{"id":34253,"date":"2022-03-29T13:00:57","date_gmt":"2022-03-29T13:00:57","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34253"},"modified":"2025-08-27T21:26:07","modified_gmt":"2025-08-27T21:26:07","slug":"rn%e4%b8%ad%e7%9a%84%e4%bb%a3%e6%95%b0%e4%b8%8e%e6%8a%95%e5%bd%b1%ef%bc%8cr3%e4%b8%ad%e7%9a%84%e5%90%91%e9%87%8f%e7%a7%af","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/rn%e4%b8%ad%e7%9a%84%e4%bb%a3%e6%95%b0%e4%b8%8e%e6%8a%95%e5%bd%b1%ef%bc%8cr3%e4%b8%ad%e7%9a%84%e5%90%91%e9%87%8f%e7%a7%af\/","title":{"rendered":"Rn\u4e2d\u7684\u4ee3\u6570\u4e0e\u6295\u5f71\uff0cR3\u4e2d\u7684\u5411\u91cf\u79ef"},"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>Rn\u4e2d\u7684\u4ee3\u6570\u4e0e\u6295\u5f71\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">{\\mathbb{R}^3}<\/span>\u4e2d\u7684\u5411\u91cf\u79ef<\/h1>\n<p style=\"text-align:center;\"><em><strong>\u6458\u8981\uff1a<\/strong><\/br>\u672c\u7cfb\u5217\u662fn\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u7cfb\u5217\u7684\u76f4\u63a5\u5ef6\u7eed\u3002\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u5c06\u56de\u987e\u4e00\u4e9b\u7ebf\u6027\u4ee3\u6570\u7684\u6982\u5ff5\uff0c\u4ee5\u5e2e\u52a9\u66f4\u597d\u5730\u7406\u89e3n\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\uff1b\u6211\u4eec\u5c06\u56de\u987e\u4e00\u4e2a\u5411\u91cf\u5728\u53e6\u4e00\u4e2a\u5411\u91cf\u4e0a\u7684\u6295\u5f71\u6982\u5ff5\uff0c\u8bc1\u660e\u52fe\u80a1\u5b9a\u7406\uff0c\u5e76\u6700\u540e\u56de\u987e<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span>\u4e2d\u7684\u5411\u91cf\u79ef\u53ca\u5176\u4e0e\u4e09\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d\u5176\u4ed6\u79ef\u7684\u5173\u7cfb\u3002<\/p>\n<p style=\"text-align:center;\"><strong>\u76ee\u5f55<\/strong><br \/>\n<a href=\"#Independencia-Lineal-Ortogonalidad-y-Proyecciones\">\u7ebf\u6027\u65e0\u5173\u3001\u6b63\u4ea4\u4e0e\u6295\u5f71<\/a><br \/>\n<a href=\"#El-Teorema-de-Pitagoras-y-la-Proyecci\u00f3n-sobre-un-Subespacio\">\u52fe\u80a1\u5b9a\u7406\u4e0e\u5b50\u7a7a\u95f4\u4e0a\u7684\u6295\u5f71<\/a><br \/>\n<a href=\"#El-Producto-Escalar-y-Vectorial-en-R3\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span>\u4e2d\u7684\u70b9\u79ef\u4e0e\u53c9\u79ef<\/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>\u7ebf\u6027\u65e0\u5173\u3001\u6b63\u4ea4\u4e0e\u6295\u5f71<\/h2>\n<h3>\u7ebf\u6027\u7ec4\u5408\u4e0e\u7ebf\u6027\u65e0\u5173<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=138s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u4e00\u4e2a\u975e\u96f6\u5411\u91cf<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> \u53ef\u4ee5\u76f8\u5bf9\u4e8e\u5176\u4ed6\u975e\u96f6\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u6784\u9020\u4e3a\u4e00\u4e2a<strong>\u7ebf\u6027\u7ec4\u5408<\/strong>\uff0c\u5982\u679c\u5b58\u5728\u4e00\u5bf9\u5b9e\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>\uff0c\u4e14\u4e0d\u540c\u65f6\u4e3a\u96f6\uff0c\u4f7f\u5f97\uff1a<\/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>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> \u53ef\u4ee5\u6784\u9020\u4e3a\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u7684\u52a0\u6743\u548c\u3002<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=609s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u7c7b\u4f3c\u5730\uff0c\u6211\u4eec\u8bf4<\/span><\/strong><\/a> \u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u662f<strong>\u7ebf\u6027\u65e0\u5173<\/strong>\u7684\uff0c\u5982\u679c <\/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>\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u7684\u7ebf\u6027\u65e0\u5173\u6027\u544a\u8bc9\u6211\u4eec\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u4e0d\u80fd\u8868\u793a\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u7684\u975e\u96f6\u6807\u91cf\u500d\u6570\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002<\/p>\n<p>\u6211\u4eec\u521a\u521a\u56de\u987e\u7684\u7ebf\u6027\u65e0\u5173\u6982\u5ff5\u53ef\u4ee5\u6269\u5c55\u5230\u66f4\u5927\u7684\u5411\u91cf\u96c6\u5408\u3002\u975e\u96f6\u5411\u91cf\u96c6\u5408 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x}_1, \\cdots, \\vec{x}_n\\}<\/span> \u88ab\u79f0\u4e3a\u7ebf\u6027\u65e0\u5173\uff0c\u5f53\u4e14\u4ec5\u5f53<\/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>\u4e24\u4e2a\u5411\u91cf\u6240\u6210\u7684\u89d2\u4e0e\u6b63\u4ea4\u6027<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=1289s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5982\u679c\u6211\u4eec\u56de\u5fc6\u67ef\u897f-\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f\uff0c<\/span><\/strong><\/a> \u5b83\u544a\u8bc9\u6211\u4eec <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> \u8003\u8651\u5230\u8fd9\u4e00\u70b9\uff0c\u5f88\u5bb9\u6613\u9a8c\u8bc1\uff0c\u5bf9\u4e8e\u4efb\u610f\u4e00\u5bf9\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span>\uff0c\u90fd\u6709\u4ee5\u4e0b\u5173\u7cfb\u6210\u7acb\uff1a<\/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>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u76f4\u89c2\u5730\u770b\u5230\u70b9\u79ef\u4e0e\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u6240\u6210\u89d2\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u56e0\u4e3a\u5b83\u4eec\u751f\u6210\u4e86\u4e00\u4e2a\u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span> \u7b49\u8ddd\u7684\u5e73\u9762\u3002\u56e0\u6b64\uff0c\u4e0d\u5931\u4e00\u822c\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b83\u4eec\u770b\u4f5c\u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span> \u4e2d\u7684\u5143\u7d20\uff0c\u76f8\u5bf9\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span> \u8f74\u7684\u89d2\u5ea6\u5206\u522b\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_y,<\/span> \u8fd9\u6837\u5411\u91cf\u53ef\u4ee5\u7528\u6781\u5750\u6807\u5f62\u5f0f\u8868\u793a\u4e3a\uff1a<\/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>\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5047\u8bbe\uff08\u518d\u6b21\u4e0d\u5931\u4e00\u822c\u6027\uff09<span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x \\lt \\theta_y,<\/span> \u7136\u540e\u8ba1\u7b97\u70b9\u79ef <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}.<\/span> \u8fd9\u6837\u6211\u4eec\u5f97\u5230\u4ee5\u4e0b\u7ed3\u679c\uff1a<\/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>\u73b0\u5728\uff0c\u53d6\u8f83\u5927\u89d2\u5ea6\u4f4d\u7f6e\u4e0e\u8f83\u5c0f\u89d2\u5ea6\u4f4d\u7f6e\u7684\u5dee\uff0c\u6211\u4eec\u5f97\u5230\u5411\u91cf\u4e4b\u95f4\u7684\u5939\u89d2\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})=\\theta_y - \\theta_x.<\/span> \u7531\u6b64\u6211\u4eec\u53ef\u4ee5\u5199\u4e3a\uff1a<\/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>\u8fd9\u91cc\u9700\u8981\u5f3a\u8c03\u7684\u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in [0, \\pi]<\/span>\n<p>\u7531\u6b64\u6211\u4eec\u53ef\u4ee5\u5c06\u67ef\u897f-\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f\u4e0e\u89d2\u5ea6\u51e0\u4f55\u8054\u7cfb\u8d77\u6765\uff0c\u5e76\u4e14\u8fd9\u8fd8\u4f7f\u6211\u4eec\u80fd\u591f\u5f97\u5230\u4e00\u4e2a\u4e25\u683c\u7684\u6b63\u4ea4\u6027\u6982\u5ff5\u3002\u5f53\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u5939\u89d2\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\pi\/2<\/span> \u5f27\u5ea6\u65f6\uff0c\u79f0\u5b83\u4eec<strong>\u6b63\u4ea4<\/strong>\uff0c\u5982\u524d\u4e00\u6bb5\u6240\u89e3\u91ca\u7684\u90a3\u6837\u3002\u8fd9\u7b49\u4ef7\u4e8e\u8bf4 <span class=\"katex-eq\" data-katex-display=\"false\">\\cos\\left(\\angle(\\vec{x},\\vec{y})\\right) = 0,<\/span> \u53c8\u7b49\u4ef7\u4e8e\u8bf4 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = 0.<\/span> \u56e0\u6b64\uff0c\u58f0\u79f0\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u6b63\u4ea4\u7b49\u4ef7\u4e8e\u8bf4 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0.<\/span>\n<h4>\u5982\u679c\u4e24\u4e2a\u975e\u96f6\u5411\u91cf\u6b63\u4ea4\uff0c\u5219\u5b83\u4eec\u7ebf\u6027\u65e0\u5173<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=2365s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u8fd9\u662f<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span>\u4e2d\u5411\u91cf\u7684\u4e00\u4e2a\u76f8\u5f53\u76f4\u89c2\u7684\u6027\u8d28<\/span><\/strong><\/a>\uff0c\u5176\u5f62\u5f0f\u5316\u8bc1\u660e\u5e76\u4e0d\u90a3\u4e48\u76f4\u63a5\uff0c\u540c\u65f6\u8fd9\u4e00\u6027\u8d28\u6709\u65f6\u4e5f\u4f1a\u5f15\u8d77\u4e00\u4e9b\u6df7\u6dc6\uff1a\u4e24\u4e2a\u5411\u91cf\u7684\u6b63\u4ea4\u6027\u8574\u542b\u5b83\u4eec\u7684\u7ebf\u6027\u65e0\u5173\u6027\uff0c\u4f46\u4e24\u4e2a\u5411\u91cf\u7684\u7ebf\u6027\u65e0\u5173\u6027\u4e0d\u4e00\u5b9a\u8574\u542b\u5b83\u4eec\u7684\u6b63\u4ea4\u6027\u3002\u8981\u770b\u540e\u8005\uff0c\u53ea\u9700\u4e00\u4e2a\u7b80\u5355\u7684\u53cd\u4f8b\uff1a<\/p>\n<p>\u5982\u679c\u6211\u4eec\u53d6\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}=(1,0)<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}=(1,1),<\/span> \u5b83\u4eec\u663e\u7136\u4e0d\u662f\u6b63\u4ea4\u7684\uff0c\u56e0\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}\\cdot\\vec{B}=1,<\/span> \u6211\u4eec\u4f1a\u53d1\u73b0\uff0c\u5982\u679c\u8fdb\u884c\u5982\u4e0b\u64cd\u4f5c\uff1a<\/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>\u90a3\u4e48\u6211\u4eec\u6709<\/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>\u56e0\u6b64\uff1a <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = 0  \\wedge \\beta=0.<\/span> \u7531\u6b64\u6211\u4eec\u5f97\u51fa\u7ed3\u8bba\uff1a<\/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>\u8fd9\u7b49\u4ef7\u4e8e\u8bf4 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}<\/span> \u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002\u7531\u6b64\u975e\u5e38\u6e05\u695a\u5730\u8868\u660e\uff0c\u7ebf\u6027\u65e0\u5173\u5e76\u4e0d\u610f\u5473\u7740\u6b63\u4ea4\u3002\u7136\u800c\uff0c\u6b63\u4ea4\u786e\u5b9e\u610f\u5473\u7740\u7ebf\u6027\u65e0\u5173\uff0c\u8fd9\u6b63\u662f\u6211\u5c06\u5728\u4e0b\u9762\u6b63\u5f0f\u8bc1\u660e\u7684\u3002\u4e3a\u6b64\u6211\u4eec\u8003\u8651\u4ee5\u4e0b\u524d\u63d0\u96c6\u5408\uff1a<\/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>\u57fa\u4e8e\u6b64\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u4ee5\u4e0b\u63a8\u7406\uff1a<\/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;{;\\;\u5047\u8bbe}\\\\ \\\\\n\n(2) &amp;\\mathcal{H}\\vdash \\vec{x}\\cdot\\vec{y}=0 &amp;{\\;\u5047\u8bbe} \\\\ \\\\\n\n(3) &amp;\\mathcal{H}\\vdash \\alpha\\vec{x} + \\beta\\vec{y} = \\vec{0} &amp;{\\;\u5047\u8bbe} \\\\ \\\\\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;{;\\; \u53cc\u7ebf\u6027\u6027} \\\\ \\\\\n\n(5) &amp;\\mathcal{H}\\vdash  \\alpha\\|\\vec{x}\\|^2 = 0 &amp; {;\\; \u7531(2,3,4)} \\\\ \\\\\n\n(6) &amp;\\mathcal{H}\\vdash  \\alpha  = 0 &amp; {;\\; \u7531(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; {;\\;\u53cc\u7ebf\u6027\u6027} \\\\ \\\\\n\n(8) &amp;\\mathcal{H}\\vdash \\beta\\|\\vec{y}\\|^2 = 0 &amp;{;\\;\u7531(2,3,7)} \\\\ \\\\\n\n(9) &amp;\\mathcal{H}\\vdash \\beta = 0 &amp;{;\\;\u7531(1,8)} \\\\ \\\\\n\n(10) &amp;\\mathcal{H}\\vdash \\alpha= 0 \\wedge \\beta = 0 &amp;{;\\;\\wedge-\u5f15\u5165(6,9)}\n\n\\end{array}<\/span>\n<p>\u7531\u6b64\u6211\u4eec\u5f97\u51fa\u7ed3\u8bba\uff1a<\/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>\u6700\u540e\uff0c\u5bf9\u4e0a\u8ff0\u8868\u8fbe\u5f0f\u5e94\u7528\u63a8\u7406\u5b9a\u7406\u53ef\u5f97\uff1a<\/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>\u5f97\u5230\u53cd\u65b9\u5411\u7bad\u5934\u7684\u8bc1\u660e\u662f\u5e73\u51e1\u7684\u3002<\/p>\n<p>\u4e5f\u5c31\u662f\u8bf4\uff1a\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u662f\u975e\u96f6\u4e14\u6b63\u4ea4\u7684\u5411\u91cf\uff0c\u90a3\u4e48\u5b83\u4eec\u5c31\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002<\/p>\n<h3>\u4e00\u4e2a\u5411\u91cf\u5728\u53e6\u4e00\u4e2a\u5411\u91cf\u4e0a\u7684\u6295\u5f71<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=3055s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5047\u8bbe\u6211\u4eec\u6709\u4e24\u4e2a\u975e\u96f6\u5411\u91cf<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span>\uff0c\u5b83\u4eec\u4e4b\u95f4\u5939\u89d2\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})<\/span>\uff0c\u6211\u4eec\u95ee\uff1a\u201c\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u5728\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u4e0a\u6709\u591a\u5c11\u5206\u91cf\uff1f\u201d \u6216\u8005 \u201c\u5f53\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u6295\u5f71\u5230\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u7684\u65b9\u5411\u4e0a\u65f6\uff0c\u5b83\u7684\u5f71\u5b50\u6709\u591a\u5927\uff1f\u201d \u8fd9\u4e2a\u95ee\u9898\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4e09\u89d2\u5b66\u6765\u89e3\u51b3\uff0c\u5e76\u7531\u6b64\u5b9a\u4e49\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u5728\u53e6\u4e00\u4e2a\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u4e0a\u7684\u6295\u5f71 <span class=\"katex-eq\" data-katex-display=\"false\">Proy_{\\vec{y}}(\\vec{x})<\/span>\uff0c\u8868\u8fbe\u5f0f\u4e3a\uff1a<\/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>\u5982\u679c\u6211\u4eec\u5c06\u5176\u4e0e\u524d\u9762\u6bb5\u843d\u4e2d\u7684\u7ed3\u679c\u7ed3\u5408\uff0c\u53ef\u4ee5\u5199\u4e3a\uff1a<\/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>\u56e0\u4e3a\u6211\u4eec\u8981\u8bb0\u4f4f\uff1a<\/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>\u6295\u5f71\u4e4b\u6240\u4ee5\u91cd\u8981\uff0c\u662f\u56e0\u4e3a\u5b83\u4eec\u5141\u8bb8\u6211\u4eec\u5c06\u5411\u91cf\u8868\u793a\u4e3a\u5728\u4efb\u610f\u57fa\u4e0b\u7684\u6295\u5f71\u4e4b\u548c\uff1a<\/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>\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{u}_i\\}_{i=1,\\cdots, n}<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u4e2d\u7ebf\u6027\u65e0\u5173\u5411\u91cf\u7684\u4e00\u7ec4\u57fa\uff0c\u800c\u7cfb\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_i = (\\vec{x}\\cdot\\vec{u}_i)\/\\|\\vec{u}_i\\|<\/span> \u6b63\u597d\u662f\u5411\u91cf\u5728\u8be5\u57fa\u7684\u6bcf\u4e2a\u5143\u7d20\u4e0a\u7684\u6295\u5f71\uff0c\u5e76\u6784\u6210\u4e86\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u76f8\u5bf9\u4e8e\u57fa <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{u}_i\\}_{i=1,\\cdots, n}<\/span> \u7684\u5750\u6807\u3002<\/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>\u52fe\u80a1\u5b9a\u7406\u4e0e\u5b50\u7a7a\u95f4\u4e0a\u7684\u6295\u5f71<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=254s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u52fe\u80a1\u5b9a\u7406\u662f\u4e00\u4e2a\u4f17\u6240\u5468\u77e5\u7684\u7ed3\u679c<\/span><\/strong><\/a>\uff0c\u5e76\u4e14\u6709\u65e0\u6570\u79cd\u8bc1\u660e\u65b9\u6cd5\u3002\u8be5\u5b9a\u7406\u7684\u4e00\u79cd\u53ef\u80fd\u8bc1\u660e\u6070\u597d\u6e90\u81ea\u6211\u4eec\u4e3a\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u6240\u53d1\u5c55\u7684\u5185\u5bb9\uff0c\u5e76\u4e14\u989d\u5916\u5177\u6709\u5bf9\u4efb\u610f\u7ef4\u6570\u90fd\u6210\u7acb\u7684\u7279\u70b9\u3002<\/p>\n<h3>\u8bc1\u660e\u52fe\u80a1\u5b9a\u7406<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=533s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5982\u679c\u6211\u4eec\u6709\u4e00\u4e2a\u76f4\u89d2\u4e09\u89d2\u5f62\uff0c\u4e24\u6761\u76f4\u89d2\u8fb9\u4e3a<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">b,<\/span> \u659c\u8fb9\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">c,<\/span> \u52fe\u80a1\u5b9a\u7406\u544a\u8bc9\u6211\u4eec <span class=\"katex-eq\" data-katex-display=\"false\">a^2+b^2=c^2.<\/span> \u5728\u7406\u89e3\u8fd9\u4e00\u70b9\u4e4b\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4e00\u5bf9\u6b63\u4ea4\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u6765\u8868\u793a\u6bcf\u6761\u76f4\u89d2\u8fb9\uff0c\u5e76\u5c06\u52fe\u80a1\u5b9a\u7406\u5199\u4e3a\uff1a<\/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>\u5176\u4e2d\u8868\u8fbe\u5f0f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\bot\\vec{y}<\/span> \u8868\u793a\u4e24\u4e2a\u5411\u91cf\u662f\u6b63\u4ea4\u7684\uff0c\u5373\u975e\u96f6\u4e14\u6ee1\u8db3 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0.<\/span> \u8fd9\u6837\uff0c\u5c31\u5efa\u7acb\u4e86\u5411\u91cf\u6b63\u4ea4\u6027\u4e0e\u4e24\u4e2a\u5411\u91cf\u6a21\u5e73\u65b9\u548c\u4e4b\u95f4\u7684\u53cc\u6761\u4ef6\u5173\u7cfb\u3002<\/p>\n<p>\u8fd9\u79cd\u4ee5\u5411\u91cf\u5f62\u5f0f\u8868\u793a\u7684\u52fe\u80a1\u5b9a\u7406\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u4e24\u79cd\u63a8\u7406\u6765\u8bc1\u660e\uff1a<\/p>\n<p><strong>\u9996\u5148\u6b63\u5411\u63a8\u7406\uff1a<\/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; {;\\;\u5047\u8bbe} \\\\ \\\\\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; {;\\;\u7531(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;;\\; \u6b27\u51e0\u91cc\u5f97\u8303\u6570\u4e0e\u70b9\u79ef\u7684\u6027\u8d28 &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; {;\\;\u7531(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; {;\\;\u63a8\u7406\u5b9a\u7406(4)} \\end{array}<\/span>\n<p><strong>\u73b0\u5728\u53cd\u5411\u63a8\u7406\uff1a<\/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; {;\\;\u5047\u8bbe} \\\\ \\\\\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;;\\; \u6b27\u51e0\u91cc\u5f97\u8303\u6570\u4e0e\u70b9\u79ef\u7684\u6027\u8d28 &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; {;\\;\u7531(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; {;\\;\u7531(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; {;\\;\u63a8\u7406\u5b9a\u7406(4)} \\end{array}<\/span>\n<p><strong>\u6700\u540e\uff0c\u5c06\u4e24\u4e2a\u63a8\u7406\u7ed3\u5408\u8d77\u6765\uff0c\u5c31\u5f97\u5230\u4e86\u60f3\u8981\u8bc1\u660e\u7684\u7ed3\u8bba\uff1a<\/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>\u4e2d\u4e00\u4e2a\u5411\u91cf\u5728\u5b50\u7a7a\u95f4\u4e0a\u7684\u6295\u5f71<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=1545s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u8003\u8651\u4e00\u4e2a\u5b50\u7a7a\u95f4<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span>\uff0c\u5b83\u7531<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span>\u4e2d\u7684\u4e00\u7ec4\u5355\u4f4d\u5411\u91cf\u57fa <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}<\/span> \u6784\u6210\u3002\u5982\u679c\u6211\u4eec\u53d6\u4e00\u4e2a\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\},<\/span> \u90a3\u4e48\u8be5\u5411\u91cf\u5728\u5b50\u7a7a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u4e0a\u7684\u6295\u5f71\u5b9a\u4e49\u4e3a\uff1a<\/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>\u4e00\u4e2a\u96c6\u5408\u662f\u6b63\u4ea4\u89c4\u8303\u7684\uff0c\u610f\u5473\u7740\u5176\u6240\u6709\u5143\u7d20\u5f7c\u6b64\u6b63\u4ea4\uff0c\u5e76\u4e14\u6bcf\u4e2a\u5143\u7d20\u7684\u8303\u6570\u7b49\u4e8e1\u3002<\/p>\n<p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u53ef\u4ee5\u628a\u5b83\u7406\u89e3\u4e3a\u5411\u91cf\u5728<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span>\u5b50\u7a7a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u7684\u5404\u4e2a\u5206\u91cf\u4e0a\u7684\u201c\u5f71\u5b50\u201d\u3002<\/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>\u4e2d\u4e00\u70b9\u6216\u5411\u91cf\u4e0e<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span>\u4e2d\u5b50\u7a7a\u95f4\u7684\u8ddd\u79bb<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=1974s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u4ece\u4e00\u4e2a\u5411\u91cf\u7684\u6295\u5f71\u51fa\u53d1<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a\u5982\u4e0b\u5f62\u5f0f\u7684\u5411\u91cf\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} - Proy_{H}(\\vec{x})<\/span>\n<p>\u8fd9\u6837\u5f62\u6210\u7684\u5411\u91cf\u662f\u8fde\u63a5\u5b50\u7a7a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u4e2d\u4e00\u70b9\u4e0e\u5750\u6807\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u7684\u70b9\u7684\u5411\u91cf\uff0c\u5e76\u4e14\u5b83\u5782\u76f4\u4e8e\u5b50\u7a7a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">H.<\/span> \u8fd9\u5e76\u4e0d\u96be\u8bc1\u660e\uff0c\u5982\u679c\u6211\u4eec\u53d6\u4efb\u610f\u4e00\u4e2a <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span> \u5e76\u8ba1\u7b97\u70b9\u79ef <span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z},<\/span> \u53ea\u9700\u770b\u5230\u6b64\u8fd0\u7b97\u7684\u7ed3\u679c\u4e3a\u96f6\u5373\u53ef\u3002\u6211\u4eec\u6765\u8ba1\u7b97\u4ee5\u9a8c\u8bc1\u8fd9\u4e00\u70b9\uff1a<\/p>\n<p>\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H,<\/span> \u90a3\u4e48\u5b83\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/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>\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_j\\}_{j=1}^k<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u7684\u4e00\u7ec4\u6b63\u4ea4\u89c4\u8303\u57fa\uff0c\u4e14 <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_j \\in\\mathbb{R}<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u4e2d\u7684\u7cfb\u6570\u3002\u8003\u8651\u5230\u8fd9\u4e00\u70b9\uff0c\u70b9\u79ef <span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z}<\/span> \u7684\u8ba1\u7b97\u5c06\u662f\uff1a<\/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>\u4f46\u662f\u7531\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u4e2d\u7684\u5411\u91cf\uff0c\u800c <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u662f\u5176\u5b50\u7a7a\u95f4\uff0c\u56e0\u6b64\u53ef\u4ee5\u627e\u5230 <span class=\"katex-eq\" data-katex-display=\"false\">n-k<\/span> \u4e2a\u5f7c\u6b64\u6b63\u4ea4\u89c4\u8303\u4e14\u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u4e2d\u6240\u6709\u5411\u91cf\u90fd\u6b63\u4ea4\u89c4\u8303\u7684\u5411\u91cf\uff0c\u8bb0\u4f5c <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_{k+1}, \\cdots, \\hat{v}_n\\},<\/span> \u8fd9\u6837\u5b83\u4eec\u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u7684\u57fa\u4e00\u8d77\u6784\u6210 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u7684\u4e00\u7ec4\u57fa\uff0c\u4ece\u800c\u53ef\u4ee5\u5199\u4e3a\uff1a<\/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>\u56e0\u6b64\uff0c\u4e0a\u9762\u7684\u63a8\u5bfc\u53ef\u4ee5\u7ee7\u7eed\u5982\u4e0b\uff1a<\/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>(*) \u4e3a\u96f6\u548c\uff0c\u56e0\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\{v_j\\}_{j=1}^n<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u7684\u4e00\u7ec4\u6b63\u4ea4\u89c4\u8303\u57fa\u3002<\/p>\n<p>\u7531\u6b64\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\u5b50\u7a7a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u4e0e\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u4e4b\u95f4\u7684\u8ddd\u79bb\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\|<\/span>\n<h4>\u8bc1\u660e<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=2995s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u4e3a\u4e86\u8bc1\u660e\u8fd9\u4e2a\u7ed3\u679c\uff0c\u5c06\u5c55\u793a<\/span><\/strong><\/a> \u5bf9\u6240\u6709 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span> \u603b\u662f\u6709 <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\| \\leq \\|\\vec{x} - \\vec{z}\\|,<\/span> \u4e3a\u6b64\u6211\u4eec\u5c06\u4ee5\u4e0b\u9762\u7684\u65b9\u5f0f\u4f7f\u7528\u52fe\u80a1\u5b9a\u7406\uff1a<\/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>\u8fd9\u4e2a\u6700\u540e\u7684\u7b49\u5f0f\u6210\u7acb\uff0c\u662f\u56e0\u4e3a\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} -Proy_{H}(\\vec{x})<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">Proy_{H}(\\vec{x}) - \\vec{z}<\/span> \u662f\u6b63\u4ea4\u7684\u3002\u56e0\u6b64\uff1a<\/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>\u8fd9\u6b63\u662f\u6211\u4eec\u60f3\u8981\u8bc1\u660e\u7684\u3002<\/p>\n<p>\u6709\u4e86\u8fd9\u4e2a\u7ed3\u679c\uff0c\u6211\u4eec\u53ef\u4ee5\u8bf4\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span> \u7684\u4e00\u70b9\u4e0e\u7531\u6b63\u4ea4\u89c4\u8303\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}<\/span> \u751f\u6210\u7684 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u5b50\u7a7a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u4e4b\u95f4\u7684\u8ddd\u79bb\u7531\u4e0b\u5f0f\u7ed9\u51fa\uff1a<\/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>\u4e2d\u7684\u70b9\u79ef\u4e0e\u53c9\u79ef<\/h2>\n<p><strong><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=242s\" rel=\"noopener\" target=\"_blank\"><span style=\"color: #ff0000;\">\u73b0\u5728\u6211\u4eec\u7a0d\u5fae\u6539\u53d8\u4e00\u4e0b\u5173\u6ce8\u70b9<\/span><\/a><\/strong>\uff0c\u96c6\u4e2d\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u4e2d\u7684\u5411\u91cf\u3002\u5728\u8fd9\u91cc\uff0c\u9664\u4e86\u6211\u4eec\u5df2\u7ecf\u9488\u5bf9 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u56de\u987e\u8fc7\u7684\u8fd0\u7b97\u5916\uff0c\u8fd8\u5b58\u5728\u5411\u91cf\u79ef\uff0c\u5b83\u7531\u4e24\u4e2a\u5411\u91cf\u7684\u4e58\u79ef\u4ea7\u751f\u53e6\u4e00\u4e2a\u5411\u91cf\u3002\u8fd9\u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u72ec\u6709\u7684\u8fd0\u7b97\uff08\u4e5f\u53ef\u80fd\u5b58\u5728\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^7<\/span>\uff0c\u4f46\u6b64\u5904\u4e0d\u4f5c\u5206\u6790\uff09\u3002\u901a\u5e38\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u7684\u6807\u51c6\u57fa\u5411\u91cf\u8868\u793a\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}, \\hat{y}, \\hat{z}<\/span>\uff0c\u6216\u8005 <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{\\imath}, \\hat{\\jmath}, \\hat{k}<\/span>\u3002\u9009\u62e9\u54ea\u79cd\u8868\u793a\u662f\u4e2a\u4eba\u504f\u597d\u3002<\/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>\u56e0\u6b64\uff0c\u5982\u679c\u6211\u4eec\u6709\u4e00\u4e2a\u5f62\u5982 <span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c)<\/span> \u7684\u5411\u91cf\uff0c\u5b83\u53ef\u4ee5\u7528\u4ee3\u6570\u5f62\u5f0f\u5199\u6210\uff1a<\/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>\u4e2d\u7684\u53c9\u79ef<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=330s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u8bbe <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u4e2d\u7684\u5411\u91cf\u3002<\/span><\/strong><\/a> \u5b9a\u4e49 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u7684\u53c9\u79ef <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y}<\/span> \u4e3a\uff1a<\/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>\u62c9\u683c\u6717\u65e5\u6052\u7b49\u5f0f<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=1399s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5bf9\u4e8e<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span>\u4e2d\u7684\u5411\u91cf<\/span><\/strong><\/a>\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc6\u522b\u51fa\u4e09\u7c7b\u201c\u79ef\u201d\uff1a\u70b9\u79ef <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y},<\/span> \u53c9\u79ef <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y},<\/span> \u4ee5\u53ca\u8303\u6570\u7684\u4e58\u79ef <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|\\|\\vec{y}\\|.<\/span> \u8fd9\u4e09\u79cd\u79ef\u901a\u8fc7\u62c9\u683c\u6717\u65e5\u6052\u7b49\u5f0f\u5f7c\u6b64\u8054\u7cfb\uff1a<\/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>\u62c9\u683c\u6717\u65e5\u6052\u7b49\u5f0f\u7684\u8bc1\u660e<\/h4>\n<p>\u8bbe <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u4e2d\u7684\u5411\u91cf\uff0c\u5219\u6709\uff1a<\/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>\u56e0\u6b64\uff1a<\/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>\u53e6\u4e00\u65b9\u9762\uff1a<\/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>\u6700\u540e\uff0c\u901a\u8fc7\u6bd4\u8f83\u5e26\u989c\u8272\u7684\u8868\u8fbe\u5f0f\uff0c\u5c31\u5f97\u5230\u4e86\u60f3\u8981\u8bc1\u660e\u7684\u7ed3\u8bba\u3002<\/p>\n<h3>\u53c9\u79ef\u4e0e\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=1954s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u4e4b\u524d\u6211\u4eec\u770b\u5230\u5b58\u5728\u4e00\u4e2a\u7d27\u5bc6\u5173\u7cfb<\/span><\/strong><\/a>\uff0c\u5373\u4e24\u4e2a\u5411\u91cf\u6240\u6210\u7684\u89d2\u4e0e\u70b9\u79ef\u7684\u7ed3\u679c\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u5b83\u7531\u516c\u5f0f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = \\|\\vec{x}\\|\\|\\vec{y}\\|\\cos(\\angle(\\vec{x},\\vec{y})).<\/span> \u7ed9\u51fa\u3002\u4e8b\u5b9e\u8bc1\u660e\uff0c\u5bf9\u4e8e\u53c9\u79ef\u4e5f\u5b58\u5728\u7c7b\u4f3c\u7684\u5173\u7cfb\uff0c\u516c\u5f0f\u4e3a\uff1a<\/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>\u8fd9\u4e2a\u8868\u8fbe\u5f0f\u662f\u4e0a\u9762\u8bc1\u660e\u8fc7\u7684\u62c9\u683c\u6717\u65e5\u6052\u7b49\u5f0f\u7684\u76f4\u63a5\u7ed3\u679c\uff0c\u8bc1\u660e\u8fc7\u7a0b\u5927\u81f4\u5982\u4e0b\uff1a<\/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>\u6700\u540e\uff0c\u53d6\u5e73\u65b9\u6839\u5f97\u5230\uff1a<\/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>\u4f46\u8bf7\u8bb0\u4f4f <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in[0,\\pi],<\/span> \u5728\u8be5\u533a\u95f4\u5185\u6b63\u5f26\u51fd\u6570\u59cb\u7ec8\u975e\u8d1f\uff0c\u56e0\u6b64\u53ef\u4ee5\u53bb\u6389\u7edd\u5bf9\u503c\u7b26\u53f7\uff0c\u4ece\u800c\u5f97\u5230\u60f3\u8981\u8bc1\u660e\u7684\u7ed3\u8bba\u3002<\/p>\n<p>\u7531\u8fd9\u4e2a\u8868\u8fbe\u5f0f\u6211\u4eec\u53ef\u4ee5\u76f4\u89c2\u5730\u770b\u51fa\uff0c\u8fd0\u7b97 <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\|<\/span> \u7684\u7ed3\u679c\u5c31\u662f\u7531\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u6240\u751f\u6210\u7684\u9762\u79ef\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Rn\u4e2d\u7684\u4ee3\u6570\u4e0e\u6295\u5f71\uff0c\u4e2d\u7684\u5411\u91cf\u79ef \u6458\u8981\uff1a\u672c\u7cfb\u5217\u662fn\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u7cfb\u5217\u7684\u76f4\u63a5\u5ef6\u7eed\u3002\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u5c06\u56de\u987e\u4e00\u4e9b\u7ebf\u6027\u4ee3\u6570\u7684\u6982\u5ff5\uff0c\u4ee5\u5e2e\u52a9\u66f4\u597d\u5730\u7406\u89e3n\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\uff1b\u6211\u4eec\u5c06\u56de\u987e\u4e00\u4e2a\u5411\u91cf\u5728\u53e6\u4e00\u4e2a\u5411\u91cf\u4e0a\u7684\u6295\u5f71\u6982\u5ff5\uff0c\u8bc1\u660e\u52fe\u80a1\u5b9a\u7406\uff0c\u5e76\u6700\u540e\u56de\u987e\u4e2d\u7684\u5411\u91cf\u79ef\u53ca\u5176\u4e0e\u4e09\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d\u5176\u4ed6\u79ef\u7684\u5173\u7cfb\u3002 \u76ee\u5f55 \u7ebf\u6027\u65e0\u5173\u3001\u6b63\u4ea4\u4e0e\u6295\u5f71 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