{"id":34277,"date":"2022-03-29T13:00:19","date_gmt":"2022-03-29T13:00:19","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34277"},"modified":"2025-08-27T22:20:25","modified_gmt":"2025-08-27T22:20:25","slug":"rn-%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e4%bb%a3%e6%95%b0%e5%ad%a6%e3%81%a8%e5%b0%84%e5%bd%b1%e3%80%81r3-%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e7%a9%8d","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ja\/rn-%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e4%bb%a3%e6%95%b0%e5%ad%a6%e3%81%a8%e5%b0%84%e5%bd%b1%e3%80%81r3-%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e7%a9%8d\/","title":{"rendered":"Rn \u306b\u304a\u3051\u308b\u4ee3\u6570\u5b66\u3068\u5c04\u5f71\u3001R3 \u306b\u304a\u3051\u308b\u30d9\u30af\u30c8\u30eb\u7a4d"},"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 \u306b\u304a\u3051\u308b\u4ee3\u6570\u5b66\u3068\u5c04\u5f71\u3001<span class=\"katex-eq\" data-katex-display=\"false\">{\\mathbb{R}^3}<\/span> \u306b\u304a\u3051\u308b\u30d9\u30af\u30c8\u30eb\u7a4d<\/h1>\n<p style=\"text-align:center;\"><em><strong>\u8981\u7d04:<\/strong><\/br>\u672c\u30b7\u30ea\u30fc\u30ba\u306f n \u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u95a2\u3059\u308b\u30b7\u30ea\u30fc\u30ba\u306e\u76f4\u63a5\u7684\u306a\u7d9a\u7de8\u3067\u3042\u308b\u3002\u3053\u3053\u3067\u306f n \u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u3092\u3088\u308a\u3088\u304f\u7406\u89e3\u3059\u308b\u305f\u3081\u306e\u7dda\u5f62\u4ee3\u6570\u5b66\u306e\u3044\u304f\u3064\u304b\u306e\u6982\u5ff5\u3092\u78ba\u8a8d\u3057\u3001\u3042\u308b\u30d9\u30af\u30c8\u30eb\u3092\u4ed6\u306e\u30d9\u30af\u30c8\u30eb\u306b\u5c04\u5f71\u3059\u308b\u6982\u5ff5\u3092\u898b\u76f4\u3057\u3001\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3092\u8a3c\u660e\u3057\u3001\u6700\u5f8c\u306b <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u306b\u304a\u3051\u308b\u30d9\u30af\u30c8\u30eb\u7a4d\u3068 3 \u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306e\u4ed6\u306e\u7a4d\u3068\u306e\u95a2\u4fc2\u3092\u78ba\u8a8d\u3059\u308b\u3002 <\/p>\n<p style=\"text-align:center;\"><strong>\u76ee\u6b21<\/strong><br \/>\n<a href=\"#Independencia-Lineal-Ortogonalidad-y-Proyecciones\">\u7dda\u5f62\u72ec\u7acb\u6027\u3001\u76f4\u4ea4\u6027\u304a\u3088\u3073\u5c04\u5f71<\/a><br \/>\n<a href=\"#El-Teorema-de-Pitagoras-y-la-Proyecci\u00f3n-sobre-un-Subespacio\">\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3068\u90e8\u5206\u7a7a\u9593\u3078\u306e\u5c04\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> \u306b\u304a\u3051\u308b\u5185\u7a4d\u3068\u5916\u7a4d<\/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>\u7dda\u5f62\u72ec\u7acb\u6027\u3001\u76f4\u4ea4\u6027\u304a\u3088\u3073\u5c04\u5f71<\/h2>\n<h3>\u7dda\u5f62\u7d50\u5408\u3068\u7dda\u5f62\u72ec\u7acb\u6027<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=138s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u96f6\u3067\u306a\u3044\u30d9\u30af\u30c8\u30eb<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> \u306f\u3001\u4ed6\u306e\u96f6\u3067\u306a\u3044\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u306b\u95a2\u3057\u3066 <strong>\u7dda\u5f62\u7d50\u5408<\/strong> \u3068\u3057\u3066\u69cb\u6210\u3067\u304d\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u540c\u6642\u306b\u96f6\u3067\u306f\u306a\u3044\u5b9f\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> \u304c\u5b58\u5728\u3057\u3066\u6b21\u3092\u6e80\u305f\u3059\u3068\u304d\u3067\u3042\u308b:<\/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>\u3064\u307e\u308a\u3001\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> \u306f\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u306e\u52a0\u91cd\u548c\u3068\u3057\u3066\u69cb\u7bc9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\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;\">\u540c\u69d8\u306b\u3001\u6b21\u306e\u3088\u3046\u306b\u8a00\u3046<\/span><\/strong><\/a> \u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u304c <strong>\u7dda\u5f62\u72ec\u7acb<\/strong> \u3067\u3042\u308b\u3068\u306f\u3001 <\/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>\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u306e\u7dda\u5f62\u72ec\u7acb\u6027\u3068\u306f\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u304c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u306e\uff08\u96f6\u3067\u306a\u3044\uff09\u30b9\u30ab\u30e9\u30fc\u500d\u3067\u5f97\u3089\u308c\u306a\u3044\u3053\u3068\u3001\u307e\u305f\u305d\u306e\u9006\u3082\u6210\u308a\u7acb\u305f\u306a\u3044\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002<\/p>\n<p>\u5148\u307b\u3069\u78ba\u8a8d\u3057\u305f\u7dda\u5f62\u72ec\u7acb\u6027\u306e\u6982\u5ff5\u306f\u3001\u3088\u308a\u5927\u304d\u306a\u30d9\u30af\u30c8\u30eb\u96c6\u5408\u306b\u62e1\u5f35\u3067\u304d\u308b\u3002\u96f6\u3067\u306a\u3044\u30d9\u30af\u30c8\u30eb\u306e\u96c6\u5408 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x}_1, \\cdots, \\vec{x}_n\\}<\/span> \u306f\u3001\u6b21\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u7dda\u5f62\u72ec\u7acb\u3067\u3042\u308b\u3068\u3044\u3046:<\/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>\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u6210\u3059\u89d2\u3068\u76f4\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;\">\u30b3\u30fc\u30b7\u30fc\u2013\u30b7\u30e5\u30ef\u30eb\u30c4\u306e\u4e0d\u7b49\u5f0f\u3092\u601d\u3044\u51fa\u3059\u3068\u3001<\/span><\/strong><\/a> \u3053\u308c\u306f <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> \u3092\u610f\u5473\u3059\u308b\u3002\u3053\u308c\u3092\u8003\u616e\u3059\u308b\u3068\u3001\u4efb\u610f\u306e <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span> \u306b\u5bfe\u3057\u3066\u6b21\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u5bb9\u6613\u306b\u78ba\u8a8d\u3067\u304d\u308b:<\/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>\u3053\u3053\u304b\u3089\u3001\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u304c\u6210\u3059\u89d2\u3068\u5185\u7a4d\u3068\u306e\u95a2\u4fc2\u3092\u76f4\u611f\u3067\u304d\u308b\u3002\u306a\u305c\u306a\u3089\u3001\u3053\u308c\u3089\u306f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span> \u306b\u540c\u578b\u306a\u5e73\u9762\u3092\u751f\u6210\u3059\u308b\u304b\u3089\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\u3001\u3053\u308c\u3089\u3092 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span> \u306e\u8981\u7d20\u3068\u3057\u3066\u8003\u3048\u3001\u8ef8 <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span> \u306b\u5bfe\u3059\u308b\u89d2\u5ea6\u3092\u305d\u308c\u305e\u308c <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_y<\/span> \u3068\u3059\u308c\u3070\u3001\u30d9\u30af\u30c8\u30eb\u306f\u6975\u5f62\u5f0f\u3067\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b:<\/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>\u3053\u306e\u3088\u3046\u306b\uff08\u518d\u3073\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\uff09<span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x \\lt \\theta_y<\/span> \u3068\u4eee\u5b9a\u3059\u308c\u3070\u3001\u5185\u7a4d <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}<\/span> \u3092\u8a08\u7b97\u3067\u304d\u308b\u3002\u3053\u308c\u306b\u3088\u308a\u6b21\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u308b:<\/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>\u3053\u3053\u3067\u3001\u5927\u304d\u3044\u89d2\u5ea6\u4f4d\u7f6e\u3068\u5c0f\u3055\u3044\u89d2\u5ea6\u4f4d\u7f6e\u306e\u5dee\u3092\u53d6\u308b\u3068\u3001\u30d9\u30af\u30c8\u30eb\u9593\u306e\u89d2\u5ea6 <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})=\\theta_y - \\theta_x.<\/span> \u3092\u5f97\u308b\u3002\u3053\u308c\u306b\u3088\u308a\u6b21\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b:<\/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>\u3053\u3053\u3067\u5f37\u8abf\u3059\u3079\u304d\u306f <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in [0, \\pi]<\/span> \u3067\u3042\u308b\u3002<\/p>\n<p>\u3053\u308c\u306b\u57fa\u3065\u3044\u3066\u3001\u30b3\u30fc\u30b7\u30fc\u2013\u30b7\u30e5\u30ef\u30eb\u30c4\u306e\u4e0d\u7b49\u5f0f\u3092\u89d2\u5ea6\u306e\u5e7e\u4f55\u5b66\u3068\u7d50\u3073\u4ed8\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u3001\u3055\u3089\u306b\u76f4\u4ea4\u6027\u306e\u53b3\u5bc6\u306a\u6982\u5ff5\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c <strong>\u76f4\u4ea4<\/strong> \u3067\u3042\u308b\u3068\u306f\u3001\u524d\u6bb5\u3067\u8aac\u660e\u3057\u305f\u610f\u5473\u3067\u4e21\u8005\u304c <span class=\"katex-eq\" data-katex-display=\"false\">\\pi\/2<\/span> \u30e9\u30b8\u30a2\u30f3\u306e\u89d2\u5ea6\u3092\u306a\u3059\u3068\u304d\u3067\u3042\u308b\u3002\u3053\u308c\u306f <span class=\"katex-eq\" data-katex-display=\"false\">\\cos\\left(\\angle(\\vec{x},\\vec{y})\\right) = 0<\/span> \u3068\u8a00\u3046\u3053\u3068\u3068\u540c\u5024\u3067\u3042\u308a\u3001\u3055\u3089\u306b <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = 0<\/span> \u3068\u8a00\u3046\u3053\u3068\u3068\u3082\u540c\u5024\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u306e\u76f4\u4ea4\u6027\u3092\u4e3b\u5f35\u3059\u308b\u3053\u3068\u306f\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0<\/span> \u3068\u8a00\u3046\u3053\u3068\u3068\u540c\u3058\u610f\u5473\u3092\u6301\u3064\u3002<\/p>\n<h4>\u4e8c\u3064\u306e\u96f6\u3067\u306a\u3044\u30d9\u30af\u30c8\u30eb\u304c\u76f4\u4ea4\u3059\u308b\u306a\u3089\u3001\u305d\u308c\u3089\u306f\u7dda\u5f62\u72ec\u7acb\u3067\u3042\u308b<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=2365s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u3053\u308c\u306f\u30d9\u30af\u30c8\u30eb\u306b\u95a2\u3059\u308b\u3042\u308b\u7a0b\u5ea6\u76f4\u611f\u7684\u306a\u6027\u8cea\u3067\u3042\u308b<\/span><\/strong><\/a> \u304c\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306b\u304a\u3044\u3066\u305d\u306e\u5f62\u5f0f\u7684\u306a\u8a3c\u660e\u306f\u305d\u308c\u307b\u3069\u76f4\u63a5\u7684\u3067\u306f\u306a\u3044\u3002\u307e\u305f\u3001\u3053\u306e\u6027\u8cea\u306f\u6642\u306b\u6df7\u4e71\u3092\u62db\u304f\u3053\u3068\u304c\u3042\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u76f4\u4ea4\u6027\u306f\u305d\u308c\u3089\u306e\u7dda\u5f62\u72ec\u7acb\u6027\u3092\u610f\u5473\u3059\u308b\u304c\u3001\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u7dda\u5f62\u72ec\u7acb\u6027\u306f\u5fc5\u305a\u3057\u3082\u76f4\u4ea4\u6027\u3092\u610f\u5473\u3057\u306a\u3044\u3002\u3053\u306e\u5f8c\u8005\u3092\u793a\u3059\u306b\u306f\u5358\u7d14\u306a\u53cd\u4f8b\u3067\u5341\u5206\u3067\u3042\u308b:<\/p>\n<p>\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}=(1,0)<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}=(1,1)<\/span> \u3092\u8003\u3048\u308b\u3068\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}\\cdot\\vec{B}=1<\/span> \u3067\u3042\u308b\u305f\u3081\u660e\u3089\u304b\u306b\u76f4\u4ea4\u3057\u3066\u3044\u306a\u3044\u3002\u3057\u304b\u3057\u6b21\u306e\u3088\u3046\u306b\u3059\u308b\u3068:<\/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>\u3059\u308b\u3068\u6b21\u304c\u6210\u308a\u7acb\u3064:<\/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>\u3057\u305f\u304c\u3063\u3066 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = 0  \\wedge \\beta=0<\/span> \u3068\u306a\u308b\u3002\u3053\u308c\u306b\u3088\u308a\u6b21\u306e\u7d50\u8ad6\u304c\u5f97\u3089\u308c\u308b:<\/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>\u3053\u308c\u306f\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}<\/span> \u304c\u7dda\u5f62\u72ec\u7acb\u3067\u3042\u308b\u3068\u8a00\u3046\u3053\u3068\u3068\u540c\u5024\u3067\u3042\u308b\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u7dda\u5f62\u72ec\u7acb\u6027\u304c\u76f4\u4ea4\u6027\u3092\u610f\u5473\u3059\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u3053\u3068\u304c\u975e\u5e38\u306b\u660e\u78ba\u306b\u306a\u308b\u3002\u3057\u304b\u3057\u3001\u76f4\u4ea4\u6027\u306f\u7dda\u5f62\u72ec\u7acb\u6027\u3092\u610f\u5473\u3059\u308b\u306e\u3067\u3042\u308a\u3001\u4ee5\u4e0b\u3067\u305d\u306e\u5f62\u5f0f\u7684\u306a\u8a3c\u660e\u3092\u793a\u3059\u3002\u305d\u306e\u305f\u3081\u306b\u6b21\u306e\u524d\u63d0\u96c6\u5408\u3092\u8003\u3048\u308b:<\/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>\u3053\u308c\u306b\u57fa\u3065\u3044\u3066\u6b21\u306e\u63a8\u8ad6\u3092\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u308b:<\/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;{;\\;\u524d\u63d0}\\\\ \\\\\n\n(2) &amp;\\mathcal{H}\\vdash \\vec{x}\\cdot\\vec{y}=0 &amp;{\\;\u524d\u63d0} \\\\ \\\\\n\n(3) &amp;\\mathcal{H}\\vdash \\alpha\\vec{x} + \\beta\\vec{y} = \\vec{0} &amp;{\\;\u524d\u63d0} \\\\ \\\\\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\u7dda\u5f62\u6027} \\\\ \\\\\n\n(5) &amp;\\mathcal{H}\\vdash  \\alpha\\|\\vec{x}\\|^2 = 0 &amp; {;\\; (2,3,4)\u3088\u308a} \\\\ \\\\\n\n(6) &amp;\\mathcal{H}\\vdash  \\alpha  = 0 &amp; {;\\; (1,5)\u3088\u308a} \\\\ \\\\\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\u7dda\u5f62\u6027} \\\\ \\\\\n\n(8) &amp;\\mathcal{H}\\vdash \\beta\\|\\vec{y}\\|^2 = 0 &amp;{;\\;(2,3,7)\u3088\u308a} \\\\ \\\\\n\n(9) &amp;\\mathcal{H}\\vdash \\beta = 0 &amp;{;\\;(1,8)\u3088\u308a} \\\\ \\\\\n\n(10) &amp;\\mathcal{H}\\vdash \\alpha= 0 \\wedge \\beta = 0 &amp;{;\\;\\wedge-\u5c0e\u5165(6,9)}\n\n\\end{array}<\/span>\n<p>\u3053\u308c\u306b\u3088\u308a\u6b21\u306e\u7d50\u8ad6\u304c\u5f97\u3089\u308c\u308b:<\/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\u5f8c\u306b\u3001\u3053\u306e\u6700\u5f8c\u306e\u5f0f\u306b\u5bfe\u3057\u3066\u63a8\u8ad6\u5b9a\u7406\u3092\u9069\u7528\u3059\u308b\u3068\u6b21\u304c\u5f97\u3089\u308c\u308b:<\/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>\u9006\u65b9\u5411\u306e\u77e2\u5370\u3092\u5f97\u308b\u8a3c\u660e\u306f\u81ea\u660e\u3067\u3042\u308b\u3002<\/p>\n<p>\u3059\u306a\u308f\u3061\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u304c\u96f6\u3067\u306a\u3044\u30d9\u30af\u30c8\u30eb\u304b\u3064\u76f4\u4ea4\u3059\u308b\u306a\u3089\u3070\u3001\u305d\u308c\u3089\u306f\u7dda\u5f62\u72ec\u7acb\u3067\u3042\u308b\u3002<\/p>\n<h3>\u3042\u308b\u30d9\u30af\u30c8\u30eb\u306e\u4ed6\u306e\u30d9\u30af\u30c8\u30eb\u3078\u306e\u5c04\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;\">\u96f6\u3067\u306a\u3044\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u3092\u8003\u3048\u3088\u3046<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u304c\u89d2\u5ea6 <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})<\/span> \u3092\u306a\u3059\u3068\u304d\u3001\u300c\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u306f\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u4e0a\u306b\u3069\u306e\u7a0b\u5ea6\u5b58\u5728\u3057\u3066\u3044\u308b\u304b\uff1f\u300d\u3042\u308b\u3044\u306f\u300c\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u304c\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u306e\u65b9\u5411\u306b\u5c04\u5f71\u3055\u308c\u305f\u3068\u304d\u3001\u305d\u306e\u5f71\u306e\u5927\u304d\u3055\u306f\u3069\u308c\u304f\u3089\u3044\u304b\uff1f\u300d\u3068\u554f\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u306e\u554f\u3044\u306f\u4e09\u89d2\u6cd5\u3092\u901a\u3058\u3066\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u3001\u3053\u308c\u306b\u3088\u308a\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u306e\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u3078\u306e\u5c04\u5f71 <span class=\"katex-eq\" data-katex-display=\"false\">Proy_{\\vec{y}}(\\vec{x})<\/span> \u3092\u6b21\u306e\u5f0f\u3067\u5b9a\u7fa9\u3067\u304d\u308b:<\/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>\u3053\u308c\u3092\u524d\u6bb5\u3067\u898b\u305f\u3053\u3068\u3068\u7d44\u307f\u5408\u308f\u305b\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u66f8\u3051\u308b:<\/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>\u3053\u3053\u3067\u601d\u3044\u51fa\u3059\u3079\u304d\u306f<\/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>\u5c04\u5f71\u306f\u91cd\u8981\u3067\u3042\u308b\u3002\u306a\u305c\u306a\u3089\u3001\u305d\u308c\u306b\u3088\u3063\u3066\u4efb\u610f\u306e\u57fa\u5e95\u306b\u95a2\u3057\u3066\u30d9\u30af\u30c8\u30eb\u3092\u305d\u306e\u5c04\u5f71\u306e\u548c\u3068\u3057\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3089\u3067\u3042\u308b:<\/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>\u3053\u3053\u3067 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{u}_i\\}_{i=1,\\cdots, n}<\/span> \u306f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306b\u304a\u3051\u308b\u7dda\u5f62\u72ec\u7acb\u306a\u30d9\u30af\u30c8\u30eb\u306e\u57fa\u5e95\u3067\u3042\u308a\u3001\u4fc2\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_i = (\\vec{x}\\cdot\\vec{u}_i)\/\\|\\vec{u}_i\\|<\/span> \u306f\u307e\u3055\u306b\u5404\u57fa\u5e95\u8981\u7d20\u3078\u306e\u5c04\u5f71\u3067\u3042\u308a\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306e\u57fa\u5e95 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{u}_i\\}_{i=1,\\cdots, n}<\/span> \u306b\u95a2\u3059\u308b <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u306e\u5ea7\u6a19\u3092\u69cb\u6210\u3059\u308b\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>\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3068\u90e8\u5206\u7a7a\u9593\u3078\u306e\u5c04\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;\">\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u306f<\/span><\/strong><\/a> \u3088\u304f\u77e5\u3089\u308c\u305f\u7d50\u679c\u3067\u3042\u308a\u3001\u7121\u6570\u306e\u8a3c\u660e\u304c\u5b58\u5728\u3059\u308b\u3002\u3053\u306e\u5b9a\u7406\u306e\u4e00\u3064\u306e\u8a3c\u660e\u306f\u3001\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u95a2\u3057\u3066\u6211\u3005\u304c\u3053\u308c\u307e\u3067\u5c55\u958b\u3057\u3066\u304d\u305f\u4e8b\u67c4\u304b\u3089\u751f\u3058\u3001\u3055\u3089\u306b\u4efb\u610f\u306e\u6b21\u5143\u6570\u306b\u5bfe\u3057\u3066\u6709\u52b9\u3067\u3042\u308b\u3068\u3044\u3046\u8ffd\u52a0\u7684\u306a\u5229\u70b9\u3092\u6301\u3064\u3002<\/p>\n<h3>\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u306e\u8a3c\u660e<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=533s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u76f4\u89d2\u4e09\u89d2\u5f62\u306e\u4e8c\u3064\u306e\u76f4\u89d2\u8fba<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>\u3001\u304a\u3088\u3073\u659c\u8fba <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> \u3092\u8003\u3048\u308b\u3068\u3001\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u306f <span class=\"katex-eq\" data-katex-display=\"false\">a^2+b^2=c^2<\/span> \u3067\u3042\u308b\u3068\u8a00\u3046\u3002\u3053\u306e\u3053\u3068\u3092\u7406\u89e3\u3059\u308b\u3068\u3001\u305d\u308c\u305e\u308c\u306e\u76f4\u89d2\u8fba\u3092\u76f4\u4ea4\u3059\u308b\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u306b\u3088\u3063\u3066\u8868\u3057\u3001\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3092\u6b21\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b:<\/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>\u3053\u3053\u3067\u8868\u73fe <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\bot\\vec{y}<\/span> \u306f\u4e21\u30d9\u30af\u30c8\u30eb\u304c\u76f4\u4ea4\u3059\u308b\u3053\u3068\u3092\u793a\u3059\u3002\u3064\u307e\u308a\u3001\u96f6\u3067\u306a\u304f <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0<\/span> \u3092\u6e80\u305f\u3059\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u76f4\u4ea4\u6027\u3068\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u306e\u4e8c\u4e57\u306e\u548c\u3068\u306e\u9593\u306b\u53cc\u6761\u4ef6\u7684\u306a\u95a2\u4fc2\u304c\u78ba\u7acb\u3055\u308c\u308b\u3002<\/p>\n<p>\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3092\u30d9\u30af\u30c8\u30eb\u5f62\u5f0f\u3067\u8868\u73fe\u3059\u308b\u3053\u306e\u65b9\u6cd5\u306f\u3001\u6b21\u306e\u4e8c\u3064\u306e\u63a8\u8ad6\u306b\u3088\u3063\u3066\u8a3c\u660e\u3067\u304d\u308b:<\/p>\n<p><strong>\u307e\u305a\u9806\u65b9\u5411:<\/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; {;\\;\u524d\u63d0} \\\\ \\\\\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)\u3088\u308a} \\\\ \\\\\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;;\\; \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u30ce\u30eb\u30e0\u3068\u5185\u7a4d\u306e\u6027\u8cea &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)\u3088\u308a} \\\\ \\\\\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\u8ad6\u5b9a\u7406(4)} \\end{array}<\/span>\n<p><strong>\u6b21\u306b\u9006\u65b9\u5411:<\/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; {;\\;\u524d\u63d0} \\\\ \\\\\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;;\\; \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u30ce\u30eb\u30e0\u3068\u5185\u7a4d\u306e\u6027\u8cea &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)\u3088\u308a} \\\\ \\\\\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)\u3088\u308a} \\\\ \\\\\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\u8ad6\u5b9a\u7406(4)} \\end{array}<\/span>\n<p><strong>\u305d\u3057\u3066\u6700\u5f8c\u306b\u3001\u4e21\u65b9\u306e\u63a8\u8ad6\u3092\u5408\u308f\u305b\u308b\u3068\u3001\u793a\u3057\u305f\u304b\u3063\u305f\u3053\u3068\u304c\u5f97\u3089\u308c\u308b:<\/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> \u306b\u304a\u3051\u308b\u90e8\u5206\u7a7a\u9593\u3078\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5c04\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;\">\u90e8\u5206\u7a7a\u9593\u3092\u8003\u3048\u3088\u3046<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u3092\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306e\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u306e\u57fa\u5e95 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}<\/span> \u306b\u3088\u3063\u3066\u5f62\u6210\u3055\u308c\u308b\u3068\u3059\u308b\u3002\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span> \u3092\u53d6\u308b\u3068\u3001\u7a7a\u9593 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u3078\u306e\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u306e\u5c04\u5f71\u306f\u6b21\u306e\u5f0f\u3067\u5b9a\u7fa9\u3055\u308c\u308b:<\/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>\u96c6\u5408\u304c\u6b63\u898f\u76f4\u4ea4\u7cfb\u3067\u3042\u308b\u3068\u306f\u3001\u3059\u3079\u3066\u306e\u8981\u7d20\u304c\u4e92\u3044\u306b\u76f4\u4ea4\u3057\u3001\u305d\u308c\u305e\u308c\u306e\u30ce\u30eb\u30e0\u304c 1 \u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002<\/p>\n<p>\u8a00\u3044\u63db\u3048\u308c\u3070\u3001\u3053\u308c\u306f\u30d9\u30af\u30c8\u30eb\u304c <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306e\u90e8\u5206\u7a7a\u9593 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u306e\u5404\u6210\u5206\u306b\u843d\u3068\u3059\u5f71\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3042\u308b\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> \u306e\u70b9\u307e\u305f\u306f\u30d9\u30af\u30c8\u30eb\u3068\u90e8\u5206\u7a7a\u9593\u3068\u306e\u8ddd\u96e2<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=1974s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u30d9\u30af\u30c8\u30eb\u306e\u5c04\u5f71\u304b\u3089\u51fa\u767a\u3057\u3066<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span> \u3092 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306e\u90e8\u5206\u7a7a\u9593 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u306b\u5c04\u5f71\u3059\u308b\u3068\u3001\u6b21\u306e\u5f62\u306e\u30d9\u30af\u30c8\u30eb\u3092\u69cb\u6210\u3067\u304d\u308b:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} - Proy_{H}(\\vec{x})<\/span>\n<p>\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u4f5c\u3089\u308c\u305f\u30d9\u30af\u30c8\u30eb\u306f\u3001\u90e8\u5206\u7a7a\u9593 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u306e\u3042\u308b\u70b9\u3068\u5ea7\u6a19 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u306e\u70b9\u3092\u7d50\u3073\u3001\u90e8\u5206\u7a7a\u9593 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u306b\u76f4\u4ea4\u3057\u3066\u51fa\u308b\u30d9\u30af\u30c8\u30eb\u3068\u306a\u308b\u3002\u3053\u308c\u306f\u96e3\u3057\u304f\u306a\u304f\u8a3c\u660e\u3067\u304d\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u4efb\u610f\u306e <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span> \u3092\u53d6\u308a\u3001\u5185\u7a4d <span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z}<\/span> \u3092\u8a08\u7b97\u3059\u308c\u3070\u3001\u305d\u306e\u7d50\u679c\u304c\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308c\u3070\u3088\u3044\u3002\u5b9f\u969b\u306b\u8a08\u7b97\u3057\u3066\u307f\u3088\u3046:<\/p>\n<p>\u3082\u3057 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span> \u3067\u3042\u308c\u3070\u3001\u6b21\u306e\u5f62\u3067\u8868\u3055\u308c\u308b:<\/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>\u3053\u3053\u3067 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_j\\}_{j=1}^k<\/span> \u306f <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u306e\u6b63\u898f\u76f4\u4ea4\u57fa\u5e95\u3067\u3042\u308a\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\beta_j \\in\\mathbb{R}<\/span> \u306f <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u306b\u304a\u3051\u308b <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> \u306e\u4fc2\u6570\u3067\u3042\u308b\u3002\u3053\u308c\u3092\u8003\u616e\u3059\u308b\u3068\u3001\u5185\u7a4d <span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z}<\/span> \u306e\u8a08\u7b97\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b:<\/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>\u3057\u304b\u3057 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u306f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306e\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\u3001<span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u306f\u305d\u306e\u90e8\u5206\u7a7a\u9593\u306a\u306e\u3067\u3001\u4e92\u3044\u306b\u76f4\u4ea4\u3057\u3001\u304b\u3064 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u306e\u3059\u3079\u3066\u306e\u30d9\u30af\u30c8\u30eb\u306b\u76f4\u4ea4\u3059\u308b <span class=\"katex-eq\" data-katex-display=\"false\">n-k<\/span> \u500b\u306e\u6b63\u898f\u76f4\u4ea4\u30d9\u30af\u30c8\u30eb\u3001\u3059\u306a\u308f\u3061 <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_{k+1}, \\cdots, \\hat{v}_n\\}<\/span> \u3092\u898b\u3064\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u308c\u306b\u3088\u308a <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u306e\u57fa\u5e95\u3068\u4f75\u305b\u3066 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306e\u57fa\u5e95\u3092\u5f62\u6210\u3057\u3001\u6b21\u306e\u3088\u3046\u306b\u66f8\u3051\u308b:<\/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>\u3057\u305f\u304c\u3063\u3066\u3001\u4e0a\u306e\u5c55\u958b\u306f\u6b21\u306e\u3088\u3046\u306b\u7d9a\u304f:<\/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>(*) \u30bc\u30ed\u306b\u306a\u308b\u306e\u306f <span class=\"katex-eq\" data-katex-display=\"false\">\\{v_j\\}_{j=1}^n<\/span> \u304c <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306e\u6b63\u898f\u76f4\u4ea4\u57fa\u5e95\u3060\u304b\u3089\u3067\u3042\u308b\u3002<\/p>\n<p>\u3053\u308c\u306b\u57fa\u3065\u304d\u3001\u90e8\u5206\u7a7a\u9593 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u3068\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u306e\u8ddd\u96e2\u306f\u6b21\u3067\u4e0e\u3048\u3089\u308c\u308b\u3053\u3068\u3092\u793a\u305b\u308b:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\|<\/span>\n<h4>\u8a3c\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;\">\u3053\u306e\u7d50\u679c\u3092\u8a3c\u660e\u3059\u308b\u305f\u3081\u306b<\/span><\/strong><\/a> \u4efb\u610f\u306e <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span> \u306b\u5bfe\u3057\u3066\u5e38\u306b <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\| \\leq \\|\\vec{x} - \\vec{z}\\|<\/span> \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059\u3002\u305d\u306e\u305f\u3081\u306b\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3092\u6b21\u306e\u3088\u3046\u306b\u5229\u7528\u3059\u308b:<\/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>\u3053\u306e\u6700\u5f8c\u306e\u7b49\u5f0f\u306f\u3001\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} -Proy_{H}(\\vec{x})<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">Proy_{H}(\\vec{x}) - \\vec{z}<\/span> \u304c\u76f4\u4ea4\u3059\u308b\u305f\u3081\u306b\u5f97\u3089\u308c\u308b\u3002\u3057\u305f\u304c\u3063\u3066:<\/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>\u3053\u308c\u304c\u793a\u3057\u305f\u304b\u3063\u305f\u3053\u3068\u3067\u3042\u308b\u3002<\/p>\n<p>\u3053\u306e\u7d50\u679c\u3092\u5f97\u305f\u306e\u3067\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306e\u70b9 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span> \u3068\u3001\u6b63\u898f\u76f4\u4ea4\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}<\/span> \u306b\u3088\u3063\u3066\u751f\u6210\u3055\u308c\u308b <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u306e\u90e8\u5206\u7a7a\u9593 <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u3068\u306e\u8ddd\u96e2\u306f\u6b21\u3067\u4e0e\u3048\u3089\u308c\u308b\u3068\u8a00\u3048\u308b:<\/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> \u306b\u304a\u3051\u308b\u5185\u7a4d\u3068\u5916\u7a4d<\/h2>\n<p><strong><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=242s\" rel=\"noopener\" target=\"_blank\"><span style=\"color: #ff0000;\">\u3053\u3053\u3067\u5c11\u3057\u8996\u70b9\u3092\u5909\u3048\u3066<\/span><\/a><\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u306e\u30d9\u30af\u30c8\u30eb\u306b\u6ce8\u76ee\u3057\u3088\u3046\u3002\u3053\u3053\u3067\u306f\u3001\u3053\u308c\u307e\u3067\u4e00\u822c\u306b <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> \u3067\u898b\u3066\u304d\u305f\u6f14\u7b97\u306b\u52a0\u3048\u3066\u3001\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u7a4d\u304b\u3089\u5225\u306e\u30d9\u30af\u30c8\u30eb\u3092\u5f97\u308b\u5916\u7a4d\u304c\u53ef\u80fd\u3067\u3042\u308b\u3002\u3053\u308c\u306f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u7279\u6709\u306e\u6f14\u7b97\u3067\u3042\u308a\uff08\u305d\u3057\u3066\u304a\u305d\u3089\u304f <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^7<\/span> \u306b\u3082\u5b58\u5728\u3059\u308b\u304c\u3001\u3053\u3053\u3067\u306f\u305d\u306e\u5834\u5408\u306f\u6271\u308f\u306a\u3044\uff09\u3002\u901a\u5e38\u3001<span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u306e\u6a19\u6e96\u57fa\u5e95\u30d9\u30af\u30c8\u30eb\u306f <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}, \\hat{y}, \\hat{z}<\/span> \u307e\u305f\u306f <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{\\imath}, \\hat{\\jmath}, \\hat{k}<\/span> \u3068\u8868\u3055\u308c\u308b\u3002\u3069\u3061\u3089\u3092\u7528\u3044\u308b\u304b\u306f\u500b\u4eba\u306e\u597d\u307f\u306b\u3088\u308b\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>\u3057\u305f\u304c\u3063\u3066\u3001\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c)<\/span> \u304c\u3042\u308b\u5834\u5408\u3001\u305d\u308c\u306f\u4ee3\u6570\u7684\u306b\u6b21\u306e\u3088\u3046\u306b\u66f8\u3051\u308b:<\/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> \u306b\u304a\u3051\u308b\u5916\u7a4d<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=330s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span> \u304a\u3088\u3073 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span> \u3092 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u306e\u30d9\u30af\u30c8\u30eb\u3068\u3059\u308b\u3002<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u306e\u5916\u7a4d <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y}<\/span> \u306f\u6b21\u3067\u5b9a\u7fa9\u3055\u308c\u308b:<\/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>\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\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;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u306e\u30d9\u30af\u30c8\u30eb\u306e\u5834\u5408<\/span><\/strong><\/a> \u300c\u7a4d\u300d\u306b\u306f 3 \u7a2e\u985e\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u5185\u7a4d <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}<\/span>\u3001\u5916\u7a4d <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y}<\/span>\u3001\u304a\u3088\u3073\u30ce\u30eb\u30e0\u306e\u7a4d <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|\\|\\vec{y}\\|<\/span> \u3067\u3042\u308b\u3002\u3053\u308c\u3089\u4e09\u3064\u306e\u7a4d\u306f\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u6052\u7b49\u5f0f\u306b\u3088\u3063\u3066\u76f8\u4e92\u306b\u95a2\u4fc2\u3057\u3066\u3044\u308b:<\/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>\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u6052\u7b49\u5f0f\u306e\u8a3c\u660e<\/h4>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span> \u304a\u3088\u3073 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span> \u3092 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> \u306e\u30d9\u30af\u30c8\u30eb\u3068\u3059\u308b\u3068\u3001\u6b21\u304c\u6210\u308a\u7acb\u3064:<\/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>\u3057\u305f\u304c\u3063\u3066:<\/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>\u4e00\u65b9\u3067:<\/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\u5f8c\u306b\u3001\u8272\u4ed8\u304d\u3067\u793a\u3057\u305f\u5f0f\u3092\u6bd4\u8f03\u3059\u308b\u3053\u3068\u3067\u3001\u793a\u3057\u305f\u304b\u3063\u305f\u3053\u3068\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n<h3>\u5916\u7a4d\u3068\u30d9\u30af\u30c8\u30eb\u9593\u306e\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;\">\u4ee5\u524d\u898b\u305f\u3088\u3046\u306b\u3001\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u306a\u3059\u89d2\u3068\u5185\u7a4d\u306e\u7d50\u679c\u306b\u306f\u5bc6\u63a5\u306a\u95a2\u4fc2\u304c\u3042\u308b<\/span><\/strong><\/a>\u3002\u3053\u308c\u306f\u95a2\u4fc2\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> \u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u308b\u3002\u5b9f\u306f\u3001\u5916\u7a4d\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306e\u3053\u3068\u304c\u6210\u308a\u7acb\u3061\u3001\u6b21\u306e\u95a2\u4fc2\u5f0f\u3067\u8868\u3055\u308c\u308b:<\/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>\u3053\u306e\u5f0f\u306f\u3001\u5148\u306b\u8a3c\u660e\u3057\u305f\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u6052\u7b49\u5f0f\u304b\u3089\u76f4\u63a5\u5c0e\u304b\u308c\u308b\u7d50\u679c\u3067\u3042\u308a\u3001\u8a3c\u660e\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b:<\/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\u5f8c\u306b\u5e73\u65b9\u6839\u3092\u53d6\u308b\u3068\u6b21\u306b\u81f3\u308b:<\/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>\u305f\u3060\u3057 <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in[0,\\pi]<\/span> \u3067\u3042\u308a\u3001\u3053\u306e\u7bc4\u56f2\u3067\u306f\u6b63\u5f26\u95a2\u6570\u306f\u5e38\u306b\u975e\u8ca0\u306a\u306e\u3067\u7d76\u5bfe\u5024\u3092\u5916\u3059\u3053\u3068\u304c\u3067\u304d\u3001\u793a\u3057\u305f\u304b\u3063\u305f\u3053\u3068\u306b\u5230\u9054\u3059\u308b\u3002<\/p>\n<p>\u3053\u306e\u5f0f\u304b\u3089\u3001\u6f14\u7b97 <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\|<\/span> \u306e\u7d50\u679c\u306f\u3001\u30d9\u30af\u30c8\u30eb <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> \u3068 <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> \u306b\u3088\u3063\u3066\u751f\u6210\u3055\u308c\u308b\u5e73\u884c\u56db\u8fba\u5f62\u306e\u9762\u7a4d\u3092\u4e0e\u3048\u308b\u3053\u3068\u304c\u76f4\u611f\u3067\u304d\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Rn \u306b\u304a\u3051\u308b\u4ee3\u6570\u5b66\u3068\u5c04\u5f71\u3001 \u306b\u304a\u3051\u308b\u30d9\u30af\u30c8\u30eb\u7a4d \u8981\u7d04:\u672c\u30b7\u30ea\u30fc\u30ba\u306f n \u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u95a2\u3059\u308b\u30b7\u30ea\u30fc\u30ba\u306e\u76f4\u63a5\u7684\u306a\u7d9a\u7de8\u3067\u3042\u308b\u3002\u3053\u3053\u3067\u306f n \u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u3092\u3088\u308a\u3088\u304f\u7406\u89e3\u3059\u308b\u305f\u3081\u306e\u7dda\u5f62\u4ee3\u6570\u5b66\u306e\u3044\u304f\u3064\u304b\u306e\u6982\u5ff5\u3092\u78ba\u8a8d\u3057\u3001\u3042\u308b\u30d9\u30af\u30c8\u30eb\u3092\u4ed6\u306e\u30d9\u30af\u30c8\u30eb\u306b\u5c04\u5f71\u3059\u308b\u6982\u5ff5\u3092\u898b\u76f4\u3057\u3001\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3092\u8a3c\u660e\u3057\u3001\u6700\u5f8c\u306b \u306b\u304a\u3051\u308b\u30d9\u30af\u30c8\u30eb\u7a4d\u3068 3 \u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306e\u4ed6\u306e\u7a4d\u3068\u306e\u95a2\u4fc2\u3092\u78ba\u8a8d\u3059\u308b\u3002 \u76ee\u6b21 \u7dda\u5f62\u72ec\u7acb\u6027\u3001\u76f4\u4ea4\u6027\u304a\u3088\u3073\u5c04\u5f71 \u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3068\u90e8\u5206\u7a7a\u9593\u3078\u306e\u5c04\u5f71 \u306b\u304a\u3051\u308b\u5185\u7a4d\u3068\u5916\u7a4d \u7dda\u5f62\u72ec\u7acb\u6027\u3001\u76f4\u4ea4\u6027\u304a\u3088\u3073\u5c04\u5f71 \u7dda\u5f62\u7d50\u5408\u3068\u7dda\u5f62\u72ec\u7acb\u6027 \u96f6\u3067\u306a\u3044\u30d9\u30af\u30c8\u30eb \u306f\u3001\u4ed6\u306e\u96f6\u3067\u306a\u3044\u30d9\u30af\u30c8\u30eb \u3068 \u306b\u95a2\u3057\u3066 \u7dda\u5f62\u7d50\u5408 \u3068\u3057\u3066\u69cb\u6210\u3067\u304d\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u540c\u6642\u306b\u96f6\u3067\u306f\u306a\u3044\u5b9f\u6570 \u3068 \u304c\u5b58\u5728\u3057\u3066\u6b21\u3092\u6e80\u305f\u3059\u3068\u304d\u3067\u3042\u308b: \u3064\u307e\u308a\u3001\u30d9\u30af\u30c8\u30eb \u306f\u30d9\u30af\u30c8\u30eb \u3068 \u306e\u52a0\u91cd\u548c\u3068\u3057\u3066\u69cb\u7bc9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002 \u540c\u69d8\u306b\u3001\u6b21\u306e\u3088\u3046\u306b\u8a00\u3046 \u30d9\u30af\u30c8\u30eb \u3068 \u304c \u7dda\u5f62\u72ec\u7acb \u3067\u3042\u308b\u3068\u306f\u3001 \u30d9\u30af\u30c8\u30eb \u3068 \u306e\u7dda\u5f62\u72ec\u7acb\u6027\u3068\u306f\u3001 \u304c \u306e\uff08\u96f6\u3067\u306a\u3044\uff09\u30b9\u30ab\u30e9\u30fc\u500d\u3067\u5f97\u3089\u308c\u306a\u3044\u3053\u3068\u3001\u307e\u305f\u305d\u306e\u9006\u3082\u6210\u308a\u7acb\u305f\u306a\u3044\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002 \u5148\u307b\u3069\u78ba\u8a8d\u3057\u305f\u7dda\u5f62\u72ec\u7acb\u6027\u306e\u6982\u5ff5\u306f\u3001\u3088\u308a\u5927\u304d\u306a\u30d9\u30af\u30c8\u30eb\u96c6\u5408\u306b\u62e1\u5f35\u3067\u304d\u308b\u3002\u96f6\u3067\u306a\u3044\u30d9\u30af\u30c8\u30eb\u306e\u96c6\u5408 \u306f\u3001\u6b21\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u7dda\u5f62\u72ec\u7acb\u3067\u3042\u308b\u3068\u3044\u3046: \u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u6210\u3059\u89d2\u3068\u76f4\u4ea4\u6027 \u30b3\u30fc\u30b7\u30fc\u2013\u30b7\u30e5\u30ef\u30eb\u30c4\u306e\u4e0d\u7b49\u5f0f\u3092\u601d\u3044\u51fa\u3059\u3068\u3001 \u3053\u308c\u306f \u3092\u610f\u5473\u3059\u308b\u3002\u3053\u308c\u3092\u8003\u616e\u3059\u308b\u3068\u3001\u4efb\u610f\u306e \u306b\u5bfe\u3057\u3066\u6b21\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u5bb9\u6613\u306b\u78ba\u8a8d\u3067\u304d\u308b: \u3053\u3053\u304b\u3089\u3001\u30d9\u30af\u30c8\u30eb \u3068 \u304c\u6210\u3059\u89d2\u3068\u5185\u7a4d\u3068\u306e\u95a2\u4fc2\u3092\u76f4\u611f\u3067\u304d\u308b\u3002\u306a\u305c\u306a\u3089\u3001\u3053\u308c\u3089\u306f \u306b\u540c\u578b\u306a\u5e73\u9762\u3092\u751f\u6210\u3059\u308b\u304b\u3089\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\u3001\u3053\u308c\u3089\u3092 \u306e\u8981\u7d20\u3068\u3057\u3066\u8003\u3048\u3001\u8ef8 \u306b\u5bfe\u3059\u308b\u89d2\u5ea6\u3092\u305d\u308c\u305e\u308c \u3068 \u3068\u3059\u308c\u3070\u3001\u30d9\u30af\u30c8\u30eb\u306f\u6975\u5f62\u5f0f\u3067\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b: 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