{"id":14726,"date":"2025-12-24T17:22:58","date_gmt":"2025-12-24T08:22:58","guid":{"rendered":"https:\/\/math-travel.com\/?p=14726"},"modified":"2026-03-06T01:09:27","modified_gmt":"2026-03-05T16:09:27","slug":"vector-vertical","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-b\/vector-vertical\/","title":{"rendered":"\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6\uff1a\u5185\u7a4d=0\u306b\u306a\u308b\u7406\u7531\u3068\u3001\u5782\u76f4\u306a\u30d9\u30af\u30c8\u30eb\u306e\u6c42\u3081\u65b9\u3092\u89e3\u8aac"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u30d9\u30af\u30c8\u30eb\u5206\u91ce\u304b\u3089\u300c\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u300d\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305a\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u5782\u76f4\u306e\u3068\u304d\u306b\u4ee5\u4e0b\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4<\/span><\/div>
\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

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u5782\u76f4\u306a\u95a2\u4fc2\u306b\u3042\u308b\u3068\u304d\u3001\u5185\u7a4d \\(\\vec{a} \\cdot \\vec{b}\\)\u306f<\/p>\n\n\n\n

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

\u300c\u5782\u76f4\u306e\u3068\u304d\u306f\u5185\u7a4d=0\u300d\u3068\u3057\u3066\u899a\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046\uff01<\/p>\n\n\n\n

\u5185\u7a4d\u3068\u306e\u95a2\u4fc2\u3084\u3001\u7df4\u7fd2\u554f\u984c\u3092\u4f7f\u3044\u306a\u304c\u3089\u5206\u304b\u308a\u3084\u3059\u304f\u8aac\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u306e\u3067\u3001\u82e6\u624b\u610f\u8b58\u3092\u6301\u3063\u3066\u3044\u308b\u65b9\u3082\u3001\u305c\u3072\u6700\u5f8c\u307e\u3067\u8aad\u3093\u3067\u307f\u3066\u304f\u3060\u3055\u3044\u306d\uff01<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6<\/h2>\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
\"\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6\"<\/figure>\n<\/div>\n\n\n

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u5782\u76f4\u306a\u95a2\u4fc2\u306b\u3042\u308b\u3068\u304d\u3001\u5185\u7a4d\\(\\vec{a} \\cdot \\vec{b}=0\\)\u3068\u306a\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6\u3010\u8a3c\u660e\u3011<\/h2>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6\u304c\u306a\u305c\u6210\u308a\u7acb\u3064\u306e\u304b\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\n

\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6<\/p>\n\n\n\n

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

\\(\\vec{a} \\bot \\vec{b} \\Leftrightarrow \\vec{a} \\cdot \\vec{b}=0\\)\u2026\u2460\u306e\u8a3c\u660e\u3092\u898b\u308b<\/a>\u3002<\/p>\n\n\n\n

\\(\\vec{a} \\cdot \\vec{b}=0 \\Leftrightarrow x_{1}x_{2} + y_{1}y_{2} =0\\)\u2026\u2461\u306e\u8a3c\u660e\u3092\u898b\u308b<\/a>\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef61\u306e\u8a3c\u660e<\/h3>\n\n\n\n

\u307e\u305a\u3001\\(\\vec{a}\\bot\\vec{b}\\Leftrightarrow\\vec{a} \\cdot \\vec{b}=0\\)\u2026\u2460\u3092\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

2\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}\u3001\\vec{b}\\)\u304c\u5782\u76f4\u3068\u3044\u3046\u3053\u3068\u306f\u3001\u306a\u3059\u89d2\\(\\theta\\)\u304c\\(90^{\\circ}\\)\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u3053\u306e\u3068\u304d\u3001\\(\\cos 90^{\\circ}=0\\)\u306a\u306e\u3067\u3001\u5185\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089<\/p>\n\n\n\n

\\begin{eqnarray}
\\vec{a} \\cdot \\vec{b}&=&|\\vec{a}||\\vec{b}|\\cos 90^{\\circ}\\\\
&=&|\\vec{a}||\\vec{b}|\\times0\\\\
&=&0
\\end{eqnarray}<\/p>\n\n\n\n

\u6b21\u306b\\(\\vec{a} \\cdot \\vec{b}=0 \\rightarrow \\vec{a}\\bot\\vec{b}\\)\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\\(\\vec{a} \\cdot \\vec{b}=0\\)\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u3001\u5185\u7a4d\u306e\u5b9a\u7fa9\u3088\u308a\u3001<\/p>\n\n\n\n

\\[\\vec{a} \\cdot \\vec{b}=\\left|\\vec{a}\\right|\\left|\\vec{b}\\right|cos\\theta=0\\]<\/p>\n\n\n\n

\u3044\u307e\u3001\\(\\vec{a}\\neq0, \\vec{b}\\neq0\\)\u3067\u3042\u308b\u304b\u3089\u3001\\(\\cos \\theta=0\\)<\/p>\n\n\n\n

\u3088\u3063\u3066\u3001\\(\\theta=90^{\\circ}\\)\u3068\u306a\u308a\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\\(\\vec{a}\u3001\\vec{b}\\)\u304c\u5782\u76f4\u306b\u4ea4\u308f\u308b\u3068\u8a00\u3048\u308b\u3002<\/p>\n\n\n\n

\u4ee5\u4e0a\u304b\u3089\u3001\\(\\vec{a} \\bot \\vec{b} \\Leftrightarrow \\vec{a} \\cdot \\vec{b}=0\\)<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef62\u306e\u8a3c\u660e<\/h3>\n\n\n\n

\\(\\vec{a} \\cdot \\vec{b}=0 \\rightarrow x_{1}x_{2}+y_{1}y_{2}=0\\)\u304b\u3089\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u5185\u7a4d\u306e\u5b9a\u7fa9\u3088\u308a\u3001<\/p>\n\n\n\n

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

\u3088\u3063\u3066\u3001\\(\\vec{a} \\cdot \\vec{b}=0\\)\u3067\u3042\u308b\u3068\u304d\u3001\\(x_{1}x_{2}+y_{1}y_{2}=0\\)<\/p>\n\n\n\n

\u6b21\u306b\\(x_{1}x_{2}+y_{1}y_{2}=0\\rightarrow\\vec{a} \\cdot \\vec{b}=0\\)\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u3061\u3089\u3082\u5185\u7a4d\u306e\u5b9a\u7fa9\u3088\u308a\u3001<\/p>\n\n\n\n

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

\\(x_{1}x_{2}+y_{1}y_{2}=0\\)\u3067\u3042\u308b\u304b\u3089\u3001\\(\\vec{a} \\cdot \\vec{b}=0\\)<\/p>\n\n\n\n

\u4ee5\u4e0a\u304b\u3089\u3001\\(\\vec{a} \\cdot \\vec{b}=0\\Leftrightarrow\\ x_{1}x_{2}+y_{1}y_{2}=0\\)<\/p>\n\n\n\n

a\u30d9\u30af\u30c8\u30eb\u306b\u5782\u76f4\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/h2>\n\n\n\n

\u3053\u3053\u304b\u3089\u306f\u5782\u76f4\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/span>\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u3088\u304f\u51fa\u984c\u3055\u308c\u308b\u554f\u984c\u3067\u300c\u3042\u308b\u30d9\u30af\u30c8\u30eb\u306b\u5782\u76f4\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3092\u6c42\u3081\u306a\u3055\u3044\u3002\u300d\u3068\u3044\u3046\u554f\u984c\u304c\u3042\u308b\u306e\u3067\u5fc5\u305a\u62bc\u3055\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/span><\/p>\n\n\n\n

\u307e\u305a\u306f\u3001\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u306b\u3064\u3044\u3066\u5fa9\u7fd2\u3057\u307e\u3057\u3087\u3046\u3002
\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3068\u306f\u3001\u4e0b\u56f3\u306e\u3088\u3046\u306a\u5927\u304d\u3055\u304c1\u306e\u30d9\u30af\u30c8\u30eb\u3067\u3059\u3002<\/p>\n\n\n

\n
\"\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\"<\/figure>\n<\/div>\n\n\n

\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u7fa9\u3092\u304a\u3055\u3048\u306a\u304c\u3089\u3001\u4f8b\u984c\u3092\u898b\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3059\u306e\u3067\u3001\\(|\\vec{t}|=1\\)\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001<\/p>\n\n\n\n

\\[x^{2}+y^{2}=1\\]<\/p>\n\n\n\n

\u307e\u305f\u3001\\(\\vec{t}\\)\u304c\\(\\vec{a}\\)\u3068\u5782\u76f4\u3067\u3042\u308b\u3068\u304d\u30012\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f0\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

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

\\begin{eqnarray}
\\vec{a} \\cdot \\vec{t}&=&0\\\\
\\left(2,1\\right) \\cdot \\left(x,y\\right)&=&0\\\\
2x+y&=&0\\\\
y&=&-2x
\\end{eqnarray}<\/p>\n\n\n\n

\u5148\u307b\u3069\u306e\u3001\\(x^{2}+y^{2}=1\\)\u3092\u4f7f\u3063\u3066\u3001<\/p>\n\n\n\n

\\begin{eqnarray}
x^{2}+{(-2x)}^{2}&=&1\\\\
5x^{2}&=&1\\\\
\\displaystyle x^{2}=\\frac{1}{5}\\\\
\\displaystyle x&=&\\pm \\frac{1}{\\sqrt5}\\\\
\\displaystyle x&=&\\pm \\frac{\\sqrt5}{5}
\\end{eqnarray}<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001\u6c42\u3081\u308b\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u306f\u3001<\/p>\n\n\n\n

\\(\\displaystyle \\vec{t}=(\\frac{\\sqrt5}{5},-\\frac{2\\sqrt5}{5}),(-\\frac{\\sqrt5}{5},\\frac{2\\sqrt5}{5})\\)<\/p>\n\n\n\n

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

1\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306b\u5bfe\u3057\u3066\u5782\u76f4\u306a\u30d9\u30af\u30c8\u30eb\u306f2\u3064\u3042\u308a\u307e\u3059\u306e\u3067\u3001\u4e21\u65b9\u3068\u3082\u7b54\u3048\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5e73\u884c\u6761\u4ef6<\/h2>\n\n\n\n

\u5782\u76f4\u6761\u4ef6\u306b\u3042\u308f\u305b\u3066\u3001\u30d9\u30af\u30c8\u30eb\u306e\u5e73\u884c\u6761\u4ef6\u3082\u77e5\u3063\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n

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

\u5225\u8a18\u4e8b\u3067\u306f\u3001\u3053\u306e\u5e73\u884c\u6761\u4ef6\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059\u306e\u3067\u3001\u305c\u3072\u30c1\u30a7\u30c3\u30af\u3057\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6\u3000\u307e\u3068\u3081<\/h2>\n\n\n\n

\u4eca\u56de\u306f\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6\u306b\u3064\u3044\u3066\u5b66\u7fd2\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u5782\u76f4\u6761\u4ef6\u3067\u3082\u5185\u7a4d\u306f\u6b20\u304b\u305b\u306a\u3044\u306e\u3067\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

\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f\u3053\u3061\u3089\u306e\u8a18\u4e8b\u3067\u8a73\u3057\u304f\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059\u3002<\/span><\/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":"

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