{"id":14675,"date":"2025-12-24T17:22:58","date_gmt":"2025-12-24T08:22:58","guid":{"rendered":"https:\/\/math-travel.com\/?p=14675"},"modified":"2026-03-06T01:08:28","modified_gmt":"2026-03-05T16:08:28","slug":"vector-angle","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-b\/vector-angle\/","title":{"rendered":"\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u306e\u6c42\u3081\u65b9\u30fc\u5e73\u9762\u30fb\u7a7a\u9593\u3069\u3061\u3089\u3082\u5185\u7a4d\u306e\u516c\u5f0f\u3067\u4e00\u767a\u89e3\u6c7a\uff01"},"content":{"rendered":"\n
\u4eca\u56de\u89e3\u6c7a\u3059\u308b\u60a9\u307f<\/span><\/div>
\n

\u300c\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3063\u3066\u306a\u306b\uff1f\u300d
\u300c\u89d2\u5ea6\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\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u300d<\/span>\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u3042\u308b\u3068\u304d\u30012\u30d9\u30af\u30c8\u30eb\u306e\u9593\u306b\u306f\u89d2\u304c\u751f\u307e\u308c\u307e\u3059\u3002\u3053\u306e\u89d2\u306e\u3053\u3068\u3092\u201c\u306a\u3059\u89d2\u201d\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2<\/span><\/div>
\n

\"2\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u306e\u56f3\"<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3068\u306f\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u540c\u58eb\u3092\u91cd\u306d\u305f\u5834\u5408\u306b\u4f5c\u3089\u308c\u308b\\(180^\\circ\\)\u4ee5\u4e0b\u306e\u89d2\u5ea6\u306e\u3053\u3068\u3002<\/p>\n<\/div><\/div>\n\n\n\n

\u306a\u3059\u89d2\u306f180\u5ea6\u3092\u8d85\u3048\u306a\u3044\u306e\u3082\u30dd\u30a4\u30f3\u30c8\u3067\u3059\u3002<\/span><\/p>\n\n\n\n

\u672c\u8a18\u4e8b\u3067\u306f\u300c\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u300d\u3092\u7df4\u7fd2\u554f\u984c\u3082\u4ea4\u3048\u306a\u304c\u3089\u89e3\u8aac<\/span>\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5834\u5408\u3068\u3001\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5834\u5408\u3069\u3061\u3089\u3082\u3057\u3063\u304b\u308a\u30de\u30b9\u30bf\u30fc\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\uff01<\/p>\n\n\n\n

\u305d\u308c\u3067\u306f\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3068\u306f\u3069\u3053\uff1f<\/h2>\n\n\n\n

\u305d\u3082\u305d\u3082\u3001\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3068\u306f\u3069\u306e\u89d2\u306e\u3053\u3068\u3067\u3057\u3087\u3046\u304b\uff1f<\/p>\n\n\n\n

\u5b9f\u306f\u3001\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8003\u3048\u308b\u30eb\u30fc\u30eb<\/span>\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n

\n
\"2\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u306e\u56f3\"<\/figure>\n<\/div>\n\n\n
\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2<\/span><\/div>
\n

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u540c\u58eb\u3092\u91cd\u306d\u305f\u5834\u5408\u306b\u4f5c\u3089\u308c\u308b\\(180^\\circ\\)\u4ee5\u4e0b\u306e\u89d2\u5ea6\u306e\u3053\u3068\u3002<\/p>\n<\/div><\/div>\n\n\n\n

\u5fc5\u305a180\u5ea6\u3088\u308a\u5c0f\u3055\u3044\u65b9\u306e\u89d2<\/span>\u3092\u201c\u306a\u3059\u89d2\u201d\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305f\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u300c\u59cb\u70b9\u300d\u306b\u3064\u3044\u3066\u3082\u8003\u3048\u65b9\u304c\u3042\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\u59cb\u70b9\u3068\u306f\u3001\\(\\vec{AB}\\)\u3060\u3063\u305f\u3089\u70b9A\u3001\\(\\vec{BA}\\)\u3060\u3063\u305f\u3089\u70b9B\u306e\u3088\u3046\u306b\u3001\u30d9\u30af\u30c8\u30eb\u306e\u5de6\u5074\u306e\u6587\u5b57\u3067\u8868\u3055\u308c\u308b\u70b9\u306e\u3053\u3068\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\"<\/figure>\n<\/div>\n\n\n

\u5148\u307b\u3069\u8ff0\u3079\u305f\u3088\u3046\u306b\u3001\u30d9\u30af\u30eb\u306e\u306a\u3059\u89d2\u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u540c\u58eb<\/span>\u3092\u91cd\u306d\u3066\u6c42\u3081\u308b\u3082\u306e\u3067\u3057\u305f\u3002
\u3067\u3059\u306e\u3067\u3001\u4e0b\u56f3\u306e\u3088\u3046\u306b\u59cb\u70b9\u304c\u91cd\u306a\u3063\u3066\u3044\u306a\u3044\u5834\u5408\u306f\u3001\u5e73\u884c\u79fb\u52d5\u3057\u3066\u59cb\u70b9\u540c\u58eb\u3092\u91cd\u306d\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n

\n
\"\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\"<\/figure>\n<\/div>\n\n\n

\u4e0a\u306e\u56f3\u3092\u898b\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u3053\u306e\u5834\u5408\u3001\\(\\vec{AB}\\)\u3068\\(\\vec{BC}\\)\u306e\u306a\u3059\u89d2\u306f\u4f55\u5ea6\u306b\u306a\u308b\u3067\u3057\u3087\u3046\u304b\uff1f<\/p>\n\n\n\n

\u30d1\u30c3\u3068\u898b\u305f\u3060\u3051\u3060\u3068\u300c\\(40^\\circ\\)\u300d\u3068\u7b54\u3048\u305d\u3046\u306b\u306a\u308a\u307e\u3059\u304c\u3001\\(\\vec{AB}\\)\u3068\\(\\vec{BC}\\)\u306f\u3001\u59cb\u70b9\u540c\u58eb\u304c\u91cd\u306a\u3063\u3066\u3044\u306a\u3044\u305f\u3081\u3001\u3053\u306e\u307e\u307e\u3067\u306f\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3002
\u30d9\u30af\u30c8\u30eb\u3092\u5e73\u884c\u79fb\u52d5\u3057\u3066\u3001\u59cb\u70b9\u540c\u58eb\u3092\u91cd\u306d\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n

\n
\"\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u2460\"<\/figure>\n<\/div>\n\n\n

\\(\\vec{AB}\\)\u3092\u5e73\u884c\u79fb\u52d5\u3057\u3001\u59cb\u70b9\u540c\u58eb\u3092\u91cd\u306d\u308b\u3068\u4e0a\u306e\u3088\u3046\u306a\u56f3\u306b\u306a\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

\u3088\u3063\u3066\u3001\\(\\vec{AB}\\)\u3068\\(\\vec{BC}\\)\u306e\u306a\u3059\u89d2\u306f\u3001\\(\\vec{BA’}\\)\u3068\\(\\vec{BC}\\)\u306e\u306a\u3059\u89d2\u306b\u306a\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002
\u3067\u3059\u306e\u3067\u3001\u7b54\u3048\u306f\\(180^\\circ – 40^\\circ=140^\\circ\\)<\/span>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u306e\u898b\u51fa\u3057\u306e\u307e\u3068\u3081<\/span><\/div>
\n
    \n
  • \u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3068\u306f\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u540c\u58eb\u3092\u91cd\u306d\u305f\u5834\u5408\u306b\u4f5c\u3089\u308c\u308b\\(180^\\circ\\)\u4ee5\u4e0b\u306e\u89d2\u5ea6\u306e\u3053\u3068\u3002<\/li>\n\n\n\n
  • 2\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u304c\u91cd\u306a\u3063\u3066\u3044\u306a\u3044\u5834\u5408\u306f\u3001\u4e00\u65b9\u306e\u30d9\u30af\u30c8\u30eb\u3092\u5e73\u884c\u79fb\u52d5\u3057\u3066\u59cb\u70b9\u3092\u91cd\u306d\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n

    \u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3092\u6c42\u3081\u308b<\/h2>\n\n\n\n

    \u3067\u306f\u3001\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u306b\u3064\u3044\u3066\u5206\u304b\u3063\u305f\u3068\u3053\u308d\u3067\u3001\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u306e\u5927\u304d\u3055\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5834\u5408<\/h3>\n\n\n\n
    \u4f8b\u984c\u2460<\/span><\/div>
    \n

    2\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{AB}\\),\\(\\vec{BC}\\)\u306e\u306a\u3059\u89d2\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \"\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u2461\"<\/p>\n<\/div><\/div>\n\n\n\n

    \u3053\u306e\u5834\u5408\u306f\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u304c\u91cd\u306a\u3063\u3066\u3044\u306a\u3044\u305f\u3081\u3001\\(\\vec{AB}\\)\u3092\u5e73\u884c\u79fb\u52d5\u3057\u30662\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u3092\u91cd\u306d\u307e\u3059\u3002<\/p>\n\n\n

    \n
    \"\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u2462\"<\/figure>\n<\/div>\n\n\n

    \u4e0a\u306e\u56f3\u3088\u308a\u3001\\(\\vec{AB}\\),\\(\\vec{BC}\\)\u306e\u306a\u3059\u89d2\u306f\u3001\\(\\vec{BC}\\)\u3068\\(\\vec{BA’}\\)\u3068\u306a\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

    \u3088\u3063\u3066\u3001\u7b54\u3048\u306f\\(180^\\circ\uff0d100^\\circ=80^\\circ\\)<\/span><\/p>\n\n\n\n

    \u4f8b\u984c\u2461<\/span><\/div>
    \n

    2\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a},\\vec{b}\\)\u306e\u306a\u3059\u89d2\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \\(|\\vec{a}|=\\sqrt3,|\\vec{b}|=2,\\vec{a} \\cdot \\vec{b}=2\\)<\/p>\n\n\n\n

    \"\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\"<\/p>\n<\/div><\/div>\n\n\n\n

    \u3053\u306e\u5834\u5408\u306f\u30d9\u30af\u30c8\u30eb\\(\\vec{a},\\vec{b}\\)\u306e\u59cb\u70b9\u306f\u91cd\u306a\u3063\u3066\u3044\u308b\u305f\u3081\u3001\u5e73\u884c\u79fb\u52d5\u306f\u5fc5\u8981\u3042\u308a\u307e\u305b\u3093\u3002
    \u554f\u984c\u3067\u4e0e\u3048\u3089\u308c\u305f\u60c5\u5831\u3092\u4f7f\u3063\u3066\u3001\\(\\vec{a},\\vec{b}\\)\u306e\u306a\u3059\u89d2\\(\\theta\\)\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n

    \u554f\u984c\u3067\u306f\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u3068\u5185\u7a4d\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u307e\u3059\u306d\u3002
    \u3053\u3053\u3067\u5185\u7a4d\u306e\u5b9a\u7fa9\u306b\u3064\u3044\u3066\u5fa9\u7fd2\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \\(\\vec{a},\\vec{b}\\)\u306e\u306a\u3059\u89d2\u3092\\(\\theta\\)\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

    \\[\\vec{a} \\cdot \\vec{b}=|\\vec{a}||\\vec{b}|\\cos{\\theta}\\]<\/p>\n\n\n\n

    \u3053\u3053\u306b\u3001\u554f\u984c\u3067\u4e0e\u3048\u3089\u308c\u305f\u6570\u5024\u3092\u5f53\u3066\u306f\u3081\u308b\u3068<\/p>\n\n\n\n

    \\begin{eqnarray}
    2&=&\\sqrt3 \\cdot 2 \\cdot \\cos{\\theta}\\\\
    \\cos{\\theta}&=&\\frac{1}{\\sqrt3}
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3088\u3063\u3066\u3001\\(\\vec{a},\\vec{b}\\)\u306e\u306a\u3059\u89d2\u306f\\(\\theta=60^\\circ\\)\u3068\u306a\u308b\u3002<\/p>\n\n\n\n

    \u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5834\u5408<\/h3>\n\n\n\n

    \u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5834\u5408\u3082\u3001\u8003\u3048\u65b9\u306f\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u6642\u3068\u540c\u69d8\u3067\u3059\u3002<\/span><\/p>\n\n\n\n

    \u305f\u3060\u3001\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3092\u6c42\u3081\u308b\u554f\u984c\u306f\u3001\u5185\u7a4d\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u554f\u984c\u304c\u591a\u3044\u3067\u3059\u306e\u3067\u3001\u305d\u3061\u3089\u3092\u4f8b\u984c\u3068\u3057\u3066\u89e3\u3044\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \u4f8b\u984c\u2462<\/span><\/div>
    \n

    \\(\\vec{a}=(1,2,-3), \\vec{b}=(2,-3,1)\\)\u3067\u3042\u308b\u3068\u304d\u3001\\(\\vec{a},\\vec{b}\\)\u306e\u306a\u3059\u89d2\\(\\theta\\)\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<\/div><\/div>\n\n\n\n

    \u3053\u3053\u3067\u306f\u3001\\(\\vec{a},\\vec{b}\\)\u306e\u6210\u5206\u8868\u793a\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

    \u5185\u7a4d\u3092\u4f7f\u3044\u306a\u304c\u3089\u3001\\(\\vec{a},\\vec{b}\\)\u306e\u306a\u3059\u89d2\\(\\theta\\)\u3092\u6c42\u3081\u3089\u308c\u305d\u3046\u3067\u3059\u3002<\/p>\n\n\n\n

    \u554f\u984c\u3067\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u6570\u5b57\u304b\u3089\u3001\u5206\u304b\u308b\u3082\u306e\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \\begin{eqnarray}
    |\\vec{a}|&=&\\sqrt{1^2+2^2+{(-3)}^2}=\\sqrt{14}\\\\
    |\\vec{b}|&=&\\sqrt{2^2+{(-3)}^2+1^2}=\\sqrt{14}\\\\
    \\vec{a} \\cdot \\vec{b}&=&1\\times2+2 \\times (-3) \\times (-3) \\times 1=-7
    \\end{eqnarray}<\/p>\n\n\n\n

    \\(\\vec{a} \\cdot \\vec{b} =|\\vec{a}||\\vec{b}|\\cos{\\theta}\\)\u3088\u308a<\/p>\n\n\n\n

    \\begin{eqnarray}
    -7&=&\\sqrt{14}\\times\\sqrt{14}\\times\\cos{\\theta}\\\\
    \\cos{\\theta}&=&-\\frac{1}{2}
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3088\u3063\u3066\u3001\\(\\theta=120\u00b0\\)<\/span><\/p>\n\n\n\n

    \u89d2\u5ea6\u3092\u6c42\u3081\u3088\u3046\u300a\u7df4\u7fd2\u554f\u984c\u300b<\/h2>\n\n\n\n

    \u306a\u3059\u89d2\u306e\u6c42\u3081\u65b9\u3082\u5206\u304b\u3063\u305f\u3068\u3053\u308d\u3067\u3001\u3044\u304f\u3064\u304b\u7df4\u7fd2\u554f\u984c\u306b\u6311\u6226\u3057\u307e\u3057\u3087\u3046\uff01<\/p>\n\n\n\n

    \u7df4\u7fd2\u554f\u984c\u2460<\/span><\/div>
    \n

    \\(\\vec{a}=(-2,1),\\vec{b}=(6,-3)\\)\u3067\u3042\u308b\u3068\u304d\u3001\\(\\vec{a},\\vec{b}\\)\u306e\u306a\u3059\u89d2\\(\\theta\\)\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<\/div><\/div>\n\n\n\n

    \n
    \u89e3\u7b54\u3092\u30c1\u30a7\u30c3\u30af\u3059\u308b<\/span><\/i><\/i><\/span><\/summary>
    \n

    \\(|\\vec{a}|=\\sqrt{{(-2)}^2+1^2}=\\sqrt5\\)
    \\(|\\vec{b}|=\\sqrt{6^2+{(-3)}^2}=\\sqrt{45}=3\\sqrt5\\)<\/p>\n\n\n\n

    \u307e\u305f\u3001\\(\\vec{a},\\vec{b}\\)\u306e\u5185\u7a4d\\(\\vec{a} \\cdot \\vec{b}\\)\u306f\u3001<\/p>\n\n\n\n

    \\[\\vec{a} \\cdot \\vec{b}=(-2) \\times 6+1 \\times (-3)=-15\\]<\/p>\n\n\n\n

    \\(\\vec{a} \\cdot \\vec{b}=|\\vec{a}||\\vec{b}|\\cos{\\theta}\\)\u3088\u308a\u3001<\/p>\n\n\n\n

    \\[-15=\\sqrt5\\times3\\sqrt5\\times\\cos{\\theta}\\]<\/p>\n\n\n\n

    \u3086\u3048\u306b\u3001\\[\\cos{\\theta=-1}\\]<\/p>\n\n\n\n

    \u3088\u3063\u3066\u3001\\(\\theta=180\u00b0\\)<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n

    \u7df4\u7fd2\u554f\u984c\u2461<\/span><\/div>
    \n

    \\(\\vec{a}=(1,2),\\vec{b}=(t,1)\\)\u306e\u3068\u304d\u3001\\(\\vec{a},\\vec{b}\\)\u304c\u5782\u76f4\u3068\u306a\u308b\u3088\u3046\u306a\u5b9a\u6570\\(t\\)\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/p>\n<\/div><\/div>\n\n\n\n

    \n
    \u89e3\u7b54\u3092\u30c1\u30a7\u30c3\u30af\u3059\u308b<\/span><\/i><\/i><\/span><\/summary>
    \n

    \\(|\\vec{a}|=\\sqrt{1^{2}+2^{2}}=\\sqrt5\\)
    \\(|\\vec{b}|=\\sqrt{t^{2}+1^{2}}=\\sqrt{t^{2}+1}\\)<\/p>\n\n\n\n

    \u307e\u305f\u3001\\(\\vec{a} \\cdot \\vec{b}=1 \\times t+2 \\times 1=t+2\\)<\/p>\n\n\n\n

    \\(\\vec{a},\\vec{b}\\)\u304c\u5782\u76f4\u3067\u306a\u3089\u3070\u3001\\(\\cos \\theta=0\\)\u3068\u306a\u308a\u3001<\/p>\n\n\n\n

    \\[\\vec{a} \\cdot \\vec{b}=0\\]<\/p>\n\n\n\n

    \u3088\u3063\u3066\u3001<\/p>\n\n\n\n

    \\[t+2=0\\]<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001\\(t=-2\\)<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n

    \u5782\u76f4\u306e\u5834\u5408\u306f\u3001\\(\\cos\\theta=0\\)\u306b\u306a\u308b\u305f\u3081\u3001\\(|\\vec{a}|,|\\vec{b}|\\)\u306f\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u305b\u3093\u3067\u3057\u305f\u306d\u3002
    \u6b21\u306e\u7ae0\u3067\u306f\u3001\u5782\u76f4\u6761\u4ef6\u3092\u4f7f\u3063\u305f\u554f\u984c\u3082\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\u306e\u5fdc\u7528<\/h2>\n\n\n\n

    \u4ee5\u524d\u3082\u5c11\u3057\u89e3\u8aac\u3057\u307e\u3057\u305f\u304c\u3001\u3053\u3053\u304b\u3089\u306f\u3001\u5185\u7a4d\u306e\u5782\u76f4\u6761\u4ef6\u3068\u5e73\u884c\u6761\u4ef6\u306b\u4ed8\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002
    \u4eca\u56de\u89e3\u8aac\u3057\u305f2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3068\u3082\u95a2\u9023\u6027\u304c\u3042\u308b\u306e\u3067\u3001\u5fa9\u7fd2\u3057\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \u5185\u7a4d\u306e\u5782\u76f4\u6761\u4ef6<\/h3>\n\n\n\n

    0\u3067\u306f\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}\u3001\\vec{b}\\)\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

    \u3053\u306e2\u30d9\u30af\u30c8\u30eb\u304c\u5782\u76f4\u306b\u4ea4\u308f\u308b\u3068\u304d\u3001<\/p>\n\n\n\n

    \\[\\vec{a}\\bot\\vec{b} \\Leftrightarrow \\vec{a} \\cdot \\vec{b}=0 \\Leftrightarrow  x_{1}x_{2}+y_{1}y_{2}\\]<\/p>\n\n\n\n

    \"\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\"<\/p>\n\n\n\n

    \u5b9f\u969b\u3001\u5782\u76f4\u6761\u4ef6\u3092\u4f7f\u3063\u305f\u4f8b\u984c\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u4f8b\u984c<\/span><\/div>
    \n

    0\u3067\u306f\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a},\\vec{b}\\)\u304c\u3042\u308a\u3001\\(2|\\vec{a}|=|\\vec{b}|\\)\u3067\u3042\u308b\u3002
    \\(3\\vec{a}+\\vec{b}\\)\u3068\\(\\vec{a}-2\\vec{b}\\)\u304c\u5782\u76f4\u3068\u306a\u308b\u6642\u3001\\(\\vec{a},\\vec{b}\\)\u306e\u306a\u3059\u89d2\\(\\theta\\)\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<\/div><\/div>\n\n\n\n

    2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u5782\u76f4\u3067\u3042\u308b\u6761\u4ef6\u3092\u4f7f\u3046\u3068\u3001<\/p>\n\n\n\n

    \\((3\\vec{a}+\\vec{b}) \\bot (\\vec{a}-2\\vec{b})\\) \u3088\u308a\u3001<\/p>\n\n\n\n

    \\[(3\\vec{a}+\\vec{b}) \\cdot (\\vec{a}-2\\vec{b})=0\\]<\/p>\n\n\n\n

    \u3086\u3048\u306b<\/p>\n\n\n\n

    \\begin{eqnarray}
    (3\\vec{a}+\\vec{b}) \\cdot (\\vec{a}-2\\vec{b})&=&3|\\vec{a}|^{2}-5\\vec{a} \\cdots \\vec{b}-2|\\vec{b}|^{2}\\\\
    &=&0
    \\end{eqnarray}<\/p>\n\n\n\n

    \\(2|\\vec{a}|=|\\vec{b}|\\)\u306a\u306e\u3067<\/p>\n\n\n\n

    \\begin{eqnarray}
    3|\\vec{a}|^{2}-5\\vec{a} \\cdot \\vec{b}-8|\\vec{a}|^{2}&=&0\\\\
    5\\vec{a} \\cdot \\vec{b}&=&-5|\\vec{a}|^{2}\\\\
    \\vec{a} \\cdot \\vec{b}&=&-|\\vec{a}|^{2}
    \\end{eqnarray}<\/p>\n\n\n\n

    \\(\\vec{a} \\cdot \\vec{b}=|\\vec{a}||\\vec{b}|\\cos{\\theta}\\)\u3067\u3042\u308b\u304b\u3089\u3001<\/p>\n\n\n\n

    \\begin{eqnarray}
    \\displaystyle \\cos{\\theta}&=&\\frac{\\vec{a} \\cdot \\vec{b}}{|\\vec{a}||\\vec{b}|}\\\\
    \\displaystyle &=&-\\frac{\\vec{a} \\cdot \\vec{b}}{|\\vec{a}| \\cdot 2|\\vec{a}|}\\\\
    \\displaystyle &=&-\\frac{1}{2}\\\\
    \\theta&=&120\u00b0
    \\end{eqnarray}<\/p>\n\n\n\n

    \u5185\u7a4d\u306e\u5e73\u884c\u6761\u4ef6<\/h3>\n\n\n\n

    0\u3067\u306f\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}=(x_{1},y_{1}),\\vec{b}=(x_{2},y_{2}\\))\u304c\u3042\u308b\u3068\u304d<\/p>\n\n\n\n

    \\[\\vec{a}\/\/\\vec{b} \\Leftrightarrow \\vec{b}=k\\vec{a}\\]<\/p>\n\n\n\n

    \\[\\vec{a}\/\/\\vec{b} \\Leftrightarrow x_{1}y_{2}-x_{2}y_{1}=0\\]<\/p>\n\n\n\n

    \u3068\u306a\u308b\u5b9f\u6570k\u304c\u3042\u308b\u3002<\/p>\n\n\n\n

    \"\u30d9\u30af\u30c8\u30eb\u306e\u5e73\u884c\"<\/p>\n\n\n\n

    \u5b9f\u969b\u3001\u5e73\u884c\u6761\u4ef6\u3092\u4f7f\u3063\u305f\u4f8b\u984c\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u4f8b\u984c<\/span><\/div>
    \n

    0\u3067\u306f\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a},\\vec{b}\\)\u304c\u3042\u308b\u3002
    \\(\\vec{a}=(1,2t),\\vec{b}=(t,t^{2}+1)\\)\u304c\u5e73\u884c\u3068\u306a\u308b\u6642\u3001\\(t\\)\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/p>\n<\/div><\/div>\n\n\n\n

    \u307e\u305a\u3001\\(\\vec{a},\\vec{b}\\)\u306f\u5e73\u884c\u3067\u3042\u308b\u304b\u3089\u3001<\/p>\n\n\n\n

    \\begin{eqnarray}
    1(t^{2}+1)-2t^{2}&=&0\\\\
    t^{2}&=&1\\\\
    t&=&\u00b11
    \\end{eqnarray}<\/p>\n\n\n\n

    [1] \\(t=1\\)\u306e\u3068\u304d
    \\(\\vec{a}=(1,2),\\vec{b}=(1,2)\\)\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

    \u3053\u306e\u3068\u304d\\(\\cos \\theta=1\\)\u3088\u308a\u3001\\(\\theta=0\u00b0\\)<\/p>\n\n\n\n

    [2] \\(t=-1\\)\u306e\u3068\u304d
    \\(\\vec{a}=(1,-2),\\vec{b}=(-1,2)\\)\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

    \u3053\u306e\u3068\u304d\\(\\cos\\theta=-1\\)\u3088\u308a\u3001\\(\\theta=180\u00b0\\)<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001\u3044\u305a\u308c\u306b\u305b\u3088\\(\\vec{a},\\vec{b}\\)\u304c\u5e73\u884c\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

    \u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3000\u307e\u3068\u3081<\/h2>\n\n\n\n

    \u4eca\u56de\u306f\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u306b\u3064\u3044\u3066\u5b66\u7fd2\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

    \u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u30fb\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u4e21\u65b9\u3067\u89d2\u5ea6\u306b\u3064\u3044\u3066\u8003\u3048\u307e\u3057\u305f\u304c\u3001\u307e\u305a\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u304b\u3089\u7406\u89e3\u3092\u9032\u3081\u308b\u3068\u5206\u304b\u308a\u3084\u3059\u3044\u3067\u3059\uff01<\/span><\/p>\n\n\n\n

    \u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2<\/span><\/div>
    \n

    \"2\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u306e\u56f3\"<\/p>\n\n\n\n

    \u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3068\u306f\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u540c\u58eb\u3092\u91cd\u306d\u305f\u5834\u5408\u306b\u4f5c\u3089\u308c\u308b\\(180^\\circ\\)\u4ee5\u4e0b\u306e\u89d2\u5ea6\u306e\u3053\u3068\u3002<\/p>\n\n\n\n

    \u4e0b\u56f3\u306e\u3088\u3046\u306b2\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u304c\u91cd\u306a\u3063\u3066\u3044\u306a\u3044\u5834\u5408\u306f\u3001\u4e00\u65b9\u306e\u30d9\u30af\u30c8\u30eb\u3092\u5e73\u884c\u79fb\u52d5\u3057\u3066\u59cb\u70b9\u3092\u91cd\u306d\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n\n\n\n

    \"\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u2460\"<\/p>\n<\/div><\/div>\n\n\n\n

    \u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\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\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u300d\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002 2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u3042\u308b\u3068\u304d\u30012\u30d9\u30af\u30c8\u30eb\u306e\u9593\u306b\u306f\u89d2\u304c\u751f\u307e\u308c\u307e\u3059\u3002\u3053\u306e\u89d2\u306e\u3053\u3068\u3092\u201c\u306a\u3059\u89d2\u201d\u3068\u547c\u3073\u307e\u3059\u3002 \u306a\u3059\u89d2\u306f180\u5ea6\u3092\u8d85\u3048\u306a\u3044\u306e\u3082\u30dd\u30a4\u30f3\u30c8\u3067\u3059 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