{"id":14709,"date":"2025-12-24T17:22:58","date_gmt":"2025-12-24T08:22:58","guid":{"rendered":"https:\/\/math-travel.com\/?p=14709"},"modified":"2026-03-06T01:09:14","modified_gmt":"2026-03-05T16:09:14","slug":"vector-triangle","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-b\/vector-triangle\/","title":{"rendered":"\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u516c\u5f0f\uff08\u30d9\u30af\u30c8\u30eb\uff09\u306e\u4f7f\u3044\u65b9\u306f\uff1f\u6210\u5206\u8868\u793a\u3068\u5185\u7a4d\u3067\u306e\u89e3\u6cd5\u3068\u8a3c\u660e"},"content":{"rendered":"\n
\u300c\u30d9\u30af\u30c8\u30eb\u306e\u9762\u7a4d\u516c\u5f0f\u3063\u3066\u306a\u306b\uff1f\u300d
\u300c\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u3092\u3069\u3046\u3084\u3063\u3066\u6c42\u3081\u308b\u306e\uff1f\u300d<\/p>\n<\/div><\/div>\n\n\n\n
\u4eca\u56de\u306f\u6570\u5b66B\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u300c\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u516c\u5f0f\u300d<\/span>\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \u30d9\u30af\u30c8\u30eb\u3092\u7528\u3044\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306f\u4ee5\u4e0b\u306e2\u3064\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \u2460\u30d9\u30af\u30c8\u30eb\u8868\u793a\u3067\u306e\u516c\u5f0f<\/p>\n\n\n\n \\(\\triangle OAB\\)\u304c\u3042\u308a\u3001\\(\\vec{OA}=\\vec{a}, \\vec{OB}=\\vec{b}\\)\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\\(\\triangle OAB\\)\u306e\u9762\u7a4dS\u306f\u3001<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2} \\sqrt{|\\vec{a}|^{2}|\\vec{b}|^{2}-(\\vec{a} \\cdot \\vec{b})^{2}}\\]<\/p>\n\n\n\n \u2461\u6210\u5206\u8868\u793a\u3067\u306e\u516c\u5f0f<\/p>\n\n\n\n \\(\\vec{OA}=\\vec{a}=(a_{1},a_{2}), \\vec{OB}=\\vec{b}=(b_{1},b_{2})\\)\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\\(\\triangle OAB\\)\u306e\u9762\u7a4dS\u306f\u3001<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2}|a_{1}b_{2}-a_{2}b_{1}|\\]<\/p>\n<\/div><\/div>\n\n\n\n \u3069\u3061\u3089\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u304b\u306f\u554f\u984c\u6587\u3067\u4e0e\u3048\u3089\u308c\u305f\u60c5\u5831\u304b\u3089\u5224\u65ad\u3057\u307e\u3057\u3087\u3046\uff01<\/span><\/p>\n\n\n\n \u672c\u8a18\u4e8b\u3067\u306f\u30d9\u30af\u30c8\u30eb\u3092\u7528\u3044\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u516c\u5f0f<\/span>\u306b\u3064\u3044\u3066\u5b66\u7fd2\u3057\u307e\u3059\u3002 \u305d\u308c\u3067\u306f\u30d9\u30af\u30c8\u30eb\u3092\u4f7f\u3063\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306e\u6c42\u3081\u65b9\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n \u30d9\u30af\u30c8\u30eb\u3092\u7528\u3044\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u516c\u5f0f\u306b\u306f\u4ee5\u4e0b\u306e2\u3064\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n 1\u3064\u305a\u3064\u3086\u3063\u304f\u308a\u7406\u89e3\u3092\u6df1\u3081\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n \\(\\triangle OAB\\)\u304c\u3042\u308a\u3001\\(\\vec{OA}=\\vec{a}, \\vec{OB}=\\vec{b}\\)\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\\(\\triangle OAB\\)\u306e\u9762\u7a4dS\u306f\u3001<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2} \\sqrt{|\\vec{a}|^{2}|\\vec{b}|^{2}-(\\vec{a} \\cdot \\vec{b})^{2}}\\]<\/p>\n\n\n\n \u3067\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n\n\n\n \u3053\u3053\u3067\u306f\u3001\u30d9\u30af\u30c8\u30eb\u3092\u305d\u308c\u305e\u308c\u6210\u5206\u8868\u793a\u3067\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \\(\\triangle OAB\\)\u304c\u3042\u308a\u3001\\(\\vec{OA}=\\vec{a}=(a_{1},a_{2}), \\vec{OB}=\\vec{b}=(b_{1},b_{2})\\)\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\\(\\triangle OAB\\)\u306e\u9762\u7a4dS\u306f\u3001<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2}|a_{1}b_{2}-a_{2}b_{1}|\\]<\/p>\n\n\n\n \u3067\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n\n\n\n \u30d9\u30af\u30c8\u30eb\u3092\u7528\u3044\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306f\u3001\u5148\u307b\u3069\u7d39\u4ecb\u3057\u305f\u516c\u5f0f\u3067\u6c42\u3081\u3089\u308c\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002 \\(|\\vec{OA}|=3,|\\vec{OB}|=4,\\vec{OA} \\cdot \\vec{OB}=7\\)\u306e\u3068\u304d\u3001\\(\\triangle OAB\\)\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088\u3002<\/p>\n\n\n\n \u3053\u306e\u554f\u984c\u306e\u5834\u5408\u3001\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u3068\u5185\u7a4d\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u306e\u3067\u3001\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306e\u516c\u5f0f\u2460\u30d9\u30af\u30c8\u30eb\u8868\u793a\u3067\u306e\u516c\u5f0f\u304c\u4f7f\u3048\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \\begin{eqnarray} \u3053\u308c\u3067\u3001\u9762\u7a4d\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \\(O(0,0),A(3,2),B(1,4)\\)\u306e\u3068\u304d\u3001\\(\\triangle OAB\\)\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088\u3002<\/p>\n\n\n\n \u3053\u306e\u554f\u984c\u306e\u5834\u5408\u3001\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u8868\u793a\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u306e\u3067\u3001\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306e\u516c\u5f0f\u2461\u6210\u5206\u8868\u793a\u3067\u306e\u516c\u5f0f\u304c\u4f7f\u3048\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \\(\\vec{OA}=(3,2),\\vec{OB}=(1,4)\\)\u3067\u3042\u308a\u3001<\/p>\n\n\n\n \\begin{eqnarray} \u3053\u3053\u304b\u3089\u306f\u3001\u5148\u307b\u3069\u306e2\u3064\u306e\u9762\u7a4d\u516c\u5f0f\u306e\u8a3c\u660e\u3092\u3057\u3066\u3044\u304d\u307e\u3059\u3002 \u307e\u305a\u306f\u3001\u516c\u5f0f\u306e\u5fa9\u7fd2\u3067\u3059\u3002<\/p>\n\n\n\n \\(\\triangle OAB\\)\u304c\u3042\u308a\u3001\\(\\vec{OA}=\\vec{a}, \\vec{OB}=\\vec{b}\\)\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\\(\\triangle OAB\\)\u306e\u9762\u7a4dS\u306f<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2}\\sqrt{|\\vec{a}|^{2} |\\vec{b}|^{2}-(\\vec{a} \\cdot \\vec{b})^{2}}\\]<\/p>\n\n\n\n \u3092\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n \u8a3c\u660e\uff1a \\[\\displaystyle S=\\frac{1}{2}|\\vec{a}||\\vec{b}| \\sin\\theta\\]<\/p>\n\n\n\n \\(0<\\theta<180^\\circ \\)\u3088\u308a\u3001\\(\\sin\\theta>0\\)\u3067\u3042\u308b\u304b\u3089\u3001<\/p>\n\n\n\n \\(\\sin^{2}{\\theta}+\\cos^{2}{\\theta}=1\\)\u3088\u308a<\/p>\n\n\n\n \\[\\sin \\theta=\\sqrt{1-\\cos^{2}{\\theta}}\\]<\/p>\n\n\n\n \u3053\u308c\u3092\u5148\u307b\u3069\u306e\u5f0f\u306b\u5f53\u3066\u306f\u3081\u308b\u3068\u3001<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2}|\\vec{a}||\\vec{b}|\\sqrt{1-\\cos^{2}{\\theta}}\\]<\/p>\n\n\n\n \u3053\u3053\u3067\u3001\u5185\u7a4d\u306e\u516c\u5f0f \\(\\vec{a} \\cdot \\vec{b}=|\\vec{a}||\\vec{b}| \\cos{\\theta}\\)\u304b\u3089<\/p>\n\n\n\n \\[\\displaystyle \\cos{\\theta}=\\frac{\\vec{a} \\cdot \\vec{b}}{|\\vec{a}||\\vec{b}|}\\]<\/p>\n\n\n\n \\begin{eqnarray} \u307e\u305a\u306f\u3001\u516c\u5f0f\u306e\u5fa9\u7fd2\u3067\u3059\u3002<\/p>\n\n\n\n \\(\\triangle OAB\\)\u304c\u3042\u308a\u3001\\(\\vec{OA}=\\vec{a}=(a_{1},a_{2}), \\vec{OB}=\\vec{b}=(b_{1},b_{2})\\)\u3068\u3059\u308b\u3002<\/p>\n\n\n\n \u3053\u306e\u3068\u304d\\(\\triangle OAB\\)\u306e\u9762\u7a4dS\u306f\u3001<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2}|a_{1}b_{2}-a_{2}b_{1}|\\]<\/p>\n\n\n\n \u3067\u306f\u3001\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n \u8a3c\u660e\uff1a \u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u3068\u3001\u5185\u7a4d\u304b\u3089<\/p>\n\n\n\n \\(|\\vec{a}|^{2}={a_{1}}^{2}+{a_{2}}^{2}\\) \u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n\n\n\n \u3053\u306e\u3068\u304d\u3001<\/p>\n\n\n\n \\(\\displaystyle S=\\frac{1}{2}\\sqrt{|\\vec{a}|^{2}|\\vec{b}|^{2}-(\\vec{a} \\cdot \\vec{b})^{2}}\\)\u3000\uff08\u2460\u30d9\u30af\u30c8\u30eb\u8868\u793a\u3067\u306e\u516c\u5f0f\u3088\u308a\uff09<\/p>\n\n\n\n \\begin{eqnarray} \u3053\u3053\u304b\u3089\u306f\u30d9\u30af\u30c8\u30eb\u3092\u7528\u3044\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u554f\u984c\u30922\u554f\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \\(\\vec{OA}=3,\\ \\vec{OB}=4,\\ \\angle AOB=30^\\circ\\)\u3067\u3042\u308b\u3002\\(\\triangle OAB\\)\u306e\u9762\u7a4dS\u3092\u6c42\u3081\u3088\u3002<\/p>\n\n\n\n \u307e\u305a\u3001\\(\\vec{OA} \\cdot \\vec{OB}\\)\u3092\u6c42\u3081\u308b\u3002<\/p>\n\n\n\n \\begin{eqnarray} \u3053\u3053\u3067<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2}\\sqrt{|\\vec{OA}|^{2}|\\vec{OB}|^{2}-(\\vec{OA} \\cdot \\vec{OB})^{2}}\\]<\/p>\n\n\n\n \u3088\u308a\u3001<\/p>\n\n\n\n \\begin{eqnarray} \u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\u3001\\(A(2,4),B(4,1),C(-1,0)\\)\u304c\u3042\u308b\u3002\u3053\u306e\u3068\u304d\\(\\triangle ABC\\)\u306e\u9762\u7a4dS\u3092\u6c42\u3081\u3088<\/p>\n\n\n\n \u2461\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u305f\u3081\u306b\u3001\u3069\u3053\u304b1\u70b9\u3092\u5e73\u884c\u79fb\u52d5\u3055\u305b\u3066\u539f\u70b9\u306b\u6301\u3063\u3066\u304d\u307e\u3059\u3002<\/p>\n\n\n\n 3\u70b9\u3092\u540c\u3058\u65b9\u5411\u306b\u3001\u540c\u3058\u8ddd\u96e2\u5e73\u884c\u79fb\u52d5\u3057\u3066\u3082\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306e\u5927\u304d\u3055\u306f\u5909\u308f\u3089\u306a\u3044\u305f\u3081\u3001\u3053\u306e\u65b9\u6cd5\u3067\u89e3\u3044\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u4eca\u56de\u306f\u3001\u70b9C(-1,0)\u3092\u539f\u70b9\u306b\u5e73\u884c\u79fb\u52d5\u3055\u305b\u307e\u3059\u3002\uff08x\u65b9\u5411\u306b+1\u5e73\u884c\u79fb\u52d5\uff09 \u6c42\u3081\u305f\u3044\u4e09\u89d2\u5f62\u306e\u9762\u7a4dS\u306f\u3001\\(\\triangle OA^{\\prime}B^{\\prime}\\)\u306e\u9762\u7a4d\u3068\u540c\u3058\u3067\u3042\u308b\u304b\u3089\u3001<\/p>\n\n\n\n \\begin{eqnarray} \u4eca\u56de\u306f\u30d9\u30af\u30c8\u30eb\u3092\u7528\u3044\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306b\u3064\u3044\u3066\u5b66\u7fd2\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u30d9\u30af\u30c8\u30eb\u3092\u7528\u3044\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u516c\u5f0f\u306b\u306f\u4ee5\u4e0b\u306e2\u3064\u304c\u3042\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u2460\u30d9\u30af\u30c8\u30eb\u8868\u793a\u3067\u306e\u516c\u5f0f<\/p>\n\n\n\n \\(\\triangle OAB\\)\u304c\u3042\u308a\u3001\\(\\vec{OA}=\\vec{a}, \\vec{OB}=\\vec{b}\\)\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\\(\\triangle OAB\\)\u306e\u9762\u7a4dS\u306f\u3001<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2} \\sqrt{|\\vec{a}|^{2}|\\vec{b}|^{2}-(\\vec{a} \\cdot \\vec{b})^{2}}\\]<\/p>\n\n\n\n \u3067\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n\n\n\n \u2781\u6210\u5206\u8868\u793a\u3067\u306e\u516c\u5f0f<\/p>\n\n\n\n \u3053\u3053\u3067\u306f\u3001\u30d9\u30af\u30c8\u30eb\u3092\u305d\u308c\u305e\u308c\u6210\u5206\u8868\u793a\u3067\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \\(\\triangle OAB\\)\u304c\u3042\u308a\u3001\\(\\vec{OA}=\\vec{a}=(a_{1},a_{2}), \\vec{OB}=\\vec{b}=(b_{1},b_{2})\\)\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\\(\\triangle OAB\\)\u306e\u9762\u7a4dS\u306f\u3001<\/p>\n\n\n\n \\[\\displaystyle S=\\frac{1}{2}|a_{1}b_{2}-a_{2}b_{1}|\\]<\/p>\n\n\n\n \u3067\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<\/div><\/div>\n\n\n\n \u30d9\u30af\u30c8\u30eb\u3067\u9762\u7a4d\u3092\u6c42\u3081\u308b\u306b\u306f\u3001\u300c\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u300d<\/span>\u306b\u3064\u3044\u3066\u3082\u3057\u3063\u304b\u308a\u3068\u7406\u89e3\u3057\u3066\u304a\u304f\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \u305d\u308c\u3067\u306f\u6700\u5f8c\u307e\u3067\u3054\u89a7\u3044\u305f\u3060\u304d\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u307f\u3093\u306a\u306e\u52aa\u529b\u304c\u5831\u308f\u308c\u307e\u3059\u3088\u3046\u306b\uff01<\/p>\n","protected":false},"excerpt":{"rendered":" \u4eca\u56de\u306f\u6570\u5b66B\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u300c\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u516c\u5f0f\u300d\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002 \u30d9\u30af\u30c8\u30eb\u3092\u7528\u3044\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306f\u4ee5\u4e0b\u306e2\u3064\u304c\u3042\u308a\u307e\u3059\u3002 \u3069\u3061\u3089\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u304b\u306f\u554f\u984c\u6587\u3067\u4e0e\u3048\u3089\u308c\u305f\u60c5\u5831\u304b\u3089\u5224\u65ad\u3057\u307e\u3057\u3087\u3046\uff01 \u672c\u8a18\u4e8b\u3067\u306f\u30d9\u30af\u30c8\u30eb\u3092 […]<\/p>\n","protected":false},"author":1,"featured_media":14716,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","footnotes":""},"categories":[16,225],"tags":[17,14,11],"class_list":["post-14709","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-vector","category-math-b","tag-17","tag-b","tag-11"],"yoast_head":"\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n
\u516c\u5f0f\u30922\u3064\u7d39\u4ecb\u3057\u307e\u3059\u306e\u3067\u3001\u305c\u3072\u554f\u984c\u306b\u5408\u308f\u305b\u3066\u4f7f\u3044\u3053\u306a\u305b\u308b\u3088\u3046\u306b\u306a\u3063\u3066\u304f\u3060\u3055\u3044\u306d\uff01<\/p>\n\n\n\n\u30d9\u30af\u30c8\u30eb\u306e\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u516c\u5f0f<\/h2>\n\n\n\n
\n
\u2460\u30d9\u30af\u30c8\u30eb\u8868\u793a\u3067\u306e\u516c\u5f0f<\/h3>\n\n\n
<\/figure>\n<\/div>\n\n\n\u2461\u6210\u5206\u8868\u793a\u3067\u306e\u516c\u5f0f<\/h3>\n\n\n
<\/figure>\n<\/div>\n\n\n\u30d9\u30af\u30c8\u30eb\u3092\u4f7f\u3063\u305f\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306e\u6c42\u3081\u65b9<\/h2>\n\n\n\n
\u3053\u3053\u3067\u306f\u3001\u5b9f\u969b\u306b\u3069\u3046\u7b54\u3048\u3092\u5c0e\u304d\u3060\u3057\u3066\u3044\u304f\u306e\u304b\u3001\u624b\u9806\u3092\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
<\/p>\n<\/div><\/div>\n\n\n\n
\\displaystyle S&=&\\frac{1}{2}\\sqrt{|\\vec{OA}|^{2}|\\vec{OB}|^{2}-(\\vec{OA} \\cdot \\vec{OB})^{2}}\\\\
\\displaystyle &=&\\frac{1}{2}\\sqrt{3^{2} \\cdot 5^{2}-7^{2}}\\\\
\\displaystyle &=&\\frac{1}{2} \\cdot 4\\sqrt{11}\\\\
&=&2\\sqrt{11}
\\end{eqnarray}<\/p>\n\n\n\n
<\/p>\n<\/div><\/div>\n\n\n\n
\\displaystyle S&=&\\frac{1}{2}|a_{1}b_{2}-a_{2}b_{1}|\\\\
\\displaystyle &=&\\frac{1}{2}|3 \\cdot 4-2 \\cdot 1|\\\\
\\displaystyle &=&\\frac{1}{2} \\cdot 10\\\\
&=&5
\\end{eqnarray}<\/p>\n\n\n\n\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u516c\u5f0f\u300a\u8a3c\u660e\u300b<\/h2>\n\n\n\n
\u2460\u2461\u305d\u308c\u305e\u308c\u5206\u3051\u3066\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n\u2460\u30d9\u30af\u30c8\u30eb\u8868\u793a\u3067\u306e\u516c\u5f0f\u300a\u8a3c\u660e\u300b<\/h3>\n\n\n
<\/figure>\n<\/div>\n\n\n
\\(\\angle AOB=\\theta\\)\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u9762\u7a4dS\u306f\u3001<\/p>\n\n\n\n
\\displaystyle S&=&\\frac{1}{2}|\\vec{a}||\\vec{b}|\\sqrt{1-\\left(\\frac{\\vec{a} \\cdot \\vec{b}}{|\\vec{a}||\\vec{b}|}\\right)^{2}}\\\\
\\displaystyle &=&\\frac{1}{2}\\sqrt{|\\vec{a}|^{2}|\\vec{b}|^{2}-{(\\vec{a} \\cdot \\vec{b})}^{2}}
\\end{eqnarray}<\/p>\n\n\n\n\u2461\u6210\u5206\u8868\u793a\u3067\u306e\u516c\u5f0f\u300a\u8a3c\u660e\u300b<\/h3>\n\n\n
<\/figure>\n<\/div>\n\n\n
\\(\\vec{OA}=\\vec{a}= (a_{1},a_{2}), \\vec{OB}=\\vec{b}=(b_{1},b_{2})\\)\u3067\u3042\u308b\u3068\u304d\u3001<\/p>\n\n\n\n
\\(|\\vec{b}|^{2}={b_{1}}^{2}+{b_{2}}^{2}\\)
\\(\\vec{a} \\cdot \\vec{b}=a_{1}b_{1}+a_{2}b_{2}\\)<\/p>\n\n\n\n
\\displaystyle S&=&\\frac{1}{2}\\sqrt{({a_{1}}^{2}+{a_{2}}^{2})({b_{1}}^{2}+{b_{2}}^{2})-(a_{1}b_{1}+a_{2}b_{2})^{2}}\\\\
\\displaystyle &=&\\frac{1}{2}\\sqrt{{a_{1}}^{2}{b_{2}}^{2}-2a_{1}b_{1}a_{2}b_{2}+{a_{2}}^{2}{b_{1}}^{2}}\\\\
\\displaystyle &=&\\frac{1}{2}\\sqrt{{(a_{1}b_{2}-a_{2}b_{1})}^{2}}\\\\
\\displaystyle &=&\\frac{1}{2}|(a_{1}b_{2}-a_{2}b_{1})|
\\end{eqnarray}<\/p>\n\n\n\n\u30d9\u30af\u30c8\u30eb\u306e\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u300a\u7df4\u7fd2\u554f\u984c\u300b<\/h2>\n\n\n\n
<\/p>\n<\/div><\/div>\n\n\n\n\u89e3\u7b54\u3092\u30c1\u30a7\u30c3\u30af\u3059\u308b<\/span><\/i><\/i><\/span><\/summary>
\\vec{OA} \\cdot \\vec{OB}&=&|\\vec{OA}||\\vec{OB}|\\cos 30^\\circ\\\\
&=&3 \\times 4 \\times \\frac{\\sqrt{3}}{2}\\\\
&=&6\\sqrt3
\\end{eqnarray}<\/p>\n\n\n\n
\\displaystyle S&=&\\frac{1}{2}\\sqrt{3^{2} \\cdot 4^{2}-(6 \\sqrt{3})^{2}}\\\\
\\displaystyle &=&\\frac{1}{2}\\sqrt{9 \\cdot 16-108}\\\\
&=&3
\\end{eqnarray}<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n<\/span><\/i><\/i><\/span><\/summary>
<\/p>\n<\/div><\/div>\n\n\n\n\u89e3\u7b54\u3092\u30c1\u30a7\u30c3\u30af\u3059\u308b<\/span><\/i><\/i><\/span><\/summary>
\u4ed6\u306e\u70b9\u3082\u3001\u540c\u3058\u3088\u3046\u306bx\u65b9\u5411\u306b+1\u5e73\u884c\u79fb\u52d5\u3055\u305b\u308b\u3068\u3001\u70b9A(2,4)\u2192\u70b9A'(3,4)\u3001\u70b9B(4,1)\u2192\u70b9B'(5,1)\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n
<\/figure>\n<\/div>\n\n\n
\\displaystyle S&=&\\frac{1}{2}|3 \\cdot1-5 \\cdot4|\\\\
\\displaystyle &=&\\frac{1}{2} \\cdot 17\\\\
\\displaystyle &=&\\frac{17}{2}
\\end{eqnarray}<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n\u30d9\u30af\u30c8\u30eb\u306e\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u3000\u307e\u3068\u3081<\/h2>\n\n\n\n
\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n