{"id":6988,"date":"2021-07-23T23:22:56","date_gmt":"2021-07-23T14:22:56","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6988"},"modified":"2025-04-09T19:09:53","modified_gmt":"2025-04-09T10:09:53","slug":"euclidean-spaces","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/euclidean-spaces\/","title":{"rendered":"\uc720\ud074\ub9ac\ub4dc \uacf5\uac04"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p><!--\n\\[\n\\newcommand{\\vecf}{{\\mathbf{f}}}\n\\newcommand{\\vecL}{{\\mathbf{L}}}\n\\newcommand{\\vecR}{{\\mathbb{R}}}\n\\newcommand{\\imI}{\\boldsymbol{i}}\n\\]\n-->\n<\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 8\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\\(d\\)\uac00 \uc591\uc758 \uc815\uc218\uc774\uace0 \\(\\mathbb{R}^d\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ub370\uce74\ub974\ud2b8 \uacf1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0<br \/>\n\\[\\mathbf{x} = \\left( x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_d \\right) ,\\quad<br \/>\n\\mathbf{y} = \\left( y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_d \\right) \\]<br \/>\n\uac00 \\(\\mathbb{R}^d\\)\uc758 \uc6d0\uc18c\uc774\uba70 \\(k\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c <span class=\"defined\">\ubca1\ud130\ud569<\/span> \\(\\mathbf{x} + \\mathbf{y}\\)\uc640 <span class=\"defined\">\uc2a4\uce7c\ub77c\uacf1<\/span> \\(k\\mathbf{x}\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\mathbf{x} + \\mathbf{y} &#038;= \\left( x_1 + y_1 ,\\, x_2 + y_2 ,\\, \\cdots ,\\, x_d + y_d \\right) ,\\\\[6pt]<br \/>\nk\\mathbf{x} &#038;= \\left( kx_1 ,\\, kx_2 ,\\, \\cdots ,\\, kx_d \\right) .<br \/>\n\\end{align}\\]<br \/>\n\uc9d1\ud569 \\(\\mathbf{R}^d\\)\uc5d0 \uc774\uc640 \uac19\uc740 \ub450 \uc5f0\uc0b0\uc774 \uc8fc\uc5b4\uc9c4 \uacf5\uac04\uc744 <span class=\"defined\">\\(\\boldsymbol{d}\\)\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04<\/span>(Euclidean \\(d\\)-space) \ub610\ub294 \uac04\ub2e8\ud788 <span class=\"defined\">\uc720\ud074\ub9ac\ub4dc \uacf5\uac04<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\mathbb{R}^d\\)\uc758 \uc6d0\uc18c\ub97c <span class=\"defined\">\ubca1\ud130<\/span>(vector)\ub77c\uace0 \ubd80\ub974\uace0, \\(k\\)\ub97c <span class=\"defined\">\uc2a4\uce7c\ub77c<\/span>(scalar)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(\\mathbf{v}\\)\uac00 \\(\\mathbb{R}^d\\)\uc758 \uc6d0\uc18c\ub77c\uace0 \ud558\uc790. \ud3b8\uc758\uc0c1 \\(-1\\mathbf{v}\\)\ub97c \uac04\ub2e8\ud788 \\(-\\mathbf{v}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \ubaa8\ub4e0 \uc131\ubd84\uc774 \\(0\\)\uc778 \ubca1\ud130<br \/>\n\\[\\mathbf{0} = (0,\\,0,\\,\\cdots,\\,0)\\]<br \/>\n\uc744 <span class=\"defined\">\uc601\ubca1\ud130<\/span>(zero vector)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\mathbf{0}\\)\uc740 \ubca1\ud130\uc758 \ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0\uc774\uba70, \\(-\\mathbf{v}\\)\ub294 \ub367\uc148\uc5d0 \ub300\ud55c \\(\\mathbf{v}\\)\uc758 \uc5ed\uc6d0\uc774\ub2e4.<\/p>\n<p>\\(\\mathbf{x},\\) \\(\\mathbf{y},\\) \\(\\mathbf{z}\\)\uac00 \\(\\mathbb{R}^d\\)\uc758 \ubca1\ud130\uc774\uace0 \\(a,\\) \\(b\\)\uac00 \uc2a4\uce7c\ub77c\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\( \\mathbf{x} + ( \\mathbf{y} + \\mathbf{z} ) = ( \\mathbf{x} + \\mathbf{y} ) + \\mathbf{z} \\)<\/li>\n<li>\\( \\mathbf{x} + \\mathbf{y} = \\mathbf{y} + \\mathbf{x} \\)<\/li>\n<li>\\( \\mathbf{x} + \\mathbf{0} = \\mathbf{x} \\)<\/li>\n<li>\\( \\mathbf{x} + (-\\mathbf{x} ) = \\mathbf{0} \\)<\/li>\n<li>\\( a(b\\mathbf{x} ) = (ab)\\mathbf{x} \\)<\/li>\n<li>\\( 1\\mathbf{x} = \\mathbf{x} \\)<\/li>\n<li>\\( a(\\mathbf{x} + \\mathbf{y} ) = a\\mathbf{x} + b\\mathbf{y} \\)<\/li>\n<li>\\( (a+b)\\mathbf{x} = a\\mathbf{x} + b\\mathbf{x} \\)<\/li>\n<\/ul>\n<p>\uc0ac\uc2e4, \\(K\\)\uac00 \uccb4(field)\uc774\uace0 \\(V\\)\uc5d0 \uc704 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub450 \uc5f0\uc0b0 \\( + : V \\times V \\rightarrow V\\)\uc640 \\(\\cdot : K \\times V \\rightarrow V\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(V\\)\ub97c <span class=\"defined\">\\(\\boldsymbol{K}\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04<\/span>(vector space over \\(K\\))\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \uacbd\uc6b0 \\(V\\)\uc758 \uc6d0\uc18c\ub97c <span class=\"defined\">\ubca1\ud130<\/span>\ub77c\uace0 \ubd80\ub974\uace0, \\(K\\)\uc758 \uc6d0\uc18c\ub97c <span class=\"defined\">\uc2a4\uce7c\ub77c<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uae30\uc800<\/h2>\n<p>\ub2e4\uc74c\uacfc \uac19\uc740 \\(\\mathbb{R}^3\\)\uc758 \uc138 \ubca1\ud130\ub97c \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\mathbf{i} = (1,\\,0,\\,0),\\quad \\mathbf{j} = (0,\\,1,\\,0),\\quad \\mathbf{k} = (0,\\,0,\\,1).\\]<br \/>\n\\(\\mathbf{R}^3\\)\uc758 \uc784\uc758\uc758 \ubca1\ud130 \\(\\mathbf{x}\\)\uc5d0 \ub300\ud558\uc5ec \uc2a4\uce7c\ub77c \\(x_1 ,\\) \\(x_2 ,\\) \\(x_3 \\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(\\mathbf{x}\\)\uac00<br \/>\n\\[\\mathbf{x} = x_1 \\mathbf{i} + x_2 \\mathbf{j} + x_3 \\mathbf{k}\\]<br \/>\n\uc758 \uaf34\ub85c \uc720\uc77c\ud558\uac8c \ud45c\ud604\ub41c\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uc138 \ubca1\ud130 \\(\\mathbf{i},\\) \\(\\mathbf{j},\\) \\(\\mathbf{k}\\)\ub97c \\(\\mathbf{R}^3\\)\uc758 <span class=\"defined\">\ud45c\uc900\uae30\uc800<\/span>(standard basis)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 \\(\\mathbb{R}^3\\)\uc758 \ud45c\uc900\uae30\uc800\uac00 \\(3\\)\uac1c\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uc73c\ubbc0\ub85c, \\(\\mathbf{R}^3\\)\uc758 <span class=\"defined\">\ucc28\uc6d0<\/span>\uc744 \\(3\\)\uc774\ub77c\uace0 \uc815\uc758\ud55c\ub2e4. \\(\\mathbf{i},\\) \\(\\mathbf{j},\\) \\(\\mathbf{k}\\)\ub97c <span class=\"defined\">\ud45c\uc900\ub2e8\uc704\ubca1\ud130<\/span>(standard unit vector)\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uacfc \uac19\uc740 \\(\\mathbb{R}^2\\)\uc758 \uc138 \ubca1\ud130\ub97c \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\mathbf{i} = (1,\\,0),\\quad \\mathbf{j} = (0,\\,1).\\]<br \/>\n\uc774 \ubca1\ud130\ub294 \\(\\mathbb{R}^2\\)\uc758 \uae30\uc800\uac00 \ub41c\ub2e4. \uc989 \\(\\mathbb{R}^2\\)\uc758 \uc784\uc758\uc758 \ubca1\ud130 \\(\\mathbf{x}\\)\uc5d0 \ub300\ud558\uc5ec \uc2a4\uce7c\ub77c \\(x_1 ,\\) \\(x_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \\(\\mathbf{x}\\)\uac00<br \/>\n\\[\\mathbf{x} = x_1 \\mathbf{i} + x_2 \\mathbf{j}\\]<br \/>\n\uc758 \uaf34\ub85c \uc720\uc77c\ud558\uac8c \ud45c\ud604\ub41c\ub2e4. \ub354\ubd88\uc5b4 \\(\\mathbb{R}^2\\)\uc758 \ucc28\uc6d0\uc744 \\(2\\)\ub77c\uace0 \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\uc0ac\uc2e4 \uae30\uc800\ub294 \uc720\ud074\ub9ac\ub4dc \ubca1\ud130\uacf5\uac04\ubfd0\ub9cc \uc544\ub2c8\ub77c \ubaa8\ub4e0 \ubca1\ud130\uacf5\uac04\uc5d0 \ub300\ud574\uc11c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \\(V\\)\uac00 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(\\mathbf{v}_1 ,\\) \\(\\mathbf{v}_2 ,\\) \\(\\cdots ,\\) \\(\\mathbf{v}_d\\)\uac00 \\(V\\)\uc758 \ubca1\ud130\uc774\uba70, \uc5b4\ub290\uac83\ub3c4 \uc601\ubca1\ud130\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(V\\)\uc758 \uc784\uc758\uc758 \ubca1\ud130 \\(\\mathbf{v}\\)\uc5d0 \ub300\ud558\uc5ec \uc2a4\uce7c\ub77c \\(a_1,\\) \\(a_2 ,\\) \\(\\cdots,\\) \\(a_d\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \\(\\mathbf{v}\\)\uac00<br \/>\n\\[\\mathbf{v} = a_1 \\mathbf{v}_1 + a_2 \\mathbf{v}_2 + \\cdots + a_d \\mathbf{v}_d\\]<br \/>\n\uc758 \uaf34\ub85c \uc720\uc77c\ud558\uac8c \ud45c\ud604\ub41c\ub2e4\uba74, \\(\\mathbf{v}_1 ,\\) \\(\\mathbf{v}_2 ,\\) \\(\\cdots ,\\) \\(\\mathbf{v}_d\\)\ub97c \\(V\\)\uc758 <span class=\"defined\">\uae30\uc800<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uadf8\ub9ac\uace0 \uae30\uc800\ub97c \uc774\ub8e8\ub294 \ubca1\ud130\uc758 \uac1c\uc218\uc778 \\(d\\)\uac00 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 <span class=\"defined\">\ucc28\uc6d0<\/span>\uc778 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \ubca1\ud130\uacf5\uac04\uc5d0 \uae30\uc800\uac00 \uc874\uc7ac\ud560 \ub54c, \uadf8 \uacf5\uac04\uc758 \uc5b4\ub5a0\ud55c \uae30\uc800\ub97c \ud0dd\ud558\ub4e0 \uae30\uc800\ub97c \uc774\ub8e8\ub294 \ubca1\ud130\uc758 \uac1c\uc218\uac00 \uc77c\uc815\ud568\uc774 \uc54c\ub824\uc838 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubca1\ud130\uacf5\uac04\uc758 \ucc28\uc6d0\uc740 \uc798 \uc815\uc758\ub41c\ub2e4.<\/p>\n<p>\uc774 \uc808\uc5d0\uc11c\ub294 \uae30\uc800\uc758 \uac1c\uc218\uac00 \uc720\ud55c\uc778 \uacbd\uc6b0\ub9cc \uc815\uc758\ud558\uc600\ub2e4. \uc0ac\uc2e4, \uae30\uc800\uc758 \uac1c\uc218\uac00 \ubb34\ud55c\uc778 \uacbd\uc6b0\ub3c4 \uc815\uc758\ud560 \uc218 \uc788\uc73c\uba70, \ucc28\uc6d0\uc774 \ubb34\ud55c\uc778 \ubca1\ud130\uacf5\uac04\ub3c4 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uadf8\uc640 \uad00\ub828\ub41c \ub17c\uc758\ub294 \uc774 \ucc45\uc758 \ubc94\uc704\ub97c \ubc97\uc5b4\ub098\ubbc0\ub85c, \ub354 \uc774\uc0c1 \uae4a\uac8c \uc124\uba85\ud558\uc9c0 \uc54a\uaca0\ub2e4. \uad00\uc2ec \uc788\ub294 \uc0ac\ub78c\uc740 \uc120\ud615\ub300\uc218\ud559 \uad50\uc7ac\ub97c \ucc38\uace0\ud558\uae30 \ubc14\ub780\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc120\ud615\ubcc0\ud658<\/h2>\n<p>\\(V\\)\uc640 \\(W\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\uace0, \\(T: V \\rightarrow W\\)\uac00 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(V\\)\uc758 \uc784\uc758\uc758 \ubca1\ud130 \\(\\mathbf{x},\\) \\(\\mathbf{y}\\)\uc640 \uc2a4\uce7c\ub77c \\(k\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\nT(\\mathbf{x} + \\mathbf{y}) &#038;= T(\\mathbf{x}) + T(\\mathbf{y}), \\\\[6pt]<br \/>\nT(k\\mathbf{x}) &#038;= kT(\\mathbf{x})<br \/>\n\\end{align}\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uba74, \u201c\\(T\\)\uac00 <span class=\"defined\">\uc120\ud615<\/span>\uc774\ub2e4(linear)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\uc778 \ud568\uc218\ub97c <span class=\"defined\">\uc120\ud615\ubcc0\ud658<\/span>(linear transformation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(V\\)\uc640 \\(W\\)\uac00 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(T : V \\rightarrow W\\)\uac00 \uc77c\ub300\uc77c \ub300\uc751\uc778 \uc120\ud615\ubcc0\ud658\uc774\uba74 \\(T\\)\ub97c <span class=\"defined\">\ub3d9\ud615\uc0ac\uc0c1<\/span>(isomorphism)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub450 \ubca1\ud130\uacf5\uac04 \\(V,\\) \\(W\\) \uc0ac\uc774\uc5d0 \uc774\uc640 \uac19\uc740 \ub3d9\ud615\uc0ac\uc0c1\uc774 \uc874\uc7ac\ud560 \ub54c, \u201c\\(V\\)\uc640 \\(W\\)\uac00 <span class=\"defined\">\ub3d9\ud615<\/span>\uc774\ub2e4(isomorphic)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\ub450 \ubca1\ud130\uacf5\uac04\uc774 \ub3d9\ud615\uc77c \ub54c \ub450 \uacf5\uac04\uc758 \ucc28\uc6d0\uc774 \uac19\uc74c\uc774 \uc54c\ub824\uc838 \uc788\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc810\uacf1\uacfc \uac00\uc704\uacf1<\/h2>\n<p>\\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\)\uac00 \\(\\mathbb{R}^d\\)\uc758 \ubca1\ud130\uc774\uace0<br \/>\n\\[\\mathbf{x} = \\left( x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_d \\right) ,\\quad<br \/>\n\\mathbf{y} = \\left( y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_d \\right) \\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\)\uc758 <span class=\"defined\">\uc810\uacf1<\/span>(dot product)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\mathbf{x} \\cdot \\mathbf{y} = x_1 y_1 + x_2 y_2 + \\cdots + x_d y_d .\\]<br \/>\n\uc810\uacf1\uc744 <span class=\"defined\">\ub0b4\uc801<\/span>(inner product)\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\ubca1\ud130 \\(\\mathbf{x}\\)\uc758 <span class=\"defined\">\uae38\uc774<\/span>(length)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\lvert\\mathbf{x}\\rvert = \\sqrt{\\mathbf{x} \\cdot \\mathbf{x}} .\\]<br \/>\n\ubca1\ud130\uc758 \uae38\uc774\ub97c <span class=\"defined\">\ub178\ub984<\/span>(norm)\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4. \ucc45\uc5d0 \ub530\ub77c\uc11c \\(\\lvert\\mathbf{x}\\rvert\\)\ub97c \\(\\lVert\\mathbf{x}\\rVert\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \uae38\uc774\uac00 \\(1\\)\uc778 \ubca1\ud130\ub97c <span class=\"defined\">\ub2e8\uc704\ubca1\ud130<\/span>(unit vector)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(d=2\\)\uc774\uac70\ub098 \\(d=3\\)\uc77c \ub54c, \uc601\ubca1\ud130\uac00 \uc544\ub2cc \ub450 \ubca1\ud130 \\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\) \uc0ac\uc774\uc758 \uac01\uc744 \\(\\theta\\)\ub77c\uace0 \ud558\uba74<br \/>\n\\[\\mathbf{x} \\cdot \\mathbf{y} = \\lvert\\mathbf{x}\\rvert \\lvert\\mathbf{y}\\rvert \\cos\\theta\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc77c\ubc18\uc801\uc73c\ub85c \\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\)\uac00 \\(\\mathbb{R}^d\\)\uc758 \ubca1\ud130\uc774\uace0 \uc601\ubca1\ud130\uac00 \uc544\ub2d0 \ub54c(\\(d\\)\uac00 \\(2\\) \ub610\ub294 \\(3\\)\uc774\ub77c\ub294 \uc870\uac74\uc774 \uc5c6\uc744 \ub54c), \\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\) \uc0ac\uc774\uc758 <span class=\"defined\">\uac01<\/span>\uc744<br \/>\n\\[\\mathbf{x} \\cdot \\mathbf{y} = \\lvert\\mathbf{x}\\rvert \\lvert\\mathbf{y}\\rvert \\cos\\theta\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\uace0 \\(0\\le\\theta\\le \\pi\\)\uc758 \ubc94\uc704\uc5d0 \uc788\ub294 \uc2e4\uc218 \\(\\theta\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(\\mathbf{x} \\cdot \\mathbf{y} = 0\\)\uc774\uba74, \u201c\\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\)\uac00 <span class=\"defined\">\uc9c1\uad50\ud55c\ub2e4<\/span>(orthogonal)\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(\\mathbf{x} \\perp \\mathbf{y}\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc601\ubca1\ud130\ub294 \uc784\uc758\uc758 \ubca1\ud130\uc640 \uc9c1\uad50\ud55c\ub2e4.<\/p>\n<p>\ub9cc\uc57d \uc2a4\uce7c\ub77c \\(k\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\mathbf{x} = k\\mathbf{y}\\) \ub610\ub294 \\(\\mathbf{y} = k\\mathbf{x}\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74, \u201c\\(\\mathbf{x}\\)\uc640 \\(\\mathbf{y}\\)\uac00 \ud3c9\ud589\ud558\ub2e4(parallel)\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c \\(\\mathbf{x} \/\\!\/ \\mathbf{y}\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc774 \uacbd\uc6b0 \\(\\mathbf{x}\\cdot\\mathbf{y} = \\pm\\lvert\\mathbf{x}\\rvert \\lvert\\mathbf{y}\\rvert\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc601\ubca1\ud130\ub294 \uc784\uc758\uc758 \ubca1\ud130\uc640 \ud3c9\ud589\ud558\ub2e4.<\/p>\n<p>\\(d=3\\)\uc77c \ub54c, \\(\\mathbf{R}^3\\)\uc5d0 \ub610 \ub2e4\ub978 \uc720\uc6a9\ud55c \uc5f0\uc0b0\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[\\mathbf{x} = (x_1 ,\\, x_2 ,\\, x_3 ),\\quad \\mathbf{y} = (y_1 ,\\, y_2 ,\\, y_3 )\\]<br \/>\n\uc77c \ub54c, \uc774 \ub450 \ubca1\ud130\uc758 <span class=\"defined\">\uac00\uc704\uacf1<\/span>(cross product)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\mathbf{x} \\times \\mathbf{y}<br \/>\n&#038;=<br \/>\n\\left\\lvert \\begin{array}{ccc} \\mathbf{i} \\,&#038;\\, \\mathbf{j} \\,&#038;\\, \\mathbf{k} \\\\ x_1 \\,&#038;\\, x_2 \\,&#038;\\, x_3 \\\\ y_1 \\,&#038;\\, y_2 \\,&#038;\\, y_3 \\end{array} \\right\\rvert<br \/>\n\\\\[5pt]<br \/>\n&#038;=<br \/>\n\\left( \\left\\lvert \\begin{array}{cc} x_2 \\,&#038;\\, x_3 \\\\ y_2 \\,&#038;\\, y_3 \\end{array} \\right\\rvert ,\\,<br \/>\n\\left\\lvert \\begin{array}{cc} x_3 \\,&#038;\\, x_1 \\\\ y_3 \\,&#038;\\, y_1 \\end{array} \\right\\rvert ,\\,<br \/>\n\\left\\lvert \\begin{array}{cc} x_1 \\,&#038;\\, x_2 \\\\ y_1 \\,&#038;\\, y_2 \\end{array} \\right\\rvert \\right)<br \/>\n\\\\[5pt]<br \/>\n&#038;= \\left( x_2 y_3 &#8211; x_3 y_2 ,\\, x_3 y_1 &#8211; x_1 y_3 ,\\, x_1 y_2 &#8211; x_2 y_1 \\right) .<br \/>\n\\end{align}\\]<br \/>\n\uac00\uc704\uacf1\uc744 <span class=\"defined\">\uc678\uc801<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4. \uac00\uc704\uacf1\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\mathbf{x} \\times \\mathbf{y} \\perp \\mathbf{x} \\quad\\text{and}\\quad \\mathbf{x} \\times \\mathbf{y} \\perp \\mathbf{y}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\\(\\mathbb{R}^3\\)\uc758 \ud45c\uc900\uae30\uc800\uc6d0\uc18c\ub294 \uac00\uc704\uacf1\uacfc \uad00\ub828\ud558\uc5ec \ub2e4\uc74c \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<br \/>\n\\[\\mathbf{i}\\times\\mathbf{j} = \\mathbf{k},\\quad \\mathbf{j}\\times\\mathbf{k} = \\mathbf{i},\\quad \\mathbf{k}\\times\\mathbf{i} = \\mathbf{j} .\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(\\mathbf{i},\\) \\(\\mathbf{j},\\) \\(\\mathbf{k}\\)\ub294 \uc9c1\uad50\uc871\uc744 \uc774\ub8ec\ub2e4. \uc989 \ubca1\ud130\ub4e4\uc774 \uc30d\ub9c8\ub2e4 \uc11c\ub85c \uc9c1\uad50\ud558\uace0 \uadf8 \uc5b4\ub290\uac83\ub3c4 \uc601\ubca1\ud130\uac00 \uc544\ub2d0 \ub54c, \uadf8 \ubca1\ud130\ub4e4\uc744 <span class=\"defined\">\uc9c1\uad50\uc871<\/span>(orthogonal family)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub354\uc6b1\uc774 \\(\\mathbf{i},\\) \\(\\mathbf{j},\\) \\(\\mathbf{k}\\)\ub294 \ubaa8\ub450 \uae38\uc774\uac00 \\(1\\)\uc778 \ubca1\ud130\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774\ub4e4 \ubca1\ud130\ub4e4\uc740 <span class=\"defined\">\uc815\uaddc\uc9c1\uad50\uc871<\/span>(orthonormal family)\uc744 \uc774\ub8ec\ub2e4.<\/p>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/exponential-and-logarithmic-functions\">\uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/definition-of-a-sequence\">\uc218\uc5f4\uc758 \uc815\uc758<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 8\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \\(d\\)\uac00 \uc591\uc758 \uc815\uc218\uc774\uace0 \\(\\mathbb{R}^d\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ub370\uce74\ub974\ud2b8 \uacf1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\mathbf{x} = \\left( x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_d \\right) ,\\, \\mathbf{y} = \\left( y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_d \\right) \\) \uac00 \\(\\mathbb{R}^d\\)\uc758 \uc6d0\uc18c\uc774\uba70 \\(k\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ubca1\ud130\ud569 \\(\\mathbf{x} + \\mathbf{y}\\)\uc640 \uc2a4\uce7c\ub77c\uacf1 \\(k\\mathbf{x}\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. \\(\\begin{align} \\mathbf{x} + \\mathbf{y}&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":18,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-6988","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6988","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=6988"}],"version-history":[{"count":21,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6988\/revisions"}],"predecessor-version":[{"id":9165,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6988\/revisions\/9165"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6988"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}