\u093f\u092f\u0928 \u0915\u093e \u0915\u094b\u0923 \u092c\u0928\u093e\u0924\u0947 \u0939\u0948\u0902, \u0909\u0938 \u0905\u0930\u094d\u0925 \u092e\u0947\u0902 \u091c\u093f\u0938\u0947 \u092a\u093f\u091b\u0932\u0947 \u0905\u0928\u0941\u091a\u094d\u091b\u0947\u0926 \u092e\u0947\u0902 \u0938\u092e\u091d\u093e\u092f\u093e \u0917\u092f\u093e \u0925\u093e\u0964 \u092f\u0939 \u0915\u0939\u0928\u0947 \u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\cos\\left(\\angle(\\vec{x},\\vec{y})\\right) = 0,<\/span> \u091c\u094b \u0915\u093f \u0907\u0938 \u0915\u0939\u0928\u0947 \u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = 0.<\/span> \u0907\u0938\u0940 \u0915\u093e\u0930\u0923 \u092f\u0939 \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0915\u0940 \u0932\u092e\u094d\u092c\u0935\u0924\u0924\u093e \u0915\u0940 \u092a\u0941\u0937\u094d\u091f\u093f \u0915\u0930\u0928\u093e \u0907\u0938 \u0915\u0939\u0928\u0947 \u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0.<\/span>\n<h4>\u092f\u0926\u093f \u0926\u094b \u0917\u0948\u0930-\u0936\u0942\u0928\u094d\u092f \u0938\u0926\u093f\u0936 \u0932\u092e\u094d\u092c\u0935\u0924 \u0939\u0948\u0902, \u0924\u094b \u0935\u0947 \u0930\u0947\u0916\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0939\u0948\u0902<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=2365s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092f\u0939 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u093e \u090f\u0915 \u0915\u0941\u091b \u0939\u0926 \u0924\u0915 \u0938\u0939\u091c \u0917\u0941\u0923 \u0939\u0948<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u092e\u0947\u0902 \u091c\u093f\u0938\u0915\u0940 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u094d\u0930\u092e\u093e\u0923\u0928\u093e \u0907\u0924\u0928\u0940 \u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0928\u0939\u0940\u0902 \u0939\u0948, \u0914\u0930 \u092f\u0939 \u092d\u0940 \u090f\u0915 \u0917\u0941\u0923 \u0939\u0948 \u091c\u094b \u0915\u092d\u0940-\u0915\u092d\u0940 \u0925\u094b\u0921\u093c\u0940 \u092d\u094d\u0930\u092e \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0930 \u0938\u0915\u0924\u093e \u0939\u0948: \u0926\u094b \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0940 \u0932\u092e\u094d\u092c\u0935\u0924\u0924\u093e \u0909\u0928\u0915\u0947 \u092c\u0940\u091a \u0930\u0948\u0916\u093f\u0915 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e \u0915\u093e \u0938\u0902\u0915\u0947\u0924 \u0926\u0947\u0924\u0940 \u0939\u0948, \u0932\u0947\u0915\u093f\u0928 \u0926\u094b \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0930\u0948\u0916\u093f\u0915 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e \u0909\u0928\u0915\u0940 \u0932\u092e\u094d\u092c\u0935\u0924\u0924\u093e \u0915\u093e \u0906\u0935\u0936\u094d\u092f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0938\u0902\u0915\u0947\u0924 \u0928\u0939\u0940\u0902 \u0926\u0947\u0924\u0940\u0964 \u0907\u0938\u0947 \u0926\u0947\u0916\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u0938\u093e\u0927\u093e\u0930\u0923 \u092a\u094d\u0930\u0924\u093f\u0935\u093e\u0926 \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0939\u0948:<\/p>\n<p>\u092f\u0926\u093f \u0939\u092e \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}=(1,0)<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}=(1,1)<\/span> \u0932\u0947\u0902, \u091c\u094b \u0938\u094d\u092a\u0937\u094d\u091f \u0930\u0942\u092a \u0938\u0947 \u0932\u092e\u094d\u092c\u0935\u0924 \u0928\u0939\u0940\u0902 \u0939\u0948\u0902 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}\\cdot\\vec{B}=1,<\/span> \u0924\u094b \u0939\u092e \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u0915\u093f \u092f\u0926\u093f \u0939\u092e \u0915\u0930\u0947\u0902<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\alpha\\vec{A} + \\beta\\vec{B} = \\vec{0}\n\n<\/span>\n<p>\u0924\u094b \u092f\u0939 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u093f<\/p>\n<p style=\"text-align:center;\"><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>\u0914\u0930 \u0907\u0938\u0932\u093f\u090f: <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = 0  \\wedge \\beta=0.<\/span> \u0914\u0930 \u0907\u0938\u0915\u0947 \u0938\u093e\u0925 \u092f\u0939 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u0928\u093f\u0915\u0932\u0924\u093e \u0939\u0948 \u0915\u093f:<\/p>\n<p style=\"text-align:center;\"><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>\u091c\u094b \u092f\u0939 \u0915\u0939\u0928\u0947 \u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}<\/span> \u0930\u0948\u0916\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0939\u0948\u0902\u0964 \u0907\u0938\u0915\u0947 \u0938\u093e\u0925 \u092f\u0939 \u092c\u0939\u0941\u0924 \u0938\u094d\u092a\u0937\u094d\u091f \u0930\u0942\u092a \u0938\u0947 \u0938\u094d\u092a\u0937\u094d\u091f \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u092f\u0939 \u0938\u0924\u094d\u092f \u0928\u0939\u0940\u0902 \u0939\u0948 \u0915\u093f \u0930\u0948\u0916\u093f\u0915 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e \u0932\u092e\u094d\u092c\u0935\u0924\u0924\u093e \u0915\u093e \u0938\u0902\u0915\u0947\u0924 \u0926\u0947\u0924\u0940 \u0939\u0948\u0964 \u0939\u093e\u0932\u093e\u0902\u0915\u093f, \u0932\u092e\u094d\u092c\u0935\u0924\u0924\u093e \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 \u0930\u0948\u0916\u093f\u0915 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e \u0915\u093e \u0938\u0902\u0915\u0947\u0924 \u0926\u0947\u0924\u0940 \u0939\u0948 \u0914\u0930 \u092f\u0939\u0940 \u092e\u0948\u0902 \u0928\u0940\u091a\u0947 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u0930\u0942\u0902\u0917\u093e, \u0914\u0930 \u0907\u0938\u0915\u0947 \u0932\u093f\u090f \u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0915\u0947 \u0938\u092e\u0942\u0939 \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902:<\/p>\n<p style=\"text-align:center;\"><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>\u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0930\u094d\u0915 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align:center;\"><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;{;\\;\u0905\u0928\u0941\u092e\u093e\u0928}\\\\ \\\\\n\n(2) &amp;\\mathcal{H}\\vdash \\vec{x}\\cdot\\vec{y}=0 &amp;{\\;\u0905\u0928\u0941\u092e\u093e\u0928} \\\\ \\\\\n\n(3) &amp;\\mathcal{H}\\vdash \\alpha\\vec{x} + \\beta\\vec{y} = \\vec{0} &amp;{\\;\u0905\u0928\u0941\u092e\u093e\u0928} \\\\ \\\\\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;{;\\; \u0926\u094d\u0935\u093f\u0916\u0902\u0921\u0924\u093e} \\\\ \\\\\n\n(5) &amp;\\mathcal{H}\\vdash  \\alpha\\|\\vec{x}\\|^2 = 0 &amp; {;\\; (2,3,4) \u0938\u0947} \\\\ \\\\\n\n(6) &amp;\\mathcal{H}\\vdash  \\alpha  = 0 &amp; {;\\; (1,5) \u0938\u0947} \\\\ \\\\\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; {;\\;\u0926\u094d\u0935\u093f\u0916\u0902\u0921\u0924\u093e} \\\\ \\\\\n\n(8) &amp;\\mathcal{H}\\vdash \\beta\\|\\vec{y}\\|^2 = 0 &amp;{;\\;(2,3,7) \u0938\u0947} \\\\ \\\\\n\n(9) &amp;\\mathcal{H}\\vdash \\beta = 0 &amp;{;\\;(1,8) \u0938\u0947} \\\\ \\\\\n\n(10) &amp;\\mathcal{H}\\vdash \\alpha= 0 \\wedge \\beta = 0 &amp;{;\\;\\wedge-int(6,9)}\n\n\\end{array}<\/span>\n<p>\u0907\u0938\u0915\u0947 \u0938\u093e\u0925 \u0939\u092e \u092f\u0939 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u0928\u093f\u0915\u093e\u0932\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f<\/p>\n<p style=\"text-align:center;\"><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  <\/span>\n<p>\u0905\u0902\u0924\u0924\u0903, \u0907\u0938 \u0905\u0902\u0924\u093f\u092e \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u092a\u0930 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u092a\u094d\u0930\u092e\u0947\u092f \u0932\u093e\u0917\u0942 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><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>\u0935\u093f\u092a\u0930\u0940\u0924 \u0926\u093f\u0936\u093e \u092e\u0947\u0902 \u0924\u0940\u0930 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0928\u0947 \u0935\u093e\u0932\u093e \u092a\u094d\u0930\u092e\u093e\u0923 \u0924\u0941\u091a\u094d\u091b \u0939\u0948\u0964<\/p>\n<p>\u0905\u0930\u094d\u0925\u093e\u0924\u094d: \u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0917\u0948\u0930-\u0936\u0942\u0928\u094d\u092f \u0914\u0930 \u0932\u092e\u094d\u092c\u0935\u0924 \u0938\u0926\u093f\u0936 \u0939\u0948\u0902, \u0924\u094b \u0935\u0947 \u0930\u0948\u0916\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0939\u0948\u0902\u0964<\/p>\n<h3>\u090f\u0915 \u0938\u0926\u093f\u0936 \u0915\u093e \u0926\u0942\u0938\u0930\u0947 \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=3055s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0926\u094b \u0917\u0948\u0930-\u0936\u0942\u0928\u094d\u092f \u0938\u0926\u093f\u0936 \u0939\u0948\u0902<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u091c\u094b \u0906\u092a\u0938 \u092e\u0947\u0902 \u090f\u0915 \u0915\u094b\u0923 <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})<\/span> \u092c\u0928\u093e\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u0939\u092e\u0938\u0947 \u092a\u0942\u091b\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u00ab\u0915\u093f\u0924\u0928\u0940 \u092e\u093e\u0924\u094d\u0930\u093e \u092e\u0947\u0902 \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u092a\u0930 \u0938\u094d\u0925\u093f\u0924 \u0939\u0948?\u00bb \u092f\u093e \u00ab\u091c\u092c \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0915\u094b \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0915\u0940 \u0926\u093f\u0936\u093e \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0924\u094b \u0909\u0938\u0915\u0940 \u091b\u093e\u092f\u093e \u0915\u093f\u0924\u0928\u0940 \u092c\u0921\u093c\u0940 \u0939\u094b\u0924\u0940 \u0939\u0948?\u00bb \u0907\u0938 \u092a\u094d\u0930\u0936\u094d\u0928 \u0915\u094b \u0939\u092e \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u093f \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0939\u0932 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u0914\u0930 \u0907\u0938\u0915\u0947 \u0938\u093e\u0925 \u0939\u0940 \u090f\u0915 \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0915\u093e \u0926\u0942\u0938\u0930\u0947 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923 <span class=\"katex-eq\" data-katex-display=\"false\">Proy_{\\vec{y}}(\\vec{x})<\/span> \u0915\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align:center;\"><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>\u092f\u0926\u093f \u0939\u092e \u0907\u0938\u0947 \u092a\u0939\u0932\u0947 \u0915\u0947 \u0905\u0928\u0941\u091a\u094d\u091b\u0947\u0926\u094b\u0902 \u092e\u0947\u0902 \u0926\u0947\u0916\u0940 \u0917\u0908 \u092c\u093e\u0924\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u092e\u093f\u0932\u093e\u090f\u0901 \u0924\u094b \u0939\u092e \u0932\u093f\u0916 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align:center;\"><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>\u0915\u094d\u092f\u094b\u0902\u0915\u093f, \u0939\u092e\u0947\u0902 \u092f\u093e\u0926 \u0930\u0916\u0928\u093e \u091a\u093e\u0939\u093f\u090f<\/p>\n<p style=\"text-align:center;\"><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>\u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u0948\u0902 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0935\u0947 \u0939\u092e\u0947\u0902 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u094b \u0909\u0928\u0915\u0940 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923\u094b\u0902 \u0915\u0947 \u092f\u094b\u0917 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0906\u0927\u093e\u0930 \u0915\u0947 \u0938\u093e\u092a\u0947\u0915\u094d\u0937 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0930\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} = \\displaystyle \\sum_{i=1}^n \\alpha_i \\hat{u}_i<\/span>\n<p>\u091c\u0939\u093e\u0901 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{u}_i\\}_{i=1,\\cdots, n}<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u0947 \u0930\u0948\u0916\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u093e \u090f\u0915 \u0906\u0927\u093e\u0930 \u0939\u0948 \u0914\u0930 \u0917\u0941\u0923\u093e\u0902\u0915 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_i = (\\vec{x}\\cdot\\vec{u}_i)\/\\|\\vec{u}_i\\|<\/span> \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0906\u0927\u093e\u0930 \u0924\u0924\u094d\u0935 \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923 \u0939\u0948\u0902 \u0914\u0930 \u0906\u0927\u093e\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{u}_i\\}_{i=1,\\cdots, n}<\/span> \u0915\u0947 \u0938\u093e\u092a\u0947\u0915\u094d\u0937 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0915\u0947 \u0928\u093f\u0930\u094d\u0926\u0947\u0936\u093e\u0902\u0915 \u092c\u0928\u093e\u0924\u0947 \u0939\u0948\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span>\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>\u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0914\u0930 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=254s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u090f\u0915 \u0910\u0938\u093e \u092a\u0930\u093f\u0923\u093e\u092e \u0939\u0948<\/span><\/strong><\/a> \u091c\u093f\u0938\u0947 \u0938\u092d\u0940 \u091c\u093e\u0928\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u091c\u093f\u0938\u0915\u0947 \u0905\u0938\u0902\u0916\u094d\u092f \u092a\u094d\u0930\u092e\u093e\u0923 \u092e\u094c\u091c\u0942\u0926 \u0939\u0948\u0902\u0964 \u0907\u0938 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u093e \u090f\u0915 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u092a\u094d\u0930\u092e\u093e\u0923 \u0920\u0940\u0915 \u0909\u0928\u094d\u0939\u0940\u0902 \u0935\u093f\u0937\u092f\u094b\u0902 \u0938\u0947 \u0909\u092d\u0930\u0924\u093e \u0939\u0948 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0939\u092e\u0928\u0947 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u0940\u092f \u0938\u094d\u0925\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f \u0935\u093f\u0915\u0938\u093f\u0924 \u0915\u093f\u092f\u093e \u0939\u0948 \u0914\u0930 \u0905\u0924\u093f\u0930\u093f\u0915\u094d\u0924 \u0932\u093e\u092d \u092f\u0939 \u0939\u0948 \u0915\u093f \u092f\u0939 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0940 \u0906\u092f\u093e\u092e\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u092e\u093e\u0928\u094d\u092f \u0939\u0948\u0964<\/p>\n<h3>\u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=533s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092f\u0926\u093f \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u090f\u0915 \u0938\u092e\u0915\u094b\u0923 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0939\u0948 \u091c\u093f\u0938\u0915\u0947 \u0932\u092e\u094d\u092c<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">b,<\/span> \u0914\u0930 \u0915\u0930\u094d\u0923 <span class=\"katex-eq\" data-katex-display=\"false\">c,<\/span> \u0939\u0948\u0902, \u0924\u094b \u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0939\u092e\u0947\u0902 \u092c\u0924\u093e\u0924\u093e \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">a^2+b^2=c^2.<\/span> \u0907\u0938\u0947 \u0938\u092e\u091d\u0928\u0947 \u0915\u0947 \u092c\u093e\u0926 \u0939\u092e \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0932\u092e\u094d\u092c \u0915\u094b \u0926\u094b \u0932\u092e\u094d\u092c\u0935\u0924 \u0938\u0926\u093f\u0936\u094b\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0930\u0940\u0915\u0947 \u0938\u0947 \u0932\u093f\u0916 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align:center;\"><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>\u091c\u0939\u093e\u0901 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\bot\\vec{y}<\/span> \u092f\u0939 \u0907\u0902\u0917\u093f\u0924 \u0915\u0930\u0924\u0940 \u0939\u0948 \u0915\u093f \u0926\u094b\u0928\u094b\u0902 \u0938\u0926\u093f\u0936 \u0932\u092e\u094d\u092c\u0935\u0924 \u0939\u0948\u0902, \u0905\u0930\u094d\u0925\u093e\u0924\u094d: \u0917\u0948\u0930-\u0936\u0942\u0928\u094d\u092f \u0914\u0930 \u0910\u0938\u0947 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0.<\/span> \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u0926\u094b \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0940 \u0932\u092e\u094d\u092c\u0935\u0924\u0924\u093e \u0914\u0930 \u0909\u0928\u0915\u0947 \u092a\u0930\u093f\u092e\u093e\u0923\u094b\u0902 \u0915\u0947 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0947 \u092f\u094b\u0917 \u0915\u0947 \u092c\u0940\u091a \u0926\u094d\u0935\u093f-\u0936\u0930\u094d\u0924\u0940\u092f \u0938\u0902\u092c\u0902\u0927 \u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<p>\u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0915\u0947 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u094b \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u093e \u092f\u0939 \u0938\u0926\u093f\u0936 \u0930\u0942\u092a \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0926\u094b \u0924\u0930\u094d\u0915\u094b\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:<\/p>\n<p><strong>\u092a\u0939\u0932\u0947 \u0906\u0917\u0947 \u0915\u0940 \u0926\u093f\u0936\u093e:<\/strong><\/p>\n<p style=\"text-align:center;\"><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; {;\\;\u0905\u0928\u0941\u092e\u093e\u0928} \\\\ \\\\\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; {;\\;(1) \u0938\u0947} \\\\ \\\\\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;;\\; \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u0940\u092f \u092e\u093e\u0928\u0915 \u0914\u0930 \u0905\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932 \u0915\u093e \u0917\u0941\u0923 &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; {;\\;(2,3) \u0938\u0947} \\\\ \\\\\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>\u0914\u0930 \u0905\u092c \u0935\u093e\u092a\u0938\u0940 \u0915\u0940 \u0926\u093f\u0936\u093e:<\/strong><\/p>\n<p style=\"text-align:center;\"><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; {;\\;\u0905\u0928\u0941\u092e\u093e\u0928} \\\\ \\\\\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;;\\; \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u0940\u092f \u092e\u093e\u0928\u0915 \u0914\u0930 \u0905\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932 \u0915\u093e \u0917\u0941\u0923 &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; {;\\;(1,2) \u0938\u0947} \\\\ \\\\\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; {;\\;(3) \u0938\u0947} \\\\ \\\\\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>\u0914\u0930 \u0905\u0902\u0924\u0924\u0903, \u0926\u094b\u0928\u094b\u0902 \u0924\u0930\u094d\u0915\u094b\u0902 \u0915\u094b \u092e\u093f\u0932\u093e\u0915\u0930 \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0935\u0939\u0940 \u0939\u0948 \u091c\u093f\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0925\u093e:<\/strong><\/p>\n<p style=\"text-align:center;\"><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><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u0947 \u090f\u0915 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 \u092a\u0930 \u0915\u093f\u0938\u0940 \u0938\u0926\u093f\u0936 \u0915\u093e \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=1545s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u090f\u0915 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 \u0939\u0948<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u093e, \u091c\u094b \u090f\u0915\u0915 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0947 \u0906\u0927\u093e\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}<\/span> \u0938\u0947 \u092c\u0928\u093e \u0939\u0948\u0964 \u092f\u0926\u093f \u0939\u092e \u090f\u0915 \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span> \u0932\u0947\u0902, \u0924\u094b \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0915\u093e <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u0938\u094d\u0925\u093e\u0928 \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><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>\u0915\u093f\u0938\u0940 \u0938\u092e\u0942\u0939 \u0915\u093e \u0906\u0930\u094d\u0925\u094b\u0928\u0949\u0930\u094d\u092e\u0932 \u0939\u094b\u0928\u093e \u092f\u0939 \u092c\u0924\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u0909\u0938\u0915\u0947 \u0938\u092d\u0940 \u0905\u0935\u092f\u0935 \u0906\u092a\u0938 \u092e\u0947\u0902 \u0932\u092e\u094d\u092c\u0935\u0924 \u0939\u0948\u0902 \u0914\u0930 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0915\u093e \u092e\u093e\u0928\u0915 \u0907\u0915\u093e\u0908 \u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u0948\u0964<\/p>\n<p>\u092f\u0939, \u092f\u0942\u0901 \u0915\u0939\u0947\u0902 \u0924\u094b, \u0909\u0938 \u091b\u093e\u092f\u093e \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948 \u091c\u093f\u0938\u0947 \u090f\u0915 \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u0947 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u0915\u0940 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0918\u091f\u0915 \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/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><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u0947 \u0915\u093f\u0938\u0940 \u092c\u093f\u0902\u0926\u0941 \u092f\u093e \u0938\u0926\u093f\u0936 \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u0947 \u0915\u093f\u0938\u0940 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 \u0915\u0947 \u092c\u0940\u091a \u0915\u0940 \u0926\u0942\u0930\u0940<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=1974s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0915\u093f\u0938\u0940 \u0938\u0926\u093f\u0936 \u0915\u093e \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923 \u0932\u0947\u0924\u0947 \u0939\u0941\u090f<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u0947 \u090f\u0915 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u092a\u0930, \u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0930\u0942\u092a \u0915\u093e \u090f\u0915 \u0938\u0926\u093f\u0936 \u092c\u0928\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} - Proy_{H}(\\vec{x})<\/span>\n<p>\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0928\u093f\u0930\u094d\u092e\u093f\u0924 \u0938\u0926\u093f\u0936 \u0935\u0939 \u0938\u0926\u093f\u0936 \u0939\u094b\u0917\u093e \u091c\u094b \u0909\u092a\u0938\u094d\u0925\u093e\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u0915\u0947 \u090f\u0915 \u092c\u093f\u0902\u0926\u0941 \u0915\u094b \u0928\u093f\u0930\u094d\u0926\u0947\u0936\u093e\u0902\u0915 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0935\u093e\u0932\u0947 \u092c\u093f\u0902\u0926\u0941 \u0938\u0947 \u091c\u094b\u0921\u093c\u0924\u093e \u0939\u0948 \u0914\u0930 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u092a\u0930 \u0932\u092e\u094d\u092c\u0935\u0924 \u0928\u093f\u0915\u0932\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0915\u0920\u093f\u0928 \u0928\u0939\u0940\u0902 \u0939\u0948: \u092f\u0926\u093f \u0939\u092e \u0915\u094b\u0908 \u092d\u0940 \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span> \u0932\u0947\u0902 \u0914\u0930 \u0921\u0949\u091f \u0917\u0941\u0923\u0928\u092b\u0932 <span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z}<\/span> \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0947\u0902, \u0924\u094b \u092f\u0939 \u0926\u0947\u0916\u0928\u093e \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0939\u0948 \u0915\u093f \u0907\u0938 \u0915\u094d\u0930\u093f\u092f\u093e \u0915\u093e \u092a\u0930\u093f\u0923\u093e\u092e \u0936\u0942\u0928\u094d\u092f \u0939\u0948\u0964 \u091a\u0932\u093f\u090f \u0917\u0923\u0928\u093e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u092f\u0939 \u0926\u0947\u0916\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f \u0915\u094d\u092f\u093e \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 \u0910\u0938\u093e \u0939\u0948:<\/p>\n<p>\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H,<\/span> \u0924\u094b \u092f\u0939 \u0907\u0938 \u0930\u0942\u092a \u0915\u093e \u0939\u094b\u0917\u093e<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}=\\displaystyle \\sum_{j=1}^k \\beta_j\\hat{v}_j<\/span>\n<p>\u091c\u0939\u093e\u0901 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_j\\}_{j=1}^k<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u0915\u093e \u090f\u0915 \u0906\u0930\u094d\u0925\u094b\u0928\u0949\u0930\u094d\u092e\u0932 \u0906\u0927\u093e\u0930 \u0939\u0948 \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_j \\in\\mathbb{R}<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u092e\u0947\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> \u0915\u0947 \u0917\u0941\u0923\u093e\u0902\u0915 \u0939\u0948\u0902\u0964 \u0907\u0938\u0947 \u0927\u094d\u092f\u093e\u0928 \u092e\u0947\u0902 \u0930\u0916\u0924\u0947 \u0939\u0941\u090f, \u0921\u0949\u091f \u0917\u0941\u0923\u0928\u092b\u0932 <span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z}<\/span> \u0915\u0940 \u0917\u0923\u0928\u093e \u092f\u0939 \u0926\u0947\u0917\u0940:<\/p>\n<p style=\"text-align:center;\"><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>\u0932\u0947\u0915\u093f\u0928 \u091a\u0942\u0902\u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u093e \u090f\u0915 \u0938\u0926\u093f\u0936 \u0939\u0948 \u091c\u093f\u0938\u0915\u093e <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u090f\u0915 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 \u0939\u0948, \u0907\u0938\u0932\u093f\u090f <span class=\"katex-eq\" data-katex-display=\"false\">n-k<\/span> \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u093e \u090f\u0915 \u0910\u0938\u093e \u0938\u092e\u0942\u0939 \u0916\u094b\u091c\u0928\u093e \u0938\u0902\u092d\u0935 \u0939\u0948 \u091c\u094b \u0906\u092a\u0938 \u092e\u0947\u0902 \u0906\u0930\u094d\u0925\u094b\u0928\u0949\u0930\u094d\u092e\u0932 \u0939\u094b\u0902 \u0914\u0930 \u0938\u093e\u0925 \u0939\u0940 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u0915\u0947 \u0938\u092d\u0940 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0947 \u092a\u094d\u0930\u0924\u093f \u0906\u0930\u094d\u0925\u094b\u0928\u0949\u0930\u094d\u092e\u0932 \u0939\u094b\u0902, \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_{k+1}, \\cdots, \\hat{v}_n\\},<\/span> \u0924\u093e\u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u0915\u0947 \u0906\u0927\u093e\u0930 \u0915\u0947 \u0938\u093e\u0925 \u0935\u0947 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u0906\u0927\u093e\u0930 \u092c\u0928\u093e\u090f\u0902 \u0914\u0930 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0932\u093f\u0916\u093e \u091c\u093e \u0938\u0915\u0947<\/p>\n<p style=\"text-align:center;\"><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>\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u090a\u092a\u0930 \u0915\u093e \u0935\u093f\u0915\u093e\u0938 \u0928\u093f\u092e\u094d\u0928 \u0930\u0942\u092a \u092e\u0947\u0902 \u0906\u0917\u0947 \u092c\u0922\u093c\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><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>(*) \u0936\u0942\u0928\u094d\u092f \u092f\u094b\u0917 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\{v_j\\}_{j=1}^n<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u093e \u090f\u0915 \u0906\u0930\u094d\u0925\u094b\u0928\u0949\u0930\u094d\u092e\u0932 \u0906\u0927\u093e\u0930 \u0939\u0948\u0964<\/p>\n<p>\u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0939\u092e \u092f\u0939 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0909\u092a\u0938\u094d\u0925\u093e\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u0914\u0930 \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0915\u0947 \u092c\u0940\u091a \u0915\u0940 \u0926\u0942\u0930\u0940 \u0928\u093f\u092e\u094d\u0928 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u0947 \u0926\u0940 \u0917\u0908 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\|<\/span>\n<h4>\u092a\u094d\u0930\u092e\u093e\u0923<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=2995s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0907\u0938 \u092a\u0930\u093f\u0923\u093e\u092e \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092f\u0939 \u0926\u093f\u0916\u093e\u092f\u093e \u091c\u093e\u090f\u0917\u093e<\/span><\/strong><\/a> \u0915\u093f \u0938\u092d\u0940 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span> \u0915\u0947 \u0932\u093f\u090f \u0938\u0926\u0948\u0935 \u092f\u0939 \u0938\u0924\u094d\u092f \u0939\u094b\u0917\u093e \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\| \\leq \\|\\vec{x} - \\vec{z}\\|,<\/span> \u0907\u0938\u0915\u0947 \u0932\u093f\u090f \u0939\u092e \u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0928\u093f\u092e\u094d\u0928 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u0947 \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0947\u0902\u0917\u0947:<\/p>\n<p style=\"text-align:center;\"><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>\u092f\u0939 \u0905\u0902\u0924\u093f\u092e \u0938\u092e\u093e\u0928\u0924\u093e \u0907\u0938\u0932\u093f\u090f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} -Proy_{H}(\\vec{x})<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">Proy_{H}(\\vec{x}) - \\vec{z}<\/span> \u0932\u092e\u094d\u092c\u0935\u0924 \u0939\u0948\u0902\u0964 \u0914\u0930 \u0907\u0938\u0932\u093f\u090f:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\|^2 \\leq \\|\\vec{x} - \\vec{z}\\|^2<\/span>\n<p>\u091c\u094b \u0935\u0939\u0940 \u0939\u0948 \u091c\u093f\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0925\u093e\u0964<\/p>\n<p>\u0905\u092c \u0907\u0938 \u092a\u0930\u093f\u0923\u093e\u092e \u0915\u0947 \u0938\u093e\u0925, \u0939\u092e \u0915\u0939 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span> \u0915\u0947 \u0915\u093f\u0938\u0940 \u092c\u093f\u0902\u0926\u0941 \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u0947 \u090f\u0915 \u0909\u092a\u0938\u094d\u0925\u093e\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u0915\u0947 \u092c\u0940\u091a \u0915\u0940 \u0926\u0942\u0930\u0940, \u091c\u094b \u0906\u0930\u094d\u0925\u094b\u0928\u0949\u0930\u094d\u092e\u0932 \u0938\u0926\u093f\u0936\u094b\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}<\/span> \u0926\u094d\u0935\u093e\u0930\u093e \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0939\u0948, \u0928\u093f\u092e\u094d\u0928 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u0947 \u0926\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><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><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u092e\u0947\u0902 \u0905\u0926\u093f\u0936 \u0914\u0930 \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932<\/h2>\n<p><strong><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=242s\" rel=\"noopener\" target=\"_blank\"><span style=\"color: #ff0000;\">\u0905\u092c \u0939\u092e \u0905\u092a\u0928\u093e \u0926\u0943\u0937\u094d\u091f\u093f\u0915\u094b\u0923 \u0925\u094b\u0921\u093c\u093e \u092c\u0926\u0932\u0947\u0902\u0917\u0947<\/span><\/a><\/strong> \u0924\u093e\u0915\u093f \u0939\u092e <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u0915\u0947 \u0938\u0926\u093f\u0936\u094b\u0902 \u092a\u0930 \u0927\u094d\u092f\u093e\u0928 \u0915\u0947\u0902\u0926\u094d\u0930\u093f\u0924 \u0915\u0930\u0947\u0902\u0964 \u092f\u0939\u093e\u0901, \u0909\u0928 \u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u0915\u0947 \u0905\u0932\u093e\u0935\u093e \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0939\u092e\u0928\u0947 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u0915\u0947 \u0932\u093f\u090f \u092a\u0939\u0932\u0947 \u0939\u0940 \u0926\u0947\u0916\u093e \u0939\u0948, \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932 \u092d\u0940 \u0938\u0902\u092d\u0935 \u0939\u0948 \u091c\u094b \u0926\u094b \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0947 \u0917\u0941\u0923\u0928 \u0938\u0947 \u090f\u0915 \u0905\u0928\u094d\u092f \u0938\u0926\u093f\u0936 \u0926\u0947\u0924\u093e \u0939\u0948\u0964 \u092f\u0939 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> (\u0914\u0930 \u0938\u0902\u092d\u0935\u0924\u0903 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^7<\/span>, \u091c\u093f\u0938\u0915\u093e \u092e\u093e\u092e\u0932\u093e \u0939\u092e \u092f\u0939\u093e\u0901 \u0928\u0939\u0940\u0902 \u0926\u0947\u0916\u0947\u0902\u0917\u0947) \u0915\u093e \u090f\u0915 \u0935\u093f\u0936\u093f\u0937\u094d\u091f \u0917\u0941\u0923\u0928 \u0939\u0948\u0964 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u0915\u0947 \u0906\u0927\u093e\u0930 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u094b <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}, \\hat{y}, \\hat{z}<\/span> \u092f\u093e <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{\\imath}, \\hat{\\jmath}, \\hat{k}<\/span> \u0905\u0915\u094d\u0937\u0930\u094b\u0902 \u0926\u094d\u0935\u093e\u0930\u093e \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0928\u092e\u0947\u0902 \u0938\u0947 \u0915\u093f\u0938\u0940 \u090f\u0915 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0935\u094d\u092f\u0915\u094d\u0924\u093f\u0917\u0924 \u092a\u0938\u0902\u0926 \u092a\u0930 \u0928\u093f\u0930\u094d\u092d\u0930 \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/p>\n<p style=\"text-align:center;\"><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>\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u092f\u0926\u093f \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 <span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c)<\/span> \u0915\u0947 \u0930\u0942\u092a \u0915\u093e \u090f\u0915 \u0938\u0926\u093f\u0936 \u0939\u0948, \u0924\u094b \u0907\u0938\u0947 \u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u0930\u0942\u092a \u092e\u0947\u0902 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0930\u0940\u0915\u0947 \u0938\u0947 \u0932\u093f\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c) = a\\hat{x} + b\\hat{y} + c\\hat{z}<\/span>\n<h3><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u092e\u0947\u0902 \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=330s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092e\u093e\u0928 \u0932\u0947\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u0915\u0947 \u0938\u0926\u093f\u0936 \u0939\u0948\u0902\u0964<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0915\u093e <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0915\u0947 \u0938\u093e\u0925 \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y}<\/span> \u0928\u093f\u092e\u094d\u0928 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><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>\u0932\u093e\u0917\u094d\u0930\u093e\u0902\u091c \u0915\u0940 \u092a\u0939\u091a\u093e\u0928<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=1399s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u0915\u0947 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0947 \u092e\u093e\u092e\u0932\u0947 \u092e\u0947\u0902<\/span><\/strong><\/a> \u0939\u092e \u0924\u0940\u0928 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u00ab\u0917\u0941\u0923\u0928\u00bb \u092a\u0939\u091a\u093e\u0928 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902: \u0905\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y},<\/span> \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y},<\/span> \u0914\u0930 \u092e\u093e\u0928\u0915\u094b\u0902 \u0915\u093e <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|\\|\\vec{y}\\|.<\/span> \u092f\u0947 \u0924\u0940\u0928\u094b\u0902 \u0917\u0941\u0923\u0928 \u0906\u092a\u0938 \u092e\u0947\u0902 \u0932\u093e\u0917\u094d\u0930\u093e\u0902\u091c \u0915\u0940 \u092a\u0939\u091a\u093e\u0928 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u091c\u0941\u0921\u093c\u0947 \u0939\u0941\u090f \u0939\u0948\u0902<\/p>\n<p style=\"text-align:center;\"><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>\u0932\u093e\u0917\u094d\u0930\u093e\u0902\u091c \u0915\u0940 \u092a\u0939\u091a\u093e\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u093e\u0923<\/h4>\n<p>\u092e\u093e\u0928 \u0932\u0947\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u0915\u0947 \u0938\u0926\u093f\u0936 \u0939\u0948\u0902, \u0924\u092c \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><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>\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930:<\/p>\n<p style=\"text-align:center;\"><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>\u0926\u0942\u0938\u0930\u0940 \u0913\u0930:<\/p>\n<p style=\"text-align:center;\"><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>\u0905\u0902\u0924\u0924\u0903, \u0930\u0902\u0917\u094b\u0902 \u092e\u0947\u0902 \u0926\u0940 \u0917\u0908 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f\u092f\u094b\u0902 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0930\u0915\u0947 \u0935\u0939\u0940 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u093f\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0925\u093e\u0964<\/p>\n<h3>\u0915\u094d\u0930\u0949\u0938 \u0917\u0941\u0923\u0928\u092b\u0932 \u0914\u0930 \u0938\u0926\u093f\u0936\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0915\u093e \u0915\u094b\u0923<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=1954s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092a\u0939\u0932\u0947 \u0939\u092e\u0928\u0947 \u0926\u0947\u0916\u093e \u0915\u093f \u090f\u0915 \u0918\u0928\u093f\u0937\u094d\u0920 \u0938\u0902\u092c\u0902\u0927 \u092e\u094c\u091c\u0942\u0926 \u0939\u0948<\/span><\/strong><\/a> \u0926\u094b \u0938\u0926\u093f\u0936\u094b\u0902 \u0926\u094d\u0935\u093e\u0930\u093e \u092c\u0928\u093e\u090f \u0917\u090f \u0915\u094b\u0923 \u0914\u0930 \u0905\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932 \u0915\u0947 \u092a\u0930\u093f\u0923\u093e\u092e \u0915\u0947 \u092c\u0940\u091a, \u091c\u094b \u0907\u0938 \u0938\u0902\u092c\u0902\u0927 \u0926\u094d\u0935\u093e\u0930\u093e \u0926\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = \\|\\vec{x}\\|\\|\\vec{y}\\|\\cos(\\angle(\\vec{x},\\vec{y})).<\/span> \u092a\u0924\u093e \u091a\u0932\u0924\u093e \u0939\u0948 \u0915\u093f \u0915\u0941\u091b \u0907\u0938\u0940 \u0924\u0930\u0939 \u0915\u0940 \u092c\u093e\u0924 \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932 \u0915\u0947 \u0938\u093e\u0925 \u0939\u094b\u0924\u0940 \u0939\u0948 \u0914\u0930 \u092f\u0939 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927 \u0926\u094d\u0935\u093e\u0930\u093e \u0926\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><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>\u092f\u0939 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0932\u093e\u0917\u094d\u0930\u093e\u0902\u091c \u0915\u0940 \u092a\u0939\u091a\u093e\u0928 \u0915\u093e \u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u092a\u0930\u093f\u0923\u093e\u092e \u0939\u0948 \u091c\u093f\u0938\u0947 \u090a\u092a\u0930 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0925\u093e, \u092a\u094d\u0930\u092e\u093e\u0923 \u0939\u092e\u0947\u0902 \u0932\u0917\u092d\u0917 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u092e\u093f\u0932\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><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>\u0905\u0902\u0924\u0924\u0903, \u092e\u0942\u0932 \u0928\u093f\u0915\u093e\u0932\u0924\u0947 \u0939\u0941\u090f \u0939\u092e \u092a\u0939\u0941\u0901\u091a\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align:center;\"><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>\u0932\u0947\u0915\u093f\u0928 \u092f\u093e\u0926 \u0930\u0916\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in[0,\\pi],<\/span> \u0914\u0930 \u0909\u0938 \u092e\u093e\u0928 \u0915\u0947 \u0926\u093e\u092f\u0930\u0947 \u092e\u0947\u0902 \u0938\u093e\u0907\u0928 \u092b\u0932\u0928 \u0938\u0926\u0948\u0935 \u0917\u0948\u0930-\u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0939\u094b\u0924\u093e \u0939\u0948, \u0907\u0938\u0932\u093f\u090f \u0939\u092e \u092a\u0930\u093f\u092e\u093e\u0923 \u091a\u093f\u0939\u094d\u0928 \u0939\u091f\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u0909\u0938 \u092a\u0930\u093f\u0923\u093e\u092e \u092a\u0930 \u092a\u0939\u0941\u0901\u091a\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0925\u093e\u0964<\/p>\n<p>\u0907\u0938 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0938\u0947 \u0939\u092e \u092f\u0939 \u0905\u0902\u0924\u0930\u094d\u091c\u094d\u091e\u093e\u0928 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0915\u094d\u0930\u093f\u092f\u093e <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\|<\/span> \u0915\u093e \u092a\u0930\u093f\u0923\u093e\u092e \u0939\u092e\u0947\u0902 \u0938\u0926\u093f\u0936 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u0926\u094d\u0935\u093e\u0930\u093e \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0926\u0947\u0924\u093e \u0939\u0948\u0964<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0906\u0930 \u092c\u0940 \u092e\u0947\u0902 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0914\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923, \u092e\u0947\u0902 \u0938\u0926\u093f\u0936 \u0917\u0941\u0923\u0928\u092b\u0932 \u0938\u093e\u0930\u093e\u0902\u0936:\u092f\u0939 \u0936\u094d\u0930\u0943\u0902\u0916\u0932\u093e n \u0906\u092f\u093e\u092e\u094b\u0902 \u0935\u093e\u0932\u0947 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u0940\u092f \u0938\u094d\u0925\u093e\u0928 \u092a\u0930 \u0936\u094d\u0930\u0943\u0902\u0916\u0932\u093e \u0915\u0940 \u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0928\u093f\u0930\u0902\u0924\u0930\u0924\u093e \u0939\u0948\u0964 \u092f\u0939\u093e\u0901 \u0939\u092e \u0930\u0948\u0916\u093f\u0915 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u0940 \u0915\u0941\u091b \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0947\u0902\u0917\u0947 \u091c\u094b n-\u0906\u092f\u093e\u092e\u0940 \u092f\u0942\u0915\u094d\u0932\u093f\u0921\u0940\u092f \u0938\u094d\u0925\u093e\u0928 \u0915\u094b \u092c\u0947\u0939\u0924\u0930 \u0938\u092e\u091d\u0928\u0947 \u092e\u0947\u0902 \u092e\u0926\u0926 \u0915\u0930\u0924\u0940 \u0939\u0948\u0902, \u0939\u092e \u090f\u0915 \u0938\u0926\u093f\u0936 \u0915\u093e \u0926\u0942\u0938\u0930\u0947 \u092a\u0930 \u092a\u094d\u0930\u0915\u094d\u0937\u0947\u092a\u0923 \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0947\u0902\u0917\u0947, [&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":9,"footnotes":""},"categories":[577,1124],"tags":[],"class_list":["post-34257","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-577","category-1124"],"yoast_head":"<!-- This site is optimized 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