{"id":14553,"date":"2025-12-24T17:22:58","date_gmt":"2025-12-24T08:22:58","guid":{"rendered":"https:\/\/math-travel.com\/?p=14553"},"modified":"2026-03-06T01:14:28","modified_gmt":"2026-03-05T16:14:28","slug":"vector-product","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-b\/vector-product\/","title":{"rendered":"\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306e\u516c\u5f0f\u3068\u6c42\u3081\u65b9\uff01\u610f\u5473\u304c\u308f\u304b\u308c\u3070\u300c\u306a\u3059\u89d2\u300d\u306e\u8a08\u7b97\u3082\u7c21\u5358"},"content":{"rendered":"\n
\u4eca\u56de\u89e3\u6c7a\u3059\u308b\u60a9\u307f<\/span><\/div>
\n

\u300c\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3063\u3066\u306a\u306b\uff1f\u300d
\u300c\u5185\u7a4d\u306e\u516c\u5f0f\u3084\u6c42\u3081\u65b9\u304c\u77e5\u308a\u305f\u3044\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\u5185\u7a4d\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\u306b\u306f\u201c\u5185\u7a4d\u201d\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u3001\u5185\u7a4d\u306b\u3088\u3063\u30662\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u306a\u3059\u89d2\u306e\u5927\u304d\u3055\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f<\/span><\/div>
\n

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}\\)\u3001\\(\\vec{b}\\)\u304c\u3042\u308a\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u540c\u58eb\u3092\u7e4b\u3044\u3067\u3067\u304d\u305f\u89d2\u3092\u03b8\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\"\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3068\u306f\uff1f\"<\/p>\n\n\n\n

\\[\\vec{a} \\cdot \\vec{b}=|\\vec{a}||\\vec{b}| \\cos \\theta \\cdots \u2460\\]<\/p>\n\n\n\n

\\[\\vec{a} \\cdot \\vec{b}=x_{1}x_{2}+y_{1}y_{2} \\cdots \u2461\\]<\/p>\n<\/div><\/div>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f\u3068\u3066\u3082\u91cd\u8981\u306a\u516c\u5f0f\u306a\u306e\u3067<\/span>\u3001\u4eca\u56de\u3067\u5fc5\u305a\u7406\u89e3\u3057\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u672c\u8a18\u4e8b\u3067\u306f\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306e\u516c\u5f0f\u3084\u6c42\u3081\u65b9\u306b\u3064\u3044\u3066\u89e3\u8aac<\/span>\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305f\u3001\u8a18\u4e8b\u4e0b\u3067\u306f\u30d9\u30af\u30c8\u30eb\u306e\u91cd\u8981\u516c\u5f0f<\/span>\u306b\u3064\u3044\u3066\u3082\u8aac\u660e\u3057\u3066\u3044\u308b\u306e\u3067\u3001\u5408\u308f\u305b\u3066\u53c2\u8003\u306b\u3057\u3066\u3044\u305f\u3060\u3051\u308c\u3070\u3068\u601d\u3044\u307e\u3059\uff01<\/p>\n\n\n\n

\u305d\u308c\u3067\u306f\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\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\u5185\u7a4d\u3068\u306f\uff1f<\/h2>\n\n\n\n

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}\\)\u3001\\(\\vec{b}\\)\u304c\u3042\u308a\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u540c\u58eb\u3092\u7e4b\u3044\u3067\u3067\u304d\u305f\u89d2\u3092\u03b8\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3068\u306f\uff1f\"<\/figure>\n<\/div>\n\n\n

\u3053\u306e\u3068\u304d\u3001\\(\\vec{a}\\)\u3068\\(\\vec{b}\\)\u306e\u5185\u7a4d\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

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

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

\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u540c\u58eb\u3092\u639b\u3051\u3066\u3001\u3042\u3044\u3060\u306e\u89d2\u306e\\(\\cos \\theta\\)\u3092\u639b\u3051\u308b\u305f\u3081\u3001<\/p>\n\n\n\n

\\(|\\vec{a}|=0\\) \u3082\u3057\u304f\u306f \\(|\\vec{b}|=0\\)\u306e\u3068\u304d<\/p>\n\n\n\n

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

\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f<\/h2>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\u306e\u5b9a\u7fa9\u3092\u8a18\u8f09\u3057\u307e\u3057\u305f\u304c\u3001\u300c\u5177\u4f53\u7684\u306b\u3069\u3046\u4f7f\u3046\u306e\uff1f\u300d\u3068\u307e\u3060\u30a4\u30e1\u30fc\u30b8\u304c\u6e67\u304b\u306a\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u3053\u304b\u3089\u306f\u5185\u7a4d\u306b\u95a2\u4fc2\u3059\u308b\u516c\u5f0f\u3092\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u2460\u89d2\u5ea6cos\u3092\u4f7f\u3046\u516c\u5f0f\uff08\u5b9a\u7fa9\uff09<\/h3>\n\n\n\n

\u5185\u7a4d\u306e\u516c\u5f0f\u306e1\u3064\u306f\u3001\u5148\u307b\u3069\u89e3\u8aac\u3057\u305f\u5b9a\u7fa9\u3067\u3059\u3002<\/p>\n\n\n

\n
\"\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3068\u306f\uff1f\"<\/figure>\n<\/div>\n\n\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\u2460<\/span><\/div>
\n

0\u3067\u306f\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}\\)\u3001\\(\\vec{b}\\)\u306e\u59cb\u70b9\u540c\u58eb\u3092\u3064\u306a\u3044\u3067\u3067\u304d\u308b\u89d2\u3092\u03b8\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

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

\u2461\u6210\u5206\u8868\u793a\u3092\u4f7f\u3046\u516c\u5f0f<\/h3>\n\n\n\n

\u5185\u7a4d\u306e\u516c\u5f0f\u306f\u3082\u30461\u3064\u3042\u308a\u3001\u6210\u5206\u8868\u793a\u3092\u4f7f\u3063\u3066\u5185\u7a4d\u3092\u8868\u3059<\/span>\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\"<\/figure>\n<\/div>\n\n\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\u2461<\/span><\/div>
\n

\\(\\vec{a}=(x_{1},y_{1}),\\vec{b}=(x_{2},y_{2})\\)\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

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

\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u6c42\u3081\u308b<\/h2>\n\n\n\n

\u5148\u307b\u3069\u7d39\u4ecb\u3057\u305f\u516c\u5f0f\u3092\u3082\u3068\u306b\u3001\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\uff1a\u4f8b\u984c\u2460<\/span><\/div>
\n

\"\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u6c42\u3081\u308b\"<\/p>\n\n\n\n

\\(|\\vec{a}|=2,|\\vec{b}|=1\\)\u3001\u307e\u305f\\(\\vec{a},\\vec{b}\\)\u306e\u59cb\u70b9\u540c\u58eb\u3092\u7e4b\u3044\u3067\u3067\u304d\u308b\u89d2\u3092\u03b8\u3068\u3057\u3066\u3001\\(\\theta=30\u00b0\\)\u3067\u3042\u308b\u3068\u304d\u3001\\(\\vec{a}\\)\u3068\\(\\vec{b}\\)\u306e\u5185\u7a4d\\(\\vec{a} \\cdot \\vec{b}\\)\u3092\u6c42\u3081\u3088\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}\\)\u3068\\(\\vec{b}\\)\u306e\u5185\u7a4d\u306f\u3001<\/p>\n\n\n\n

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

\u3067\u3042\u308b\u304b\u3089\u3001<\/p>\n\n\n\n

\"\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u6c42\u3081\u308b<\/p>\n\n\n\n

\\[\\displaystyle \\vec{a} \\cdot \\vec{b}=2 \\times 1 \\times \\frac{\\sqrt{3}}{2}\\]<\/p>\n\n\n\n

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

\\[\\vec{a} \\cdot \\vec{b}=\\sqrt{3}\\]<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n

\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\uff1a\u4f8b\u984c\u2461<\/span><\/div>
\n

\"\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u6c42\u3081\u308b<\/p>\n\n\n\n

\\(\\vec{a}=(2,3),\\vec{b}=(1,4)\\)\u306e\u3068\u304d\u3001\\(\\vec{a}\\)\u3068\\(\\vec{b}\\)\u306e\u5185\u7a4d\\(\\vec{a} \\cdot \\vec{b}\\)\u3092\u6c42\u3081\u3088\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

\u3053\u3061\u3089\u3082\u5185\u7a4d\u306e\u516c\u5f0f\u306b\u4ee3\u5165\u3057\u3066<\/p>\n\n\n\n

\\[\\vec{a} \\cdot \\vec{b}=2 \\times 1+3 \\times 4=14\\]<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n

\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f<\/h2>\n\n\n\n

\u5148\u307b\u3069\u306f\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306b\u95a2\u3059\u308b\u516c\u5f0f\u3092\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n

\n
\"\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\"
\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f<\/figcaption><\/figure>\n<\/div>\n\n\n

\u3053\u3053\u304b\u3089\u306f\u3001\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306b\u95a2\u3059\u308b\u516c\u5f0f<\/span>\u3092\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u2460\u89d2\u5ea6cos\u3092\u4f7f\u3046\u516c\u5f0f\uff08\u5b9a\u7fa9\uff09<\/h3>\n\n\n\n

0\u3067\u306f\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}\\)\u3001\\(\\vec{b}\\)\u306e\u59cb\u70b9\u540c\u58eb\u3092\u7e4b\u3044\u3067\u3067\u304d\u308b\u89d2\u3092\u03b8\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\"<\/figure>\n<\/div>\n\n\n
\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\u2460<\/span><\/div>
\n

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

\u3053\u308c\u306f\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u6642\u3068\u540c\u3058\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u306d\uff01<\/p>\n\n\n\n

\u2461\u6210\u5206\u8868\u793a\u3092\u4f7f\u3046\u516c\u5f0f<\/h3>\n\n\n\n

\u5148\u307b\u3069\u89d2\u5ea6\\(\\cos\\)\u3092\u4f7f\u3046\u516c\u5f0f\u306f\u3001\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u6642\u3068\u540c\u3058\u3067\u3057\u305f\u304c\u3001\u6210\u5206\u8868\u793a\u3092\u4f7f\u3046\u969b\u306f\u5c11\u3057\u6ce8\u610f\u304c\u5fc5\u8981<\/span>\u3067\u3059\u3002<\/p>\n\n\n\n

\u3068\u3044\u3046\u306e\u3082\u3001\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306b\u306a\u308b\u3068\u6210\u5206\u304c1\u3064\u5897\u3048\u308b<\/span>\u304b\u3089\u3067\u3059\u3002<\/p>\n\n\n\n

\u6210\u5206\u8868\u793a\u3092\u4f7f\u3063\u305f\u3001\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f<\/figure>\n<\/div>\n\n\n
\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\u2461<\/span><\/div>
\n

\\(\\vec{a}=(x_{1},y_{1},z_{1}),\\vec{b}=(x_{2},y_{2},z_{2})\\)\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

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

\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u6c42\u3081\u308b<\/h2>\n\n\n\n

\u5148\u307b\u3069\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u3001\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\uff1a\u4f8b\u984c\u2460<\/span><\/div>
\n

\"\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u6c42\u3081\u308b\"<\/p>\n\n\n\n

\\(\\vec{a}=(1,2,3),\\vec{b}=(2,2,1)\\)\u306e\u3068\u304d\u3001\\(\\vec{a}\\)\u3068\\(\\vec{b}\\)\u306e\u5185\u7a4d\\(\\vec{a} \\cdot \\vec{b}\\)\u3092\u6c42\u3081\u3088\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
\"\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u6c42\u3081\u308b\"<\/figure>\n\n\n\n

\u5148\u307b\u3069\u306e\u516c\u5f0f\u3092\u3082\u3068\u306b\u5f53\u3066\u306f\u3081\u308b\u3068\u3001<\/p>\n\n\n\n

\\[\\vec{a} \\cdot \\vec{b}=1 \\times 2+2 \\times 2+3 \\times 1=9\\]<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n

\u5e73\u9762\u30d9\u30af\u30c8\u30eb\u306e\u6642\u3082\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u306e\u6642\u3082\u3001\u57fa\u672c\u7684\u306a\u8003\u3048\u65b9\u306f\u540c\u3058\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u306d\uff01<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306e\u6d3b\u7528<\/h2>\n\n\n\n

\u3053\u3053\u304b\u3089\u306f\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3092\u7528\u3044\u305f\u91cd\u8981\u516c\u5f0f<\/span>\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

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

\u307e\u305a\u306f\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u3053\u306b0\u3067\u306f\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a},\\vec{b}\\)\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u5185\u7a4d\u306e\u5782\u76f4\u6761\u4ef6\"<\/figure>\n<\/div>\n\n\n
\u5782\u76f4\u6761\u4ef6<\/span><\/div>
\n

\\(\\vec{a},\\vec{b}\\)\u304c\u5782\u76f4\u306b\u4ea4\u308f\u308b\u3068\u304d<\/p>\n\n\n\n

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

\u7c21\u5358\u306a\u8a3c\u660e\u3068\u3057\u3066\u306f\u5185\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089\u3001<\/p>\n\n\n\n

\\(\\vec{a} \\cdot \\vec{b}=|\\vec{a}||\\vec{b}| \\cos \\theta\\)\u3067\u3042\u308a\u3001<\/p>\n\n\n\n

\u5782\u76f4\u3067\u3042\u308b\u3068\u304d\u3001\\(\\cos \\theta=0\\)\u3068\u306a\u308a\u3001\u5782\u76f4\u6761\u4ef6\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n

\\(\\vec{a} \\cdot \\vec{b}=x_{1}x_{2}+y_{1}y_{2}=0 \\)\u3082\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n

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

\u6b21\u306f\u5e73\u884c\u306a\u3068\u304d\u306e\u6761\u4ef6\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u5185\u7a4d\u306e\u5e73\u884c\u6761\u4ef6\"<\/figure>\n<\/div>\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\u3001<\/p>\n\n\n\n

\u5782\u76f4\u6761\u4ef6<\/span><\/div>
\n

\\(\\vec{a},\\vec{b}\\)\u304c\u5e73\u884c\u306e\u3068\u304d<\/p>\n\n\n\n

\\(\\vec{a} \/\/ \\vec{b}=k \\vec{a}\\)\u3068\u306a\u308b\u5b9f\u6570k\u304c\u3042\u308b\u3002<\/p>\n\n\n\n

\u307e\u305f\u3001\\(\\vec{a} \/\/ \\vec{b}=x_{1}y_{2}-x_{2}y_{1}=0\\)\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n<\/div><\/div>\n\n\n\n

\u8a73\u3057\u304f\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u5185\u7a4d\u306e\u5e73\u884c\u6761\u4ef6\"<\/figure>\n<\/div>\n\n\n

\u753b\u50cf\u3092\u898b\u308b\u3068\u5206\u304b\u308a\u3084\u3059\u304f\u3001<\/p>\n\n\n\n

\\(\\vec{a} \/\/ \\vec{b}\\)\u306e\u3068\u304d\u3001\\(\\vec{b}=k \\vec{a}\\)\uff08k\u306f\u5b9f\u6570\uff09\u3068\u8868\u305b\u308b\u3002\u2026\u2462<\/p>\n\n\n\n

k\u306e\u5024\u304c0\u3088\u308a\u5927\u304d\u3044\u3068\u304d\u306f\u3001(\\vec{a})\u3068(\\vec{b})\u306f\u540c\u3058\u5411\u304d\u306b\u5e73\u884c\u3001
k\u306e\u5024\u304c0\u3088\u308a\u5c0f\u3055\u3044\u3068\u304d\u306f\u3001(\\vec{a})\u3068(\\vec{b})\u306f\u9006\u5411\u304d\u306b\u5e73\u884c\u3067\u3059\u3002<\/p>\n\n\n\n

\u6b21\u306b\u3001\\(\\vec{a} \/\/ \\vec{b}\\)\u306e\u3068\u304d\u3001\\(x_{1}y_{2}-x_{2}y_{1}=0\\)\u306b\u3064\u3044\u3066\u307f\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u8868\u793a\u3092\u4f7f\u3063\u3066<\/p>\n\n\n\n

\\((x_{2},y_{2})=k (x_{1},y_{1})\\)\u3068\u306a\u308b\u304b\u3089\u3001\\(x_{2}=kx_{1},y_{2}=ky_{1}\\)<\/p>\n\n\n\n

\u3053\u3053\u304b\u3089\u6bd4\u3092\u8003\u3048\u308b\u3068\u3001\\(x_{2}:y_{2}=kx_{1}:ky_{1}\\)\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u3055\u3089\u306b\u3001\u6bd4\u306e\u6027\u8cea\u3092\u4f7f\u3048\u3070\u3001\\(x_{2}:y_{2}=x_{1}:y_{1}\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u3088\u3063\u3066\u3001\\(x_{1}y_{2}-x_{2}y_{1}=0\\)\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\u3000\u8a3c\u660e<\/h2>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\u306e\u8a3c\u660e\u3092\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f<\/span><\/div>
\n

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}\\)\u3001\\(\\vec{b}\\)\u304c\u3042\u308a\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u540c\u58eb\u3092\u7e4b\u3044\u3067\u3067\u304d\u305f\u89d2\u3092\u03b8\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3068\u306f\uff1f\"<\/figure>\n<\/div>\n\n\n

\\[\\vec{a} \\cdot \\vec{b}=|\\vec{a}||\\vec{b}| \\cos \\theta \\cdots \u2460\\]<\/p>\n\n\n\n

\\[\\vec{a} \\cdot \\vec{b}=x_{1}x_{2}+y_{1}y_{2} \\cdots \u2461\\]<\/p>\n<\/div><\/div>\n\n\n\n

\u2460\u306b\u95a2\u3057\u3066\u306f\u5185\u7a4d\u306e\u5b9a\u7fa9\u306a\u306e\u3067\u3001\u305d\u3046\u3044\u3046\u3082\u306e\u3060\u3068\u3057\u3066\u899a\u3048\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u2461\u306e\u6210\u5206\u8868\u793a\u3092\u4f7f\u3046\u516c\u5f0f\u306e\u8a3c\u660e\u306f\u3001\u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

0\u3067\u306f\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}=\\vec{OA},\\vec{b}=\\vec{OB}\\)\u304c\u3042\u308a\u3001\u3053\u306e2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u306a\u3059\u89d2\u3092\u03b8\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u3001\\(\\angle{AOB}=\\theta\\)\u3068\u304a\u304f\u3002<\/p>\n\n\n

\n
\"\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f\u3000\u8a3c\u660e\"<\/figure>\n<\/div>\n\n\n


\u3053\u306e\u3068\u304d\u3001\\(\\triangle{OAB}\\)\u306b\u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u3046\u3068\u3001<\/p>\n\n\n\n

\\[{AB}^2={OA}^2+{OB}^2-2 OA\\times OB\\times \\cos \\theta\\]<\/p>\n\n\n\n

\u56f3\u3088\u308a\u3001<\/p>\n\n\n\n

\\(AB=|\\vec{b}-\\vec{a}|\u3001OA=|\\vec{a}|\u3001OB=|\\vec{b}|\\)\u3067\u3042\u308b\u304b\u3089\u3001<\/p>\n\n\n\n

\u2460\u306e\u5f0f\u306f<\/p>\n\n\n\n

\\[|\\vec{b}-\\vec{a}|^2=|\\vec{a}|^2+|\\vec{b}|^2-2ab \\cos \\theta\\]<\/p>\n\n\n\n

\u307e\u305f\u3001\\(\\vec{b}-\\vec{a}=(x_{2}-x_{1},y_{2}-y_{1})\\)\u3088\u308a\u3001
\\(AB=|\\vec{b}-\\vec{a}|=\\sqrt{{(x_{2}-x_{1})}^2+{(y_{2}-y_{1})}^2}\\)\u3067\u3042\u308b\u304b\u3089\u3001<\/p>\n\n\n\n

\\({(x_{2}-x_{1})}^2+{(y_{2}-y_{1})}^2={x_{1}}^2+{y_{1}}^2+{x_{2}}^2+{y_{2}}^2-2|\\vec{a}||\\vec{b}| \\cos \\theta\\)
\u3068\u306a\u308b\u3002<\/p>\n\n\n\n

\u3053\u3053\u3067\u4e21\u8fba\u3092\u6574\u7406\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

\\[|\\vec{a}||\\vec{b}| \\cos \\theta=x_1x_2+y_1y_2\\]<\/p>\n\n\n\n

\u3068\u306a\u308b\u3002<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001\\(\\vec{a} \\cdot \\vec{b}=x_{1}x_{2}+y_{1}y_{2}\\)<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3000\u307e\u3068\u3081<\/h2>\n\n\n\n

\u4eca\u56de\u306f\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\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\u5185\u7a4d\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\u3088\uff01<\/span><\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u516c\u5f0f<\/span><\/div>
\n

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}\\)\u3001\\(\\vec{b}\\)\u304c\u3042\u308a\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u59cb\u70b9\u540c\u58eb\u3092\u7e4b\u3044\u3067\u3067\u304d\u305f\u89d2\u3092\u03b8\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\"\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3068\u306f\uff1f\"<\/p>\n\n\n\n

\\[\\vec{a} \\cdot \\vec{b}=|\\vec{a}||\\vec{b}| \\cos \\theta \\cdots \u2460\\]<\/p>\n\n\n\n

\\[\\vec{a} \\cdot \\vec{b}=x_{1}x_{2}+y_{1}y_{2} \\cdots \u2461\\]<\/p>\n<\/div><\/div>\n\n\n\n

\u3053\u306e\u6210\u5206\u8868\u793a\u3092\u4f7f\u3048\u3070\u3001\u8db3\u3057\u7b97\u30fb\u5f15\u304d\u7b97\u306a\u3069\u8a08\u7b97\u3082\u30b9\u30e0\u30fc\u30ba\u306b\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u305c\u3072\u6d3b\u7528\u3057\u3066\u3001\u30d9\u30af\u30c8\u30eb\u3078\u306e\u7406\u89e3\u3092\u6df1\u3081\u3066\u304f\u3060\u3055\u3044\u306d\uff01<\/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\u5185\u7a4d\u300d\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002 \u30d9\u30af\u30c8\u30eb\u306b\u306f\u201c\u5185\u7a4d\u201d\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308a\u3001\u5185\u7a4d\u306b\u3088\u3063\u30662\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u306a\u3059\u89d2\u306e\u5927\u304d\u3055\u304c\u5206\u304b\u308a\u307e\u3059\u3002 \u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f\u3068\u3066\u3082\u91cd\u8981\u306a\u516c\u5f0f\u306a\u306e\u3067\u3001\u4eca\u56de\u3067 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