{"id":1767,"date":"2019-05-24T23:25:36","date_gmt":"2019-05-24T14:25:36","guid":{"rendered":"http:\/\/141.164.34.82\/?p=1767"},"modified":"2019-05-26T02:17:33","modified_gmt":"2019-05-25T17:17:33","slug":"%eb%8b%a4%ec%b0%a8%ec%9b%90-%eb%b2%a1%ed%84%b0-%ec%82%ac%ec%9d%b4%ec%9d%98-%ea%b0%81%eb%8f%84","status":"publish","type":"post","link":"http:\/\/ds.sumeun.org\/?p=1767","title":{"rendered":"\ub2e4\ucc28\uc6d0 \ubca1\ud130 \uc0ac\uc774\uc758 \uac01\ub3c4"},"content":{"rendered":"<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/wvsE8jm1GzE?start=21\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h1>\ub450 \ubca1\ud130 \\(\\vec{u}\\) \uc640 \\(\\vec{v}\\) \uc0ac\uc774\uc758 \uac01\ub3c4<\/h1>\n<p>\ub450 \\(n\\)-\ucc28\uc6d0 \ubca1\ud130 \\(\\vec{u}\\) \uc640 \\(\\vec{v}\\) \uc758 \ub0b4\uc801\uc740 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4.<\/p>\n<p>\\[\\vec{u} \\cdot \\vec{v} = u_1 v_1 + u_2 v_2 + \\cdots + u_n v_n\\]<\/p>\n<p>\uadf8\ub9ac\uace0 \ub450 \ubca1\ud130 \uc0ac\uc774\uc758 \uac01\ub3c4\ub294 \ub2e4\uc74c\uc758 \ucf54\uc0ac\uc778 2\ubc95\uce59\uc744 \ud65c\uc6a9\ud558\uc5ec \uad6c\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/p>\n<p>\ucf54\uc0ac\uc778 \uc81c 2\ubc95\uce59\uc740 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4.<\/p>\n<p>\\[|\\vec{w}|^2 = |\\vec{u}|^2 + |\\vec{v}|^2 &#8211; 2|\\vec{u}||\\vec{v}| \\cos\\theta\\]<\/p>\n<p>\uc774 \ubc95\uce59\uc740 \ud53c\ud0c0\uace0\ub77c\uc2a4 \uc815\ub9ac(\\(|\\vec{w}|^2=|\\vec{u}|^2+|\\vec{v}|^2\\))\uc758 \uc77c\ubc18\ud654\ub41c \ud615\uc2dd\uc73c\ub85c \ud53c\ud0c0\uace0\ub77c\uc2a4 \ubc95\uce59\uc740 \ube57\ubcc0\uc758 \uc0c1\ub300\uac01\uc774 <strong>\uc9c1\uac01<\/strong>\uc778 \uacbd\uc6b0\uc5d0\ub9cc \uc0ac\uc6a9\ud560 \uc218 \uc788\uc9c0\ub9cc, \ucf54\uc0ac\uc778 \uc81c 2\ubc95\uce59\uc740 \uc5b4\ub5a4 \uac01\ub3c4\uc5d0 \ub300\ud574\uc11c\ub3c4 \uc0ac\uc6a9\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4. (\\(\\vec{w}\\) \ub294 \\(\\vec{u}-\\vec{v}\\) \ub610\ub294 \\(\\vec{v}-\\vec{u}\\) \uc785\ub2c8\ub2e4.)<\/p>\n<p>\uc0ac\uc2e4 2\ucc28\uc6d0\uc5d0\uc11c\ub294 \uac01\ub3c4\ub780 \ub208\uc5d0 \ubcf4\uc774\uae30 \ub54c\ubb38\uc5d0 \uadf8 \uc758\ubbf8\ub97c \uc815\ud655\ud558\uac8c \ud30c\uc545\ud560 \uc218 \uc788\uc9c0\ub9cc \uace0\ucc28\uc6d0 \uc0c1\uc758 \ub450 \ubca1\ud130\uc5d0 \ub300\ud574 \uac01\ub3c4\ub97c \uc0c1\uc0c1\ud558\ub294 \uac83\uc740 \uc9c1\uad00\uc801\uc73c\ub85c \uc27d\uac8c \ub2e4\uac00\uc624\uc9c0 \uc54a\uc744 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \ud558\uc9c0\ub9cc \\(n\\)-\ucc28\uc6d0 \uc0c1\uc758 \ud55c \ubca1\ud130 \\(\\vec{v}\\) \uc758 \uae38\uc774\ub97c \\(\\sqrt{v_1^2 + v_2^2 + \\cdots + v_n^2}\\) \ub77c\uace0 \ud560 \ub54c, \ub450 \ubca1\ud130\ub97c \ud3ec\ud568\ud558\ub294 2\ucc28\uc6d0 \ud3c9\uba74\uc744 \uc0c1\uc0c1\ud55c\ub2e4\uba74, \uc774 \ud3c9\uba74 \uc0c1\uc5d0\uc11c\ub3c4 \uc704\uc758 \ucf54\uc0ac\uc778 \ubc95\uce59\uc774 \uc131\ub9bd\ud560 \uac83\uc774\uace0, \ub530\ub77c\uc11c \uace0\ucc28\uc6d0 \uacf5\uac04 \uc18d\uc758 \ub450 \ubca1\ud130\uc5d0 \ub300\ud574\uc11c\ub3c4 \uac01\ub3c4\ub97c \uad6c\ud560 \uc218 \uc788\uac8c \ub429\ub2c8\ub2e4.<\/p>\n<p>\ub2e4\ucc28\uc6d0 \uacf5\uac04 \uc18d\uc758 \ub450 \ubca1\ud130\uc758 \uac01\ub3c4\ub294 \uc774\ub7f0 \uc2dd\uc73c\ub85c 2\ucc28\uc6d0 \uc0c1\uc758 \uac01\ub3c4 \uac1c\ub150\uc744 \ud655\uc7a5\uc2dc\ud0a8 \uac83\uc73c\ub85c \uc0dd\uac01\ud558\uba74 \ub429\ub2c8\ub2e4.<\/p>\n<p>\uc704\uc758 \uc2dd\uc744 \ud480\uba74 \ub450 \ubca1\ud130 \\(\\vec{u}\\), \\(\\vec{v}\\) \uc758 \uc0ac\uc774\uac01\uc740 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4.<\/p>\n<p>\\[\\theta = \\cos^{-1}\\left(\\frac{|\\vec{u}|^2+|\\vec{v}|^2-|\\vec{w}|^2}{2|\\vec{u}||\\vec{v}|}\\right)\\]<\/p>\n<p>\uc774\ub97c \uac01 \ubca1\ud130\uc758 \uc131\ubd84\uc73c\ub85c \ud45c\ud604\ud55c\ub2e4\uba74 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4. (\\(\\vec{w} = \\vec{u}-\\vec{v}\\))<\/p>\n<p>\\[\\theta = \\cos^{-1}\\left(\\frac{\\sum{u_i^2}+\\sum{v_i^2}-\\sum(u_i-v_i)^2}{2\\sqrt{\\sum{u_i^2}}\\sqrt{\\sum{v_i^2}}}\\right)\\]<\/p>\n<p>\uadf8\ub9ac\uace0 \uc774\ub97c \uc798 \uc815\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uc2dd\uc744 \uc5bb\uc744 \uc218 \uc788\uc2b5\ub2c8\ub2e4. <\/p>\n<p>\\[\\theta = \\cos^{-1}\\left(\\frac{\\sum{u_i v_i}}{\\sqrt{\\sum{u_i^2}}\\sqrt{\\sum{v_i^2}}}\\right)\\]<\/p>\n<p>\uc774\ub97c R\uc5d0\uc11c \uacc4\uc0b0\ud55c\ub2e4\uba74 \ub2e4\uc74c\uacfc \uac19\uc774 \ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/p>\n<h2>R\ub85c \uad6c\ud604\ud558\ub294 \ub450 \ubca1\ud130 \uc0ac\uc774\uc758 \uac01\ub3c4<\/h2>\n<pre><code class=\"r\">u = c(1,3,2,4,7)\nv = c(2,-1,4,3,0)\n\n# inner product\nsum(u*v) \n<\/code><\/pre>\n<pre>## [1] 19\n<\/pre>\n<pre><code class=\"r\">u %*% v\n<\/code><\/pre>\n<pre>##      [,1]\n## [1,]   19\n<\/pre>\n<pre><code class=\"r\"># angle\nacos(sum(u*v)\/(sqrt(sum(u*u))*sqrt(sum(v*v))))\n<\/code><\/pre>\n<pre>## [1] 1.169858\n<\/pre>\n<p><strong>\ub77c\ub514\uc548<\/strong>\uc744 <strong>\ub3c4<\/strong>\ub85c \ubc14\uafb8\ub824\uba74 \\(\\times180\/\\pi\\) \ub97c \ud574\uc8fc\uba74 \ub41c\ub2e4.<\/p>\n<pre><code class=\"r\">rad = acos(sum(u*v)\/(sqrt(sum(u*u))*sqrt(sum(v*v)))) \ndeg = rad * 180 \/ pi\ndeg\n<\/code><\/pre>\n<pre>## [1] 67.0279\n<\/pre>\n<h3>\uc608\uc81c: 30 \ucc28\uc6d0 \uc0c1\uc758 \ub450 \ubca1\ud130 \uc0ac\uc774 \uac01\ub3c4<\/h3>\n<pre><code class=\"r\">u = rnorm(30)\nv = 2* rnorm(30)\n\n# angle\nrad = acos(sum(u*v)\/(sqrt(sum(u*u))*sqrt(sum(v*v))))\ndeg = rad * 180 \/ pi\nrad\n<\/code><\/pre>\n<pre>## [1] 1.807722\n<\/pre>\n<pre><code class=\"r\">deg\n<\/code><\/pre>\n<pre>## [1] 103.5748\n<\/pre>\n<h3>\uc0dd\uac01\ud574\ubcfc \uac70\ub9ac: \uace0\ucc28\uc6d0 \uc0c1\uc758 \ub450 \ubb34\uc791\uc704 \ubca1\ud130 \uc0ac\uc774 \uac01\ub3c4<\/h3>\n<p>\uace0\ucc28\uc6d0\uc5d0 \ub300\ud574 \uc54c\ub824\uc9c4 \uc0ac\uc2e4\uc758 \ud558\ub098\ub294 \uace0\ucc28\uc6d0 \uc0c1\uc758 \ub450 \ubb34\uc791\uc704 \ubca1\ud130\ub294 \ucc28\uc6d0\uc774 \ub192\uc544\uc9c8 \uc218\ub85d \\(90^\\circ\\) \uc5d0 \uac00\uae4c\uc6b8 \uac00\ub2a5\uc131\uc774 \ud06c\ub2e4\ub294 \uac83\uc785\ub2c8\ub2e4. \ub2e4\uc74c\uc758 \uadf8\ub798\ud504\ub294 \uc774\ub97c \ubcf4\uc5ec\uc8fc\uace0 \uc788\uc2b5\ub2c8\ub2e4. 3\ucc28\uc6d0 \uc0c1\uc758 \ubb34\uc791\uc704 \ubca1\ud130 \uc0ac\uc774 \uac01\ub3c4\ub294 \ub2e4\uc18c \uace8\uace0\ub8e8 \ubd84\ud3ec\ub418\uc5b4 \uc788\ub2e4\uba74 100\ucc28\uc6d0 \uc0c1\uc758 \ub450 \ubb34\uc791\uc704 \ubca1\ud130 \uc0ac\uc774\uc758 \uac01\ub3c4\ub294 \ub300\ubd80\ubd84 \\(90^\\circ\\) \uc5d0 \uac00\uae5d\uc2b5\ub2c8\ub2e4. \uc774\ub97c \ubcf4\ud1b5 \uace0\ucc28\uc6d0\uc774 \ub420 \uc218\ub85d \uc5b4\ub5a4 \ud55c \ubc29\ud5a5\uc5d0 \ub300\ud574 <strong>\uc9c1\uac01\uc758 \ubc29\ud5a5<\/strong>\uc774 \ub9ce\uae30 \ub54c\ubb38\uc774\ub77c\uace0 \uc124\uba85\ud569\ub2c8\ub2e4. <\/p>\n<pre><code class=\"r\">deg = data.frame(n=rep(c(3,10,100),1000), i=rep(1:1000,3), deg=NA)\nfor (n in c(3,10,100)) {\n  for (i in 1:1000) {\n    u = rnorm(n)\n    v = 2* rnorm(n)\n\n    rad = acos(sum(u*v)\/(sqrt(sum(u*u))*sqrt(sum(v*v))))\n    deg[deg$n==n &amp; deg$i == i, &quot;deg&quot;] = rad * 180 \/ pi\n    #deg[j, &quot;deg&quot;] = rad * 180 \/ pi\n  }\n}\n\nlibrary(ggplot2)\n\nggplot(data=deg, aes(x=deg)) + geom_histogram() + facet_wrap(~n) + geom_vline(xintercept=90, linetype=&#39;dotted&#39;)\n<\/code><\/pre>\n<pre>## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.\n<\/pre>\n<p><img src=\"data:image\/png;base64,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