\n$$\\boldsymbol{a} \\pm \\boldsymbol{b} = (a_x \\pm b_x) \\boldsymbol{i} +(a_y \\pm b_y) \\boldsymbol{j}+(a_z \\pm b_z) \\boldsymbol{k}$$<\/li>\n
\n$$ k\\, \\boldsymbol{a} = k\\, a_x\\, \\boldsymbol{i} + k\\, a_y\\, \\boldsymbol{j} + k\\, a_z\\, \\boldsymbol{k} $$<\/li>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d<\/h3>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3068\u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u30b9\u30ab\u30e9\u30fc\u3092\u4f5c\u308b\u6f14\u7b97\u3067\u3042\u308a\uff0c\\(\\cdot\\)\u00a0 \u3067\u8868\u3059\u3002<\/p>\n
\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u306e\u5b9a\u7fa9<\/h4>\n\n\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u7cfb\u306e\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n<\/div>\n
\n$${\\color{blue}\\boldsymbol{i}\\cdot\\boldsymbol{i}} = {\\color{blue}\\boldsymbol{j}\\cdot\\boldsymbol{j}} = {\\color{blue}\\boldsymbol{k}\\cdot\\boldsymbol{k}} =1$$$$ \\boldsymbol{i}\\cdot\\boldsymbol{j} =\\boldsymbol{j}\\cdot\\boldsymbol{i} = 0$$$$\\boldsymbol{j}\\cdot\\boldsymbol{k} =\\boldsymbol{k}\\cdot\\boldsymbol{j} = 0$$$$\\boldsymbol{k}\\cdot\\boldsymbol{i} =\\boldsymbol{i}\\cdot\\boldsymbol{k} = 0$$<\/p>\n<\/div>\n
\n\n\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u306f \\(1\\)\u3002\u81ea\u5206\u81ea\u8eab\u4ee5\u5916\u3068\u306e\u5185\u7a4d\u306f \\(0\\)\u3002<\/p>\n<\/div>\n
\n\u4e00\u822c\u306e\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u306f…<\/h4>\n
$$\\boldsymbol{a} = a_x \\, \\boldsymbol{i} + a_y \\, \\boldsymbol{j} +a_z \\, \\boldsymbol{k}$$ $$\\boldsymbol{b} = b_x \\, \\boldsymbol{i} + b_y \\, \\boldsymbol{j} +b_z \\, \\boldsymbol{k}$$ \u3068\u3059\u308b\u3068\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089\uff08\u5185\u7a4d\u306b\u3064\u3044\u3066\u5206\u914d\u5247\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u524d\u63d0\u3068\u3057\u3066\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7d50\u679c\u304c\u5c0e\u304b\u308c\u308b\u3002
\\begin{eqnarray}
\\boldsymbol{a}\\cdot\\boldsymbol{b} &=& \\left( a_x \\, \\boldsymbol{i} + a_y \\, \\boldsymbol{j} +a_z \\, \\boldsymbol{k} \\right) \\cdot \\left( b_x \\, \\boldsymbol{i} + b_y \\, \\boldsymbol{j} +b_z \\, \\boldsymbol{k}\\right) \\\\
&=& {\\color{white}{ + }\\ \\\u00a0 } a_x\u00a0 b_x\\, {\\color{blue}\\boldsymbol{i} \\cdot \\boldsymbol{i}} +
a_x\u00a0 b_y\\, \\boldsymbol{i} \\cdot \\boldsymbol{j} +
a_x\u00a0 b_z\\, \\boldsymbol{i} \\cdot \\boldsymbol{k} \\\\
&& + a_y\u00a0 b_x\\, \\boldsymbol{j} \\cdot \\boldsymbol{i} +
a_y\u00a0 b_y\\, {\\color{blue}\\boldsymbol{j} \\cdot \\boldsymbol{j}} +
a_y\u00a0 b_z\\, \\boldsymbol{j} \\cdot \\boldsymbol{k} \\\\
&& + a_z\u00a0 b_x\\, \\boldsymbol{k} \\cdot \\boldsymbol{i} +
a_z\u00a0 b_y\\, \\boldsymbol{k} \\cdot \\boldsymbol{j} +
a_z\u00a0 b_z\\, {\\color{blue}\\boldsymbol{k} \\cdot \\boldsymbol{k}} \\\\
&=& a_x b_x + a_y b_y + a_z b_z
\\end{eqnarray}
\u3059\u306a\u308f\u3061
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = a_x b_x + a_y b_y + a_z b_z$$
\u4ee5\u5f8c\u306f\u3053\u308c\u3092\u4e00\u822c\u306e\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u306e\u5b9a\u7fa9\u306b\u3059\u308b\u3002<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306b\u3064\u3044\u3066<\/p>\n
\n- \u5185\u7a4d\u8a18\u53f7\u7701\u7565\u4e0d\u53ef\uff01 \\( \\boldsymbol{a}\\cdot \\boldsymbol{b} \\) \u3092 \\( \\boldsymbol{a}\\ \\boldsymbol{b} \\) \u306a\u3093\u3066\u66f8\u3044\u305f\u3089\u30c0\u30e1\uff01<\/li>\n
- \u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u3064\uff1a \\(\\boldsymbol{a}\\cdot\\boldsymbol{b} = \\boldsymbol{b}\\cdot\\boldsymbol{a} \\)<\/li>\n
- \u5206\u914d\u5247\u304c\u6210\u308a\u7acb\u3064\uff1a \\(\\boldsymbol{a}\\cdot (\\boldsymbol{b} + \\boldsymbol{c}) = \\boldsymbol{a}\\cdot\\boldsymbol{b} + \\boldsymbol{a}\\cdot\\boldsymbol{c}\\),\u00a0 etc.<\/li>\n<\/ul>\n\n
\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055<\/h4>\n<\/div>\n\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{a}\\) \u306e\u5927\u304d\u3055\u306f\uff0c\\( |\\boldsymbol{a} | \\) \u3068\u66f8\u304d\uff0c\u5185\u7a4d\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002
$$ |\\boldsymbol{a} | \\equiv \\sqrt{\\boldsymbol{a}\\cdot\\boldsymbol{a}} = \\sqrt{a_x^2 + a_y^2 + a_z^2} $$<\/div>\n\u3053\u308c\u306f\uff0c\u539f\u70b9 $O (0, 0, 0)$ \u304b\u3089\u70b9 $P (a_x, a_y, a_z)$ \u307e\u3067\u306e\u9577\u3055\u306e2\u4e57\u304c\uff0c\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\uff08\u4e09\u5e73\u65b9\u306e\u5b9a\u7406\uff09\u304b\u3089 $a_x^2 + a_y^2 + a_z^2$ \u3068\u306a\u308b\u3053\u3068\u304b\u3089\u7406\u89e3\u3067\u304d\u308b\u3002<\/div>\n\u5185\u7a4d\u306e\u3082\u3046\u4e00\u3064\u306e\u8868\u3057\u65b9<\/h4>\n
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = |\\boldsymbol{a} | |\\boldsymbol{b} | \\cos\\theta$$ \u3053\u3053\u3067 \\(\\theta \\) \u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3002<\/p>\n
\u6559\u79d1\u66f8\u306b\u3088\u3063\u3066\u306f\uff08\u307b\u3068\u3093\u3069\u306e\u30c6\u30ad\u30b9\u30c8\u3067\u306f\uff09\uff0c\\(\\boldsymbol{a}\\cdot\\boldsymbol{b} \\equiv |\\boldsymbol{a} | |\\boldsymbol{b} | \\cos\\theta\\) \u3092\u5185\u7a4d\u306e\u300c\u5b9a\u7fa9\u300d\u3068\u3057\u3066\u3044\u308b\u3082\u306e\u3082\u3042\u308b\u304c\uff0c\u3053\u3053\u3067\u306f\u3053\u308c\u3092\u7b2c1\u7fa9\u7684\u306a\u5b9a\u7fa9\u3068\u3057\u3066\u306f\u63a1\u7528\u3057\u306a\u3044\u3002\u306a\u305c\u306a\u3089\uff0c\u3053\u306e\u5b9a\u7fa9\u306e\u53f3\u8fba\u306b\u73fe\u308c\u308b\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055 \\( |\\boldsymbol{a} |\\) \u306f\uff0c\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u306e\u5e73\u65b9\u6839\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u306f\u305a\u3067\u3042\u308a\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u3092\u4f7f\u3046\u524d\u306b\u5185\u7a4d\u3092\u5b9a\u7fa9\u3057\u3066\u304a\u304f\u3053\u3068\u304c\u5fc5\u8981\u304b\u306a\u3068\u8003\u3048\u308b\u304b\u3089\u3067\u3042\u308b\u3002<\/span><\/p>\n\u3053\u306e\u5f0f\u306b\u3064\u3044\u3066\u306f\uff0c\u3082\u3046\u4e00\u3064\u306e\u5b9a\u7fa9\u3068\u3057\u3066\u982d\u3054\u306a\u3057\u306b\u899a\u3048\u308b\u3068\u3044\u3046\u3088\u308a\u3082\uff0c\u6700\u521d\u306b\u5c0e\u3044\u305f\u5b9a\u7fa9\uff1a
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = a_x b_x + a_y b_y + a_z b_z$$ \u304b\u3089\u5c0e\u304b\u308c\u308b\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\u8aac\u660e\u306e\u6982\u8981\u306f\u4ee5\u4e0b\u306e\u901a\u308a\uff1a<\/p>\n
\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c\\( \\boldsymbol{a}, \\boldsymbol{b} \\) \u306f \\(xy\\) \u5e73\u9762\u4e0a\u306b\u3042\u308b\u3068\u3057\uff0c\u3055\u3089\u306b \\( \\boldsymbol{a} \\) \u306f\\(x\\) \u8ef8\u4e0a\uff0c\\(+x\\) \u306e\u5411\u304d\u306b\u3042\u308b\u3068\u3059\u308b\u3002\u3059\u308b\u3068\uff0c
\\begin{eqnarray}
\\boldsymbol{a} &=& (a_x, a_y, a_z) = (|\\boldsymbol{a}|, 0, 0) \\\\
\\boldsymbol{b} &=& (b_x, b_y, b_z) = (|\\boldsymbol{b}| \\cos\\theta, |\\boldsymbol{b}| \\sin\\theta, 0) \\end{eqnarray}<\/p>\n
![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/naiseki2022-640x355.png\")
<\/p>\n
\u5185\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089
\n\\begin{eqnarray}
\n\\boldsymbol{a}\\cdot\\boldsymbol{b} &=& a_x b_x + a_y b_y + a_z b_z \\\\
\n&=& a_x b_x + 0 + 0\\\\
\n&=& |\\boldsymbol{a}|\\, |\\boldsymbol{b}| \\cos\\theta
\n\\end{eqnarray} \u3044\u3063\u305f\u3093\u8a3c\u660e\u3055\u308c\u308b\u3068\uff0c\u3053\u306e\u5f0f\u306f\u5ea7\u6a19\u7cfb\u306e\u53d6\u308a\u65b9\u306b\u3088\u3089\u305a\u306b\u6210\u308a\u7acb\u3064\u306e\u3067\uff0c\\( \\boldsymbol{a}, \\boldsymbol{b} \\) \u304c \\(xy\\) \u5e73\u9762\u4e0a\u306b\u306a\u3044\u3088\u3046\u306a\u4e00\u822c\u306e\u5834\u5408\u306b\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n
\u3053\u306e\u8868\u3057\u65b9\u304b\u3089\u3059\u3050\u306b\u308f\u304b\u308b\u3053\u3068\uff1a\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb\u3067\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{a}, \\, \\boldsymbol{b} \\) \u306e\u5185\u7a4d\u304c\u30bc\u30ed\u3067\u3042\u308c\u3070\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306f\u76f4\u4ea4\u3059\u308b\u3002<\/p>\n
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = 0 \\quad\\Leftrightarrow \\quad \\cos\\theta = 0 \\quad\\Leftrightarrow\\quad \\theta = \\frac{\\pi}{2}, \\ \\\u00a0 \\mbox{i.e.,}\\ \\ \\boldsymbol{a}\\perp\\boldsymbol{b}$$<\/p>\n
\u3042\u3089\u305f\u3081\u3066\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u306e\u5b9a\u7fa9\u3092\u89e3\u91c8\u3059\u308b<\/h4>\n
\u3055\u3066\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u304c
\n$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = |\\boldsymbol{a} | |\\boldsymbol{b} | \\cos\\theta$$
\n\u3053\u3053\u3067 \\(\\theta \\) \u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\uff0c\u7279\u306b\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u306f
\n$$\\boldsymbol{a}\\cdot\\boldsymbol{a} = |\\boldsymbol{a} |^2$$
\n\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\uff0c\u3042\u3089\u305f\u3081\u3066\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306e\u5b9a\u7fa9\u3092\u89e3\u91c8\u3057\u3066\u307f\u308b\u3002<\/p>\n
\n$$\\boldsymbol{i}\\cdot\\boldsymbol{i} = \\boldsymbol{j}\\cdot\\boldsymbol{j} = \\boldsymbol{k}\\cdot\\boldsymbol{k} =1$$<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u304c $1$ \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3002\u307e\u305f\uff0c<\/p>\n
$$ \\boldsymbol{i}\\cdot\\boldsymbol{j} =\\boldsymbol{j}\\cdot\\boldsymbol{i} = 0$$$$\\boldsymbol{j}\\cdot\\boldsymbol{k} =\\boldsymbol{k}\\cdot\\boldsymbol{j} = 0$$$$\\boldsymbol{k}\\cdot\\boldsymbol{i} =\\boldsymbol{i}\\cdot\\boldsymbol{k} = 0$$<\/p>\n<\/div>\n
\n\u306f\u57fa\u672c\u30d9\u30af\u30c8\u30eb $\\boldsymbol{i}, \\, \\boldsymbol{j}, \\, \\boldsymbol{k}$ \u306f\u4e92\u3044\u306b\u76f4\u4ea4\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n
\u3053\u306e\u3088\u3046\u306b\uff0c\u5927\u304d\u3055\u304c $1$ \u3067\u4e92\u3044\u306b\u76f4\u4ea4\u3057\u3066\u3044\u308b\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u3053\u3068\u3092\u6b63\u898f\u76f4\u4ea4\u57fa\u5e95<\/strong><\/span>\u306a\u3069\u3068\u8a00\u3063\u305f\u308a\u3059\u308b\u3002<\/p>\n\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d<\/h3>\n
3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u3082\u3046\u3072\u3068\u3064\u306e\u639b\u3051\u7b97\u3002\u5916\u7a4d\u3068\u306f\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u5225\u306e\u30d9\u30af\u30c8\u30eb\u3092\u3064\u304f\u308b\u6f14\u7b97\u3002\u5916\u7a4d\u306e\u8a18\u53f7\u306f $\\times $\u3002\u639b\u3051\u7b97\u3068\u540c\u3058\u8a18\u53f7\u3060\u304c\uff0c\u7701\u7565\u4e0d\u53ef\uff01<\/strong><\/span>\u3000\u7b54\u3048\u304c\u30d9\u30af\u30c8\u30eb\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n\n\n\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5916\u7a4d\u306e\u5b9a\u7fa9<\/h4>\n
\\begin{eqnarray}
\n{\\color{blue}\\boldsymbol{i}\\times \\boldsymbol{j}} &=& \\boldsymbol{k}\u00a0 \\\\
\n{\\color{blue}\\boldsymbol{j}\\times \\boldsymbol{k}} &=& \\boldsymbol{i} \\\\
\n{\\color{blue}\\boldsymbol{k}\\times \\boldsymbol{i}} &=& \\boldsymbol{j}
\n\\end{eqnarray}
\n\u30b5\u30a4\u30af\u30ea\u30c3\u30af\uff08\u5faa\u74b0\u7684\uff09\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002\u307e\u305f\uff0c
\\begin{eqnarray}
\n{\\color{green}\\boldsymbol{j}\\times \\boldsymbol{i}} &=& -{\\color{blue}\\boldsymbol{i}\\times \\boldsymbol{j}} = -\\boldsymbol{k} \\\\
\n{\\color{green}\\boldsymbol{k}\\times \\boldsymbol{j}} &=&\u00a0 -{\\color{blue}\\boldsymbol{j}\\times \\boldsymbol{k}} = -\\boldsymbol{i} \\\\
\n{\\color{green}\\boldsymbol{i}\\times \\boldsymbol{k}} &=& -{\\color{blue}\\boldsymbol{k}\\times \\boldsymbol{i}} = -\\boldsymbol{j}
\n\\end{eqnarray}
\n\u9806\u5e8f\u3092\u5909\u3048\u308b\u3068\u8ca0\u53f7\u304c\u3064\u304f\u3053\u3068\u306b\u6ce8\u610f\u3002\u6700\u5f8c\u306b\uff0c
$$ \\boldsymbol{i} \\times \\boldsymbol{i} = \\boldsymbol{j} \\times \\boldsymbol{j} = \\boldsymbol{k} \\times \\boldsymbol{k} = \\boldsymbol{0}$$
\n\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5916\u7a4d\u306f\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb\u3002<\/p>\n
\u4e00\u822c\u306e\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5916\u7a4d\u306f…<\/h4>\n
$$\\boldsymbol{a} = a_x \\, \\boldsymbol{i} + a_y \\, \\boldsymbol{j} +a_z \\, \\boldsymbol{k}$$
$$\\boldsymbol{b} = b_x \\, \\boldsymbol{i} + b_y \\, \\boldsymbol{j} +b_z \\, \\boldsymbol{k}$$
\n\u3068\u3059\u308b\u3068\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5916\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089\uff08\u5916\u7a4d\u306b\u3064\u3044\u3066\u5206\u914d\u5247\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u524d\u63d0\u3068\u3057\u3066\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7d50\u679c\u304c\u5c0e\u304b\u308c\u308b\u3002
\\begin{eqnarray}
\\boldsymbol{a}\\times\\boldsymbol{b} &=& \\left( a_x \\, \\boldsymbol{i} + a_y \\, \\boldsymbol{j} +a_z \\, \\boldsymbol{k} \\right) \\times \\left( b_x \\, \\boldsymbol{i} + b_y \\, \\boldsymbol{j} +b_z \\, \\boldsymbol{k}\\right) \\\\
&=&{\\color{white}{+}\\ \\\u00a0 }{\\color{gray}{a_x\u00a0 b_x\\, \\boldsymbol{i} \\times \\boldsymbol{i} }}+
a_x\u00a0 b_y\\, {\\color{blue}\\boldsymbol{i} \\times \\boldsymbol{j} }+
a_x\u00a0 b_z\\, {\\color{green}\\boldsymbol{i} \\times \\boldsymbol{k} }\\\\
&& + a_y\u00a0 b_x\\, {\\color{green}\\boldsymbol{j} \\times \\boldsymbol{i} }+
{\\color{gray}{a_y\u00a0 b_y\\, \\boldsymbol{j} \\times \\boldsymbol{j} }}+
a_y\u00a0 b_z\\, {\\color{blue}\\boldsymbol{j} \\times \\boldsymbol{k}} \\\\
&& + a_z\u00a0 b_x\\, {\\color{blue}\\boldsymbol{k} \\times \\boldsymbol{i}} +
a_z\u00a0 b_y\\, {\\color{green}\\boldsymbol{k} \\times \\boldsymbol{j}} +
{\\color{gray}{a_z\u00a0 b_z\\, \\boldsymbol{k} \\times \\boldsymbol{k} }}\\\\
&=& {\\color{white}{+}\\ \\\u00a0 }
a_x\u00a0 b_y\\, {\\color{blue}\\boldsymbol{i} \\times \\boldsymbol{j} }
\\, -\\,\u00a0 a_x\u00a0 b_z\\, {\\color{blue}\\boldsymbol{k} \\times \\boldsymbol{i}} \\\\
&& \\, -\\,\u00a0 a_y\u00a0 b_x\\, {\\color{blue}\\boldsymbol{i} \\times \\boldsymbol{j}}
+
a_y\u00a0 b_z\\, {\\color{blue}\\boldsymbol{j} \\times \\boldsymbol{k}} \\\\
&& + a_z\u00a0 b_x\\, {\\color{blue}\\boldsymbol{k} \\times \\boldsymbol{i} }
-a_z\u00a0 b_y\\, {\\color{blue}\\boldsymbol{j} \\times \\boldsymbol{k} }
\\\\
&=& (a_y b_z -a_z b_y) \\,\\boldsymbol{i} + (a_z b_x -a_x b_z) \\,\\boldsymbol{j} +(a_x b_y -a_y b_x) \\,\\boldsymbol{k}
\\end{eqnarray}
\u3059\u306a\u308f\u3061
$$\\boldsymbol{a}\\times\\boldsymbol{b} = (a_y b_z -a_z b_y) \\,\\boldsymbol{i} + (a_z b_x -a_x b_z) \\,\\boldsymbol{j} +(a_x b_y -a_y b_x) \\,\\boldsymbol{k} $$ \u4ee5\u5f8c\u306f\u3053\u308c\u3092\u5916\u7a4d\u306e\u5b9a\u7fa9\u3068\u3057\u3066\uff0c\u3053\u308c\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3059\u308b\u3002<\/p>\n<\/div>\n
\n\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u306b\u3064\u3044\u3066\u306f\uff0c\u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u305f\u306a\u3044<\/h4>\n<\/div>\n$$\\boldsymbol{b} \\times \\boldsymbol{a} {\\color{red}{\\neq}} \\boldsymbol{a} \\times \\boldsymbol{b}$$<\/div>\n\u5916\u7a4d\u306e\u5b9a\u7fa9\u3092\u4f7f\u3063\u3066\u3061\u3083\u3093\u3068\u8a08\u7b97\u3059\u308b\u3068\uff0c
\\begin{eqnarray}
\\boldsymbol{b} \\times \\boldsymbol{a} &=& {\\color{white}{+}\\ \\\u00a0 }
\n(b_y a_z -b_z a_y) \\,\\boldsymbol{i} +(b_z a_x -b_x a_z) \\,\\boldsymbol{j} + (b_x a_y -b_y a_x) \\,\\boldsymbol{k} \\\\
&=& -(a_y b_z -a_z b_y) \\,\\boldsymbol{i} -(a_z b_x -a_x b_z) \\,\\boldsymbol{j} -(a_x b_y -a_y b_x) \\,\\boldsymbol{k} \\\\
&=& -\\boldsymbol{a} \\times \\boldsymbol{b}
\\end{eqnarray}<\/div>\n\n<\/div>\n
\u3064\u307e\u308a\uff0c$$\\boldsymbol{b} \\times \\boldsymbol{a} = -\\boldsymbol{a} \\times \\boldsymbol{b}$$ \u9806\u5e8f\u3092\u4ea4\u63db\u3059\u308b\u3068\u30de\u30a4\u30ca\u30b9\u304c\u3064\u304f\u3068\u3044\u3046\u3053\u3068\u3067\u300c\u53cd\u4ea4\u63db<\/strong><\/span>\u300d\u3067\u3042\u308b\u3068\u3044\u3046\u8a00\u3044\u65b9\u3082\u3042\u308b\u3002<\/p>\n\u5916\u7a4d\u306e\u53cd\u4ea4\u63db\u6027\u304b\u3089\u3059\u3050\u306b\u5c0e\u304b\u308c\u308b\u3053\u3068\u304c\u3042\u308b\u3002\u305d\u308c\u306f\uff0c\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5916\u7a4d\u306f\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u3002\u8a3c\u660e\u306f\u7c21\u5358\u3067\uff0c\u53cd\u4ea4\u63db\u3067\u3042\u308b\u3053\u3068\u304b\u3089
$$ {\\color{blue}\\boldsymbol{a}} \\times\\boldsymbol{a} = -\\boldsymbol{a} \\times {\\color{blue}\\boldsymbol{a}}$$
\n\u5de6\u8fba\u306b\u79fb\u884c\u3057\u3066
\n$$ 2 \\,\\boldsymbol{a} \\times\\boldsymbol{a} = \\boldsymbol{0} \\quad\\therefore \\boldsymbol{a} \\times\\boldsymbol{a} = \\boldsymbol{0} $$<\/p>\n\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u306b\u3064\u3044\u3066<\/p>\n
\n- \u5916\u7a4d\u8a18\u53f7\u7701\u7565\u4e0d\u53ef\uff01\u3000\\(\\boldsymbol{a}\\times\\boldsymbol{b}\\) \u3092 \\(\\boldsymbol{a}\\,\\boldsymbol{b}\\) \u306a\u3093\u3066\u7701\u7565\u3057\u3066\u66f8\u3044\u305f\u3089\u30c0\u30e1\uff01<\/li>\n
- \u4ea4\u63db\u5247\u306f\u6210\u308a\u7acb\u305f\u306a\u3044\uff1a \\( \\boldsymbol{a} \\times \\boldsymbol{b} \\neq \\boldsymbol{b} \\times \\boldsymbol{a}\\) \u3080\u3057\u308d\u53cd\u4ea4\u63db\u3067\u3042\u308b\uff1a\\( \\boldsymbol{a} \\times \\boldsymbol{b} = -\\boldsymbol{b} \\times \\boldsymbol{a}\\)<\/li>\n
- \u5206\u914d\u5247\u304c\u6210\u308a\u7acb\u3064\uff1a \\(\\boldsymbol{a}\\times\\left( \\boldsymbol{b} + \\boldsymbol{c}\\right) = \\boldsymbol{a}\\times\\boldsymbol{b} + \\boldsymbol{a}\\times\\boldsymbol{c}\\),\u00a0 etc.<\/li>\n<\/ul>\n
\u5916\u7a4d\u306e\u3082\u3046\u4e00\u3064\u306e\u8868\u3057\u65b9<\/h4>\n
$$\\boldsymbol{a} \\times \\boldsymbol{b}\u00a0 = |\\boldsymbol{a}| |\\boldsymbol{b}| \\sin\\theta \\ \\boldsymbol{n}$$ \u3053\u3053\u3067 \\(\\theta\\) \u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{a}, \\, \\boldsymbol{b} \\) \u306e\u306a\u3059\u89d2\u3067\u3042\u308a\uff0c\\(\\boldsymbol{n}\\) \u306f \\(\\boldsymbol{a}\\) \u306b\u3082 \\(\\boldsymbol{b}\\) \u306b\u3082\u76f4\u4ea4\u3057\uff0c\u5411\u304d\u304c \\(\\boldsymbol{a}\\) \u304b\u3089 \\(\\boldsymbol{b}\\) \u306b\u53f3\u306d\u3058\u3092\u56de\u3057\u305f\u3068\u304d\u306e\u9032\u884c\u65b9\u5411\u3067\u3042\u308b\u3088\u3046\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3002<\/p>\n
\u3053\u306e\u8868\u3057\u65b9\u3082\uff0c\u4e0a\u306b\u8ff0\u3079\u305f\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089\u6c42\u3081\u3089\u308c\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n
\u8aac\u660e\u306e\u6982\u8981\u306f\u4ee5\u4e0b\u306e\u901a\u308a\uff1a
\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c\\( \\boldsymbol{a}, \\boldsymbol{b} \\) \u306f \\(xy\\) \u5e73\u9762\u4e0a\u306b\u3042\u308b\u3068\u3057\uff0c\u3055\u3089\u306b \\( \\boldsymbol{a} \\) \u306f\\(x\\) \u8ef8\u4e0a\uff0c\\(+x\\) \u306e\u5411\u304d\u306b\u3042\u308b\u3068\u3059\u308b\u3002\u3059\u308b\u3068\uff0c
\\begin{eqnarray}
\\boldsymbol{a} &=& (a_x, a_y, a_z) = (|\\boldsymbol{a}|, 0, 0) \\\\
\\boldsymbol{b} &=& (b_x, b_y, b_z) = (|\\boldsymbol{b}| \\cos\\theta, |\\boldsymbol{b}| \\sin\\theta, 0) \\end{eqnarray} \u5916\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{eqnarray} \\left(\\boldsymbol{a}\\times\\boldsymbol{b} \\right)_x &=& a_y b_z -a_z b_y = 0 \\\\
\\left(\\boldsymbol{a}\\times\\boldsymbol{b} \\right)_y &=& a_z b_x -a_x b_z = 0 \\\\
\\left(\\boldsymbol{a}\\times\\boldsymbol{b} \\right)_z &=& a_x b_y -a_y b_x = |\\boldsymbol{a}|\\,|\\boldsymbol{b}| \\sin\\theta \\\\
\\therefore \\boldsymbol{a}\\times\\boldsymbol{b} &=& |\\boldsymbol{a}|\\,|\\boldsymbol{b}| \\sin\\theta \\,\\boldsymbol{k}
\n\\equiv
\n|\\boldsymbol{a}|\\,|\\boldsymbol{b}| \\sin\\theta \\,\\boldsymbol{n}
\n\\end{eqnarray}
\uff08\u3053\u3053\u3067\u306f \\(\\boldsymbol{n}\\) \u3059\u306a\u308f\u3061 \\(\\boldsymbol{k}\\) \u304c \\(\\boldsymbol{a}\\) \u306b\u3082 \\(\\boldsymbol{b}\\) \u306b\u3082\u76f4\u4ea4\u3059\u308b\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3002\uff09\u3044\u3063\u305f\u3093\u8a3c\u660e\u3055\u308c\u308b\u3068\uff0c\u3053\u306e\u5f0f\u306f\u5ea7\u6a19\u7cfb\u306e\u53d6\u308a\u65b9\u306b\u3088\u3089\u305a\u306b\u6210\u308a\u7acb\u3064\u306e\u3067\uff0c\\( \\boldsymbol{a}, \\boldsymbol{b} \\) \u304c \\(xy\\) \u5e73\u9762\u4e0a\u306b\u306a\u3044\u3088\u3046\u306a\u4e00\u822c\u306e\u5834\u5408\u306b\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n
\\(| \\boldsymbol{a} \\times \\boldsymbol{b} |\\) \u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{a}, \\, \\boldsymbol{b} \\) \u3067\u4f5c\u3089\u308c\u308b\u5e73\u884c\u56db\u8fba\u5f62\u306e\u9762\u7a4d\u3092\u8868\u3059\u3053\u3068\u306f\u4ee5\u4e0b\u306e\u56f3\u304b\u3089\u308f\u304b\u308b\u3002<\/p>\n
![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/heikou4-640x417.png\")
<\/div>\n\u307e\u305f\uff0c\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5916\u7a4d\u306f\u30bc\u30ed\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3053\u3068\u306f\u3059\u3067\u306b\u8ff0\u3079\u305f\u304c\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u5e73\u884c\u3067\u3042\u308c\u3070\uff0c\u5916\u7a4d\u306f\u30bc\u30ed\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002\u8a3c\u660e\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n
\\(\\boldsymbol{b}\\) \u304c \\(\\boldsymbol{a}\\) \u306b\u5e73\u884c\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u3042\u308b\u5b9a\u6570 \\(k\\) \u3092\u4f7f\u3063\u3066\uff0c\\(\\boldsymbol{b}= k \\boldsymbol{a}\\) \u3068\u66f8\u3051\u308b\u3068\u3044\u3046\u3053\u3068\u3002\u3057\u305f\u304c\u3063\u3066\uff0c
\n$$\\boldsymbol{a}\\times\\boldsymbol{b} = \\boldsymbol{a}\\times(k \\boldsymbol{a} ) = k \\boldsymbol{a}\\times\\boldsymbol{a} =\\boldsymbol{0}$$<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u4e09\u91cd\u7a4d<\/h3>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306b\u3088\u308b\u300c\u639b\u3051\u7b97\u300d<\/p>\n
\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d<\/h4>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u300c\u639b\u3051\u7b97\u300d\u306b\u3088\u3063\u3066\u6700\u7d42\u7684\u306b\u30b9\u30ab\u30e9\u30fc\u3092\u3064\u304f\u308b\u6f14\u7b97\u306a\u306e\u3067\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u300c\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u300d\u3068\u547c\u3076\uff08\u307e\u304e\u3089\u308f\u3057\u3044\uff09\u3002
\n$$\\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c})$$ \u3053\u308c\u304c\u30b9\u30ab\u30e9\u30fc\u3092\u3064\u304f\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u308f\u304b\u308b\u3002<\/p>\n
\n- \u5148\u306b\u304b\u3063\u3053\u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u3092\u8a08\u7b97\u3059\u308b\u306e\u3067 \\( (\\boldsymbol{b} \\times \\boldsymbol{c}) \\) \u306f\u5916\u7a4d\u306a\u306e\u3067\u30d9\u30af\u30c8\u30eb\u3092\u4f5c\u308b\u3002\u3053\u308c\u3092\u305f\u3068\u3048\u3070 \\(\\boldsymbol{d}\\) \u3068\u3057\u3088\u3046\u3002$$\\boldsymbol{b} \\times \\boldsymbol{c} = \\boldsymbol{d}$$<\/li>\n
- \u6b21\u306b\uff0c\\(\\boldsymbol{a}\\cdot\\boldsymbol{d}\\) \u3092\u8a08\u7b97\u3059\u308b\u304c\uff0c\u3053\u308c\u306f\u5185\u7a4d\u306a\u306e\u3067\u6700\u7d42\u7684\u306b\u7b54\u3048\u306f\u30b9\u30ab\u30e9\u30fc\u3002<\/li>\n<\/ol>\n\n
\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306e\u91cd\u8981\u306a\u516c\u5f0f<\/h4>\n<\/div>\n3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u9806\u756a\u3092\u30b5\u30a4\u30af\u30ea\u30c3\u30af\u306b\u5909\u3048\u3066\u3082\u5024\u306f\u5909\u308f\u3089\u306a\u3044\u3002\u3064\u307e\u308a\uff0c
\n$$ \\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c}) = \\boldsymbol{b}\\cdot (\\boldsymbol{c} \\times \\boldsymbol{a}) = \\boldsymbol{c}\\cdot (\\boldsymbol{a} \\times \\boldsymbol{b}) $$<\/div>\n<\/div>\n\u76f4\u63a5\u8a08\u7b97\u3057\u3066\uff0c\u305f\u3068\u3048\u3070 \\( \\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c}) = \\boldsymbol{b}\\cdot (\\boldsymbol{c} \\times \\boldsymbol{a})\\) \u3092\u793a\u3057\u3066\u307f\u308b\u3002<\/div>\n\\begin{eqnarray}
\n\\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c}) &=&
\na_x (\\boldsymbol{b} \\times \\boldsymbol{c})_x + a_y (\\boldsymbol{b} \\times \\boldsymbol{c})_y + a_z (\\boldsymbol{b} \\times \\boldsymbol{c})_z \\\\
\n&=& a_x ({\\color{blue}{b_y}} c_z -{\\color{green}{b_z}} c_y) + a_y ({\\color{green}{b_z}} c_x -{\\color{red}{b_x}} c_z) + a_z ({\\color{red}{b_x}} c_y -{\\color{blue}{b_y}} c_x) \\\\
\n&=& {\\color{red}{b_x}} (c_y a_z -c_z a_y) + {\\color{blue}{b_y}} (c_z a_x -c_x a_z) + {\\color{green}{b_z}} (c_x a_y -c_y a_x) \\\\
\n&=&\u00a0 b_x (\\boldsymbol{c}\\times\\boldsymbol{a})_x + b_y (\\boldsymbol{c}\\times\\boldsymbol{a})_y + b_z (\\boldsymbol{c}\\times\\boldsymbol{a})_z \\\\
\n&=& \\boldsymbol{b}\\cdot (\\boldsymbol{c}\\times\\boldsymbol{a})
\n\\end{eqnarray}<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\u4ee5\u4e0b\u306e\u56f3\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306f\uff0c3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u4f5c\u3089\u308c\u308b\u5e73\u884c\u516d\u9762\u4f53\u306e\u4f53\u7a4d $V$ \u306b\u306a\u308b\u3002<\/div>\n<\/div>\n![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/6mentai2-640x362.png\")
<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n\n\u53c2\u8003\uff1a<\/h5>\n<\/div>\n\\(\\boldsymbol{a}\\times\\boldsymbol{b}\\) \u306f \\(\\boldsymbol{a}\\) \u306b\u3082 \\(\\boldsymbol{b}\\) \u306b\u3082\u76f4\u4ea4\u3059\u308b\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3053\u3068\u3092\uff0c\u30b9\u30ab\u30e9\u30fc3\u91cd\u7a4d\u306e\u6027\u8cea\u3092\u4f7f\u3063\u3066\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/div>\n$$ {\\color{red}{\\boldsymbol{a}}} \\cdot(\\boldsymbol{a}\\times\\boldsymbol{b}) = \\boldsymbol{b}\\cdot ({\\color{red}{\\boldsymbol{a}}} \\times\\boldsymbol{a}) = 0, \\quad \\because \\boldsymbol{a} \\times\u00a0 \\boldsymbol{a} = \\boldsymbol{0}$$$$\\therefore\\ \\ \\boldsymbol{a} \\perp \\boldsymbol{a}\\times\\boldsymbol{b}$$<\/div>\n\u540c\u69d8\u306b\u3057\u3066 \\(\\boldsymbol{b}\\perp \\boldsymbol{a}\\times\\boldsymbol{b}\\) \u3082\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/div>\n\n\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d<\/h4>\n<\/div>\n3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u300c\u639b\u3051\u7b97\u300d\u306b\u3088\u3063\u3066\u6700\u7d42\u7684\u306b\u30d9\u30af\u30c8\u30eb\u3092\u3064\u304f\u308b\u6f14\u7b97\u306a\u306e\u3067\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u300c\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u300d\u3068\u547c\u3076\uff08\u3084\u3084\u3053\u3057\u3044\uff09\u3002$$ \\boldsymbol{a}\\times(\\boldsymbol{b}\\times\\boldsymbol{c})$$<\/div>\n\u3053\u308c\u304c\u30d9\u30af\u30c8\u30eb\u3092\u3064\u304f\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u308f\u304b\u308b\u3002<\/div>\n\n\n- \u5148\u306b\u304b\u3063\u3053\u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u8a08\u7b97\u3002\\( (\\boldsymbol{b}\\times\\boldsymbol{c}) \\) \u306f\u5916\u7a4d\u306a\u306e\u3067\u30d9\u30af\u30c8\u30eb\u3092\u4f5c\u308b\u3002\u3053\u308c\u3092\u305f\u3068\u3048\u3070 \\(\\boldsymbol{d}\\) \u3068\u3057\u3088\u3046\u3002<\/li>\n
- \u6b21\u306b\uff0c\\( \\boldsymbol{a}\\times\\boldsymbol{d} \\) \u3092\u8a08\u7b97\u3059\u308b\u304c\uff0c\u3053\u308c\u306f\u5916\u7a4d\u306a\u306e\u3067\u6700\u7d42\u7684\u306b\u7b54\u3048\u306f\u30d9\u30af\u30c8\u30eb\u3002<\/li>\n<\/ol>\n
<\/h5>\n
\u5916\u7a4d\u306e\u300c\u7d50\u5408\u5247\u300d\u306f\u6210\u308a\u7acb\u305f\u306a\u3044\u3002<\/b><\/u><\/span>\u3064\u307e\u308a\uff0c$$\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c}) {\\color{red}{\\neq}} (\\boldsymbol{a}\\times\\boldsymbol{b}) \\times \\boldsymbol{c}$$<\/p>\n <\/p>\n
\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306e\u91cd\u8981\u306a\u516c\u5f0f<\/h4>\n
$$\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c}) = (\\boldsymbol{a}\\cdot\\boldsymbol{c}) \\boldsymbol{b} -(\\boldsymbol{a}\\cdot\\boldsymbol{b}) \\boldsymbol{c} $$<\/p>\n
\u6210\u5206\u3092\u76f4\u63a5\u8a08\u7b97\u3057\u3066\u3053\u306e\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\u4f8b\u3068\u3057\u3066 $x$ \u6210\u5206\u306e\u8a08\u7b97\uff1a<\/p>\n
\\begin{eqnarray}
\n\\bigl\\{\\boldsymbol{a}\\times(\\boldsymbol{b}\\times\\boldsymbol{c})\\bigr\\}_x &=& a_y (\\boldsymbol{b}\\times\\boldsymbol{c})_z -a_z (\\boldsymbol{b}\\times\\boldsymbol{c})_y \\\\
\n&=& a_y ({\\color{blue}{b_x}} c_y -b_y {\\color{green}{c_x}}) -a_z (b_z {\\color{green}{c_x}} -{\\color{blue}{b_x}} c_z) \\\\
\n&=& ({\\color{white}{a_x c_x}} {\\color{white}{+}\\ \\, } a_y c_y + a_z c_z) {\\color{blue}{b_x}}\u00a0 -({\\color{white}{a_x b_x}\\\u00a0 }{\\color{white}{+}} a_y b_y + a_z b_x) {\\color{green}{c_x}} \\\\
\n&=& (a_x c_x + a_y c_y + a_z c_z) {\\color{blue}{b_x} } -(a_x b_x + a_y b_y + a_z b_x) {\\color{green}{c_x}} \\\\
\n&=& \\bigl\\{(\\boldsymbol{a}\\cdot\\boldsymbol{c}) \\boldsymbol{b} -(\\boldsymbol{a}\\cdot\\boldsymbol{b}) \\boldsymbol{c} \\bigr\\}_x
\n\\end{eqnarray}<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u6f14\u7b97\u306e\u307e\u3068\u3081<\/h3>\n
\u4ee5\u4e0a\u306e\u7d50\u679c\u3092\u307e\u3068\u3081\u308b\u3068…<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u8868\u3057\u65b9<\/h4>\n
$$ \\boldsymbol{a} = a_x \\,\\boldsymbol{i} + a_y \\,\\boldsymbol{j} + a_z \\,\\boldsymbol{k}$$ \u307e\u305f\u306f $$ \\boldsymbol{a} = (a_x, \\ a_y, \\ a_z) $$<\/tt><\/p>\n\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d<\/h4>\n
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = a_x b_x + a_y b_y + a_z b_z = |\\boldsymbol{a} | |\\boldsymbol{b} | \\cos\\theta$$<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055<\/h4>\n
$$ |\\boldsymbol{a} | \\equiv \\sqrt{\\boldsymbol{a}\\cdot\\boldsymbol{a}} = \\sqrt{a_x^2 + a_y^2 + a_z^2} $$<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d<\/h4>\n
\\begin{eqnarray}
\n\\boldsymbol{a}\\times\\boldsymbol{b} &=& (a_y b_z -a_z b_y) \\,\\boldsymbol{i} + (a_z b_x -a_x b_z) \\,\\boldsymbol{j} +(a_x b_y -a_y b_x) \\,\\boldsymbol{k} \\\\
\n&=& |\\boldsymbol{a}| |\\boldsymbol{b}| \\sin\\theta\\ \\boldsymbol{n}
\n\\end{eqnarray}
\n\\(\\boldsymbol{n}\\) \u306f \\(\\boldsymbol{a}\\) \u306b\u3082 \\(\\boldsymbol{b}\\) \u306b\u3082\u76f4\u4ea4\u3057\uff0c\u5411\u304d\u304c \\(\\boldsymbol{a}\\) \u304b\u3089 \\(\\boldsymbol{b}\\) \u306b\u53f3\u306d\u3058\u3092\u56de\u3057\u305f\u3068\u304d\u306e\u9032\u884c\u65b9\u5411\u3067\u3042\u308b\u3088\u3046\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3002<\/p>\n
\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306e\u516c\u5f0f<\/h4>\n
$$ \\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c}) = \\boldsymbol{b}\\cdot (\\boldsymbol{c} \\times \\boldsymbol{a}) = \\boldsymbol{c}\\cdot (\\boldsymbol{a} \\times \\boldsymbol{b}) $$<\/p>\n
<\/p>\n
\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306e\u516c\u5f0f<\/h4>\n
$$\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c}) = (\\boldsymbol{a}\\cdot\\boldsymbol{c}) \\boldsymbol{b} -(\\boldsymbol{a}\\cdot\\boldsymbol{b}) \\boldsymbol{c} $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u30d9\u30af\u30c8\u30eb\u306e\u8db3\u3057\u7b97\uff0c\u5f15\u304d\u7b97\uff0c\u5185\u7a4d\uff0c\u5916\u7a4d\uff0c\u4e09\u91cd\u7a4d\u3002<\/p>
\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u7cfb\u306e\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n<\/div>\n
$${\\color{blue}\\boldsymbol{i}\\cdot\\boldsymbol{i}} = {\\color{blue}\\boldsymbol{j}\\cdot\\boldsymbol{j}} = {\\color{blue}\\boldsymbol{k}\\cdot\\boldsymbol{k}} =1$$$$ \\boldsymbol{i}\\cdot\\boldsymbol{j} =\\boldsymbol{j}\\cdot\\boldsymbol{i} = 0$$$$\\boldsymbol{j}\\cdot\\boldsymbol{k} =\\boldsymbol{k}\\cdot\\boldsymbol{j} = 0$$$$\\boldsymbol{k}\\cdot\\boldsymbol{i} =\\boldsymbol{i}\\cdot\\boldsymbol{k} = 0$$<\/p>\n<\/div>\n
\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u306f \\(1\\)\u3002\u81ea\u5206\u81ea\u8eab\u4ee5\u5916\u3068\u306e\u5185\u7a4d\u306f \\(0\\)\u3002<\/p>\n<\/div>\n
\u4e00\u822c\u306e\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u306f…<\/h4>\n
$$\\boldsymbol{a} = a_x \\, \\boldsymbol{i} + a_y \\, \\boldsymbol{j} +a_z \\, \\boldsymbol{k}$$ $$\\boldsymbol{b} = b_x \\, \\boldsymbol{i} + b_y \\, \\boldsymbol{j} +b_z \\, \\boldsymbol{k}$$ \u3068\u3059\u308b\u3068\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089\uff08\u5185\u7a4d\u306b\u3064\u3044\u3066\u5206\u914d\u5247\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u524d\u63d0\u3068\u3057\u3066\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7d50\u679c\u304c\u5c0e\u304b\u308c\u308b\u3002
\\begin{eqnarray}
\\boldsymbol{a}\\cdot\\boldsymbol{b} &=& \\left( a_x \\, \\boldsymbol{i} + a_y \\, \\boldsymbol{j} +a_z \\, \\boldsymbol{k} \\right) \\cdot \\left( b_x \\, \\boldsymbol{i} + b_y \\, \\boldsymbol{j} +b_z \\, \\boldsymbol{k}\\right) \\\\
&=& {\\color{white}{ + }\\ \\\u00a0 } a_x\u00a0 b_x\\, {\\color{blue}\\boldsymbol{i} \\cdot \\boldsymbol{i}} +
a_x\u00a0 b_y\\, \\boldsymbol{i} \\cdot \\boldsymbol{j} +
a_x\u00a0 b_z\\, \\boldsymbol{i} \\cdot \\boldsymbol{k} \\\\
&& + a_y\u00a0 b_x\\, \\boldsymbol{j} \\cdot \\boldsymbol{i} +
a_y\u00a0 b_y\\, {\\color{blue}\\boldsymbol{j} \\cdot \\boldsymbol{j}} +
a_y\u00a0 b_z\\, \\boldsymbol{j} \\cdot \\boldsymbol{k} \\\\
&& + a_z\u00a0 b_x\\, \\boldsymbol{k} \\cdot \\boldsymbol{i} +
a_z\u00a0 b_y\\, \\boldsymbol{k} \\cdot \\boldsymbol{j} +
a_z\u00a0 b_z\\, {\\color{blue}\\boldsymbol{k} \\cdot \\boldsymbol{k}} \\\\
&=& a_x b_x + a_y b_y + a_z b_z
\\end{eqnarray}
\u3059\u306a\u308f\u3061
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = a_x b_x + a_y b_y + a_z b_z$$
\u4ee5\u5f8c\u306f\u3053\u308c\u3092\u4e00\u822c\u306e\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u306e\u5b9a\u7fa9\u306b\u3059\u308b\u3002<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306b\u3064\u3044\u3066<\/p>\n
- \n
- \u5185\u7a4d\u8a18\u53f7\u7701\u7565\u4e0d\u53ef\uff01 \\( \\boldsymbol{a}\\cdot \\boldsymbol{b} \\) \u3092 \\( \\boldsymbol{a}\\ \\boldsymbol{b} \\) \u306a\u3093\u3066\u66f8\u3044\u305f\u3089\u30c0\u30e1\uff01<\/li>\n
- \u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u3064\uff1a \\(\\boldsymbol{a}\\cdot\\boldsymbol{b} = \\boldsymbol{b}\\cdot\\boldsymbol{a} \\)<\/li>\n
- \u5206\u914d\u5247\u304c\u6210\u308a\u7acb\u3064\uff1a \\(\\boldsymbol{a}\\cdot (\\boldsymbol{b} + \\boldsymbol{c}) = \\boldsymbol{a}\\cdot\\boldsymbol{b} + \\boldsymbol{a}\\cdot\\boldsymbol{c}\\),\u00a0 etc.<\/li>\n<\/ul>\n\n
\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055<\/h4>\n<\/div>\n
\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{a}\\) \u306e\u5927\u304d\u3055\u306f\uff0c\\( |\\boldsymbol{a} | \\) \u3068\u66f8\u304d\uff0c\u5185\u7a4d\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002
$$ |\\boldsymbol{a} | \\equiv \\sqrt{\\boldsymbol{a}\\cdot\\boldsymbol{a}} = \\sqrt{a_x^2 + a_y^2 + a_z^2} $$<\/div>\n\u3053\u308c\u306f\uff0c\u539f\u70b9 $O (0, 0, 0)$ \u304b\u3089\u70b9 $P (a_x, a_y, a_z)$ \u307e\u3067\u306e\u9577\u3055\u306e2\u4e57\u304c\uff0c\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\uff08\u4e09\u5e73\u65b9\u306e\u5b9a\u7406\uff09\u304b\u3089 $a_x^2 + a_y^2 + a_z^2$ \u3068\u306a\u308b\u3053\u3068\u304b\u3089\u7406\u89e3\u3067\u304d\u308b\u3002<\/div>\n\u5185\u7a4d\u306e\u3082\u3046\u4e00\u3064\u306e\u8868\u3057\u65b9<\/h4>\n
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = |\\boldsymbol{a} | |\\boldsymbol{b} | \\cos\\theta$$ \u3053\u3053\u3067 \\(\\theta \\) \u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3002<\/p>\n
\u6559\u79d1\u66f8\u306b\u3088\u3063\u3066\u306f\uff08\u307b\u3068\u3093\u3069\u306e\u30c6\u30ad\u30b9\u30c8\u3067\u306f\uff09\uff0c\\(\\boldsymbol{a}\\cdot\\boldsymbol{b} \\equiv |\\boldsymbol{a} | |\\boldsymbol{b} | \\cos\\theta\\) \u3092\u5185\u7a4d\u306e\u300c\u5b9a\u7fa9\u300d\u3068\u3057\u3066\u3044\u308b\u3082\u306e\u3082\u3042\u308b\u304c\uff0c\u3053\u3053\u3067\u306f\u3053\u308c\u3092\u7b2c1\u7fa9\u7684\u306a\u5b9a\u7fa9\u3068\u3057\u3066\u306f\u63a1\u7528\u3057\u306a\u3044\u3002\u306a\u305c\u306a\u3089\uff0c\u3053\u306e\u5b9a\u7fa9\u306e\u53f3\u8fba\u306b\u73fe\u308c\u308b\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055 \\( |\\boldsymbol{a} |\\) \u306f\uff0c\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u306e\u5e73\u65b9\u6839\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u306f\u305a\u3067\u3042\u308a\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u3092\u4f7f\u3046\u524d\u306b\u5185\u7a4d\u3092\u5b9a\u7fa9\u3057\u3066\u304a\u304f\u3053\u3068\u304c\u5fc5\u8981\u304b\u306a\u3068\u8003\u3048\u308b\u304b\u3089\u3067\u3042\u308b\u3002<\/span><\/p>\n
\u3053\u306e\u5f0f\u306b\u3064\u3044\u3066\u306f\uff0c\u3082\u3046\u4e00\u3064\u306e\u5b9a\u7fa9\u3068\u3057\u3066\u982d\u3054\u306a\u3057\u306b\u899a\u3048\u308b\u3068\u3044\u3046\u3088\u308a\u3082\uff0c\u6700\u521d\u306b\u5c0e\u3044\u305f\u5b9a\u7fa9\uff1a
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = a_x b_x + a_y b_y + a_z b_z$$ \u304b\u3089\u5c0e\u304b\u308c\u308b\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n\u8aac\u660e\u306e\u6982\u8981\u306f\u4ee5\u4e0b\u306e\u901a\u308a\uff1a<\/p>\n
\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c\\( \\boldsymbol{a}, \\boldsymbol{b} \\) \u306f \\(xy\\) \u5e73\u9762\u4e0a\u306b\u3042\u308b\u3068\u3057\uff0c\u3055\u3089\u306b \\( \\boldsymbol{a} \\) \u306f\\(x\\) \u8ef8\u4e0a\uff0c\\(+x\\) \u306e\u5411\u304d\u306b\u3042\u308b\u3068\u3059\u308b\u3002\u3059\u308b\u3068\uff0c
\\begin{eqnarray}
\\boldsymbol{a} &=& (a_x, a_y, a_z) = (|\\boldsymbol{a}|, 0, 0) \\\\
\\boldsymbol{b} &=& (b_x, b_y, b_z) = (|\\boldsymbol{b}| \\cos\\theta, |\\boldsymbol{b}| \\sin\\theta, 0) \\end{eqnarray}<\/p>\n<\/p>\n\u5185\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089
\n\\begin{eqnarray}
\n\\boldsymbol{a}\\cdot\\boldsymbol{b} &=& a_x b_x + a_y b_y + a_z b_z \\\\
\n&=& a_x b_x + 0 + 0\\\\
\n&=& |\\boldsymbol{a}|\\, |\\boldsymbol{b}| \\cos\\theta
\n\\end{eqnarray} \u3044\u3063\u305f\u3093\u8a3c\u660e\u3055\u308c\u308b\u3068\uff0c\u3053\u306e\u5f0f\u306f\u5ea7\u6a19\u7cfb\u306e\u53d6\u308a\u65b9\u306b\u3088\u3089\u305a\u306b\u6210\u308a\u7acb\u3064\u306e\u3067\uff0c\\( \\boldsymbol{a}, \\boldsymbol{b} \\) \u304c \\(xy\\) \u5e73\u9762\u4e0a\u306b\u306a\u3044\u3088\u3046\u306a\u4e00\u822c\u306e\u5834\u5408\u306b\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n\u3053\u306e\u8868\u3057\u65b9\u304b\u3089\u3059\u3050\u306b\u308f\u304b\u308b\u3053\u3068\uff1a\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb\u3067\u306a\u30442\u3064\u306e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{a}, \\, \\boldsymbol{b} \\) \u306e\u5185\u7a4d\u304c\u30bc\u30ed\u3067\u3042\u308c\u3070\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306f\u76f4\u4ea4\u3059\u308b\u3002<\/p>\n
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = 0 \\quad\\Leftrightarrow \\quad \\cos\\theta = 0 \\quad\\Leftrightarrow\\quad \\theta = \\frac{\\pi}{2}, \\ \\\u00a0 \\mbox{i.e.,}\\ \\ \\boldsymbol{a}\\perp\\boldsymbol{b}$$<\/p>\n
\u3042\u3089\u305f\u3081\u3066\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u306e\u5b9a\u7fa9\u3092\u89e3\u91c8\u3059\u308b<\/h4>\n
\u3055\u3066\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u304c
\n$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = |\\boldsymbol{a} | |\\boldsymbol{b} | \\cos\\theta$$
\n\u3053\u3053\u3067 \\(\\theta \\) \u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\uff0c\u7279\u306b\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u306f
\n$$\\boldsymbol{a}\\cdot\\boldsymbol{a} = |\\boldsymbol{a} |^2$$
\n\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\uff0c\u3042\u3089\u305f\u3081\u3066\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306e\u5b9a\u7fa9\u3092\u89e3\u91c8\u3057\u3066\u307f\u308b\u3002<\/p>\n\n$$\\boldsymbol{i}\\cdot\\boldsymbol{i} = \\boldsymbol{j}\\cdot\\boldsymbol{j} = \\boldsymbol{k}\\cdot\\boldsymbol{k} =1$$<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u304c $1$ \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3002\u307e\u305f\uff0c<\/p>\n
$$ \\boldsymbol{i}\\cdot\\boldsymbol{j} =\\boldsymbol{j}\\cdot\\boldsymbol{i} = 0$$$$\\boldsymbol{j}\\cdot\\boldsymbol{k} =\\boldsymbol{k}\\cdot\\boldsymbol{j} = 0$$$$\\boldsymbol{k}\\cdot\\boldsymbol{i} =\\boldsymbol{i}\\cdot\\boldsymbol{k} = 0$$<\/p>\n<\/div>\n
\n\u306f\u57fa\u672c\u30d9\u30af\u30c8\u30eb $\\boldsymbol{i}, \\, \\boldsymbol{j}, \\, \\boldsymbol{k}$ \u306f\u4e92\u3044\u306b\u76f4\u4ea4\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n
\u3053\u306e\u3088\u3046\u306b\uff0c\u5927\u304d\u3055\u304c $1$ \u3067\u4e92\u3044\u306b\u76f4\u4ea4\u3057\u3066\u3044\u308b\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u3053\u3068\u3092\u6b63\u898f\u76f4\u4ea4\u57fa\u5e95<\/strong><\/span>\u306a\u3069\u3068\u8a00\u3063\u305f\u308a\u3059\u308b\u3002<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d<\/h3>\n
3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u3082\u3046\u3072\u3068\u3064\u306e\u639b\u3051\u7b97\u3002\u5916\u7a4d\u3068\u306f\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u5225\u306e\u30d9\u30af\u30c8\u30eb\u3092\u3064\u304f\u308b\u6f14\u7b97\u3002\u5916\u7a4d\u306e\u8a18\u53f7\u306f $\\times $\u3002\u639b\u3051\u7b97\u3068\u540c\u3058\u8a18\u53f7\u3060\u304c\uff0c\u7701\u7565\u4e0d\u53ef\uff01<\/strong><\/span>\u3000\u7b54\u3048\u304c\u30d9\u30af\u30c8\u30eb\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n
\n\n\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5916\u7a4d\u306e\u5b9a\u7fa9<\/h4>\n
\\begin{eqnarray}
\n{\\color{blue}\\boldsymbol{i}\\times \\boldsymbol{j}} &=& \\boldsymbol{k}\u00a0 \\\\
\n{\\color{blue}\\boldsymbol{j}\\times \\boldsymbol{k}} &=& \\boldsymbol{i} \\\\
\n{\\color{blue}\\boldsymbol{k}\\times \\boldsymbol{i}} &=& \\boldsymbol{j}
\n\\end{eqnarray}
\n\u30b5\u30a4\u30af\u30ea\u30c3\u30af\uff08\u5faa\u74b0\u7684\uff09\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002\u307e\u305f\uff0c
\\begin{eqnarray}
\n{\\color{green}\\boldsymbol{j}\\times \\boldsymbol{i}} &=& -{\\color{blue}\\boldsymbol{i}\\times \\boldsymbol{j}} = -\\boldsymbol{k} \\\\
\n{\\color{green}\\boldsymbol{k}\\times \\boldsymbol{j}} &=&\u00a0 -{\\color{blue}\\boldsymbol{j}\\times \\boldsymbol{k}} = -\\boldsymbol{i} \\\\
\n{\\color{green}\\boldsymbol{i}\\times \\boldsymbol{k}} &=& -{\\color{blue}\\boldsymbol{k}\\times \\boldsymbol{i}} = -\\boldsymbol{j}
\n\\end{eqnarray}
\n\u9806\u5e8f\u3092\u5909\u3048\u308b\u3068\u8ca0\u53f7\u304c\u3064\u304f\u3053\u3068\u306b\u6ce8\u610f\u3002\u6700\u5f8c\u306b\uff0c
$$ \\boldsymbol{i} \\times \\boldsymbol{i} = \\boldsymbol{j} \\times \\boldsymbol{j} = \\boldsymbol{k} \\times \\boldsymbol{k} = \\boldsymbol{0}$$
\n\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5916\u7a4d\u306f\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb\u3002<\/p>\n\u4e00\u822c\u306e\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5916\u7a4d\u306f…<\/h4>\n
$$\\boldsymbol{a} = a_x \\, \\boldsymbol{i} + a_y \\, \\boldsymbol{j} +a_z \\, \\boldsymbol{k}$$
$$\\boldsymbol{b} = b_x \\, \\boldsymbol{i} + b_y \\, \\boldsymbol{j} +b_z \\, \\boldsymbol{k}$$
\n\u3068\u3059\u308b\u3068\uff0c\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5916\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089\uff08\u5916\u7a4d\u306b\u3064\u3044\u3066\u5206\u914d\u5247\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u524d\u63d0\u3068\u3057\u3066\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7d50\u679c\u304c\u5c0e\u304b\u308c\u308b\u3002
\\begin{eqnarray}
\\boldsymbol{a}\\times\\boldsymbol{b} &=& \\left( a_x \\, \\boldsymbol{i} + a_y \\, \\boldsymbol{j} +a_z \\, \\boldsymbol{k} \\right) \\times \\left( b_x \\, \\boldsymbol{i} + b_y \\, \\boldsymbol{j} +b_z \\, \\boldsymbol{k}\\right) \\\\
&=&{\\color{white}{+}\\ \\\u00a0 }{\\color{gray}{a_x\u00a0 b_x\\, \\boldsymbol{i} \\times \\boldsymbol{i} }}+
a_x\u00a0 b_y\\, {\\color{blue}\\boldsymbol{i} \\times \\boldsymbol{j} }+
a_x\u00a0 b_z\\, {\\color{green}\\boldsymbol{i} \\times \\boldsymbol{k} }\\\\
&& + a_y\u00a0 b_x\\, {\\color{green}\\boldsymbol{j} \\times \\boldsymbol{i} }+
{\\color{gray}{a_y\u00a0 b_y\\, \\boldsymbol{j} \\times \\boldsymbol{j} }}+
a_y\u00a0 b_z\\, {\\color{blue}\\boldsymbol{j} \\times \\boldsymbol{k}} \\\\
&& + a_z\u00a0 b_x\\, {\\color{blue}\\boldsymbol{k} \\times \\boldsymbol{i}} +
a_z\u00a0 b_y\\, {\\color{green}\\boldsymbol{k} \\times \\boldsymbol{j}} +
{\\color{gray}{a_z\u00a0 b_z\\, \\boldsymbol{k} \\times \\boldsymbol{k} }}\\\\
&=& {\\color{white}{+}\\ \\\u00a0 }
a_x\u00a0 b_y\\, {\\color{blue}\\boldsymbol{i} \\times \\boldsymbol{j} }
\\, -\\,\u00a0 a_x\u00a0 b_z\\, {\\color{blue}\\boldsymbol{k} \\times \\boldsymbol{i}} \\\\
&& \\, -\\,\u00a0 a_y\u00a0 b_x\\, {\\color{blue}\\boldsymbol{i} \\times \\boldsymbol{j}}
+
a_y\u00a0 b_z\\, {\\color{blue}\\boldsymbol{j} \\times \\boldsymbol{k}} \\\\
&& + a_z\u00a0 b_x\\, {\\color{blue}\\boldsymbol{k} \\times \\boldsymbol{i} }
-a_z\u00a0 b_y\\, {\\color{blue}\\boldsymbol{j} \\times \\boldsymbol{k} }
\\\\
&=& (a_y b_z -a_z b_y) \\,\\boldsymbol{i} + (a_z b_x -a_x b_z) \\,\\boldsymbol{j} +(a_x b_y -a_y b_x) \\,\\boldsymbol{k}
\\end{eqnarray}
\u3059\u306a\u308f\u3061
$$\\boldsymbol{a}\\times\\boldsymbol{b} = (a_y b_z -a_z b_y) \\,\\boldsymbol{i} + (a_z b_x -a_x b_z) \\,\\boldsymbol{j} +(a_x b_y -a_y b_x) \\,\\boldsymbol{k} $$ \u4ee5\u5f8c\u306f\u3053\u308c\u3092\u5916\u7a4d\u306e\u5b9a\u7fa9\u3068\u3057\u3066\uff0c\u3053\u308c\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3059\u308b\u3002<\/p>\n<\/div>\n\n\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u306b\u3064\u3044\u3066\u306f\uff0c\u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u305f\u306a\u3044<\/h4>\n<\/div>\n
$$\\boldsymbol{b} \\times \\boldsymbol{a} {\\color{red}{\\neq}} \\boldsymbol{a} \\times \\boldsymbol{b}$$<\/div>\n\u5916\u7a4d\u306e\u5b9a\u7fa9\u3092\u4f7f\u3063\u3066\u3061\u3083\u3093\u3068\u8a08\u7b97\u3059\u308b\u3068\uff0c
\\begin{eqnarray}
\\boldsymbol{b} \\times \\boldsymbol{a} &=& {\\color{white}{+}\\ \\\u00a0 }
\n(b_y a_z -b_z a_y) \\,\\boldsymbol{i} +(b_z a_x -b_x a_z) \\,\\boldsymbol{j} + (b_x a_y -b_y a_x) \\,\\boldsymbol{k} \\\\
&=& -(a_y b_z -a_z b_y) \\,\\boldsymbol{i} -(a_z b_x -a_x b_z) \\,\\boldsymbol{j} -(a_x b_y -a_y b_x) \\,\\boldsymbol{k} \\\\
&=& -\\boldsymbol{a} \\times \\boldsymbol{b}
\\end{eqnarray}<\/div>\n\n<\/div>\n
\u3064\u307e\u308a\uff0c$$\\boldsymbol{b} \\times \\boldsymbol{a} = -\\boldsymbol{a} \\times \\boldsymbol{b}$$ \u9806\u5e8f\u3092\u4ea4\u63db\u3059\u308b\u3068\u30de\u30a4\u30ca\u30b9\u304c\u3064\u304f\u3068\u3044\u3046\u3053\u3068\u3067\u300c\u53cd\u4ea4\u63db<\/strong><\/span>\u300d\u3067\u3042\u308b\u3068\u3044\u3046\u8a00\u3044\u65b9\u3082\u3042\u308b\u3002<\/p>\n
\u5916\u7a4d\u306e\u53cd\u4ea4\u63db\u6027\u304b\u3089\u3059\u3050\u306b\u5c0e\u304b\u308c\u308b\u3053\u3068\u304c\u3042\u308b\u3002\u305d\u308c\u306f\uff0c\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5916\u7a4d\u306f\u30bc\u30ed\u30fb\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u3002\u8a3c\u660e\u306f\u7c21\u5358\u3067\uff0c\u53cd\u4ea4\u63db\u3067\u3042\u308b\u3053\u3068\u304b\u3089
$$ {\\color{blue}\\boldsymbol{a}} \\times\\boldsymbol{a} = -\\boldsymbol{a} \\times {\\color{blue}\\boldsymbol{a}}$$
\n\u5de6\u8fba\u306b\u79fb\u884c\u3057\u3066
\n$$ 2 \\,\\boldsymbol{a} \\times\\boldsymbol{a} = \\boldsymbol{0} \\quad\\therefore \\boldsymbol{a} \\times\\boldsymbol{a} = \\boldsymbol{0} $$<\/p>\n\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u306b\u3064\u3044\u3066<\/p>\n
- \n
- \u5916\u7a4d\u8a18\u53f7\u7701\u7565\u4e0d\u53ef\uff01\u3000\\(\\boldsymbol{a}\\times\\boldsymbol{b}\\) \u3092 \\(\\boldsymbol{a}\\,\\boldsymbol{b}\\) \u306a\u3093\u3066\u7701\u7565\u3057\u3066\u66f8\u3044\u305f\u3089\u30c0\u30e1\uff01<\/li>\n
- \u4ea4\u63db\u5247\u306f\u6210\u308a\u7acb\u305f\u306a\u3044\uff1a \\( \\boldsymbol{a} \\times \\boldsymbol{b} \\neq \\boldsymbol{b} \\times \\boldsymbol{a}\\) \u3080\u3057\u308d\u53cd\u4ea4\u63db\u3067\u3042\u308b\uff1a\\( \\boldsymbol{a} \\times \\boldsymbol{b} = -\\boldsymbol{b} \\times \\boldsymbol{a}\\)<\/li>\n
- \u5206\u914d\u5247\u304c\u6210\u308a\u7acb\u3064\uff1a \\(\\boldsymbol{a}\\times\\left( \\boldsymbol{b} + \\boldsymbol{c}\\right) = \\boldsymbol{a}\\times\\boldsymbol{b} + \\boldsymbol{a}\\times\\boldsymbol{c}\\),\u00a0 etc.<\/li>\n<\/ul>\n
\u5916\u7a4d\u306e\u3082\u3046\u4e00\u3064\u306e\u8868\u3057\u65b9<\/h4>\n
$$\\boldsymbol{a} \\times \\boldsymbol{b}\u00a0 = |\\boldsymbol{a}| |\\boldsymbol{b}| \\sin\\theta \\ \\boldsymbol{n}$$ \u3053\u3053\u3067 \\(\\theta\\) \u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{a}, \\, \\boldsymbol{b} \\) \u306e\u306a\u3059\u89d2\u3067\u3042\u308a\uff0c\\(\\boldsymbol{n}\\) \u306f \\(\\boldsymbol{a}\\) \u306b\u3082 \\(\\boldsymbol{b}\\) \u306b\u3082\u76f4\u4ea4\u3057\uff0c\u5411\u304d\u304c \\(\\boldsymbol{a}\\) \u304b\u3089 \\(\\boldsymbol{b}\\) \u306b\u53f3\u306d\u3058\u3092\u56de\u3057\u305f\u3068\u304d\u306e\u9032\u884c\u65b9\u5411\u3067\u3042\u308b\u3088\u3046\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3002<\/p>\n
\u3053\u306e\u8868\u3057\u65b9\u3082\uff0c\u4e0a\u306b\u8ff0\u3079\u305f\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089\u6c42\u3081\u3089\u308c\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n
\u8aac\u660e\u306e\u6982\u8981\u306f\u4ee5\u4e0b\u306e\u901a\u308a\uff1a
\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c\\( \\boldsymbol{a}, \\boldsymbol{b} \\) \u306f \\(xy\\) \u5e73\u9762\u4e0a\u306b\u3042\u308b\u3068\u3057\uff0c\u3055\u3089\u306b \\( \\boldsymbol{a} \\) \u306f\\(x\\) \u8ef8\u4e0a\uff0c\\(+x\\) \u306e\u5411\u304d\u306b\u3042\u308b\u3068\u3059\u308b\u3002\u3059\u308b\u3068\uff0c
\\begin{eqnarray}
\\boldsymbol{a} &=& (a_x, a_y, a_z) = (|\\boldsymbol{a}|, 0, 0) \\\\
\\boldsymbol{b} &=& (b_x, b_y, b_z) = (|\\boldsymbol{b}| \\cos\\theta, |\\boldsymbol{b}| \\sin\\theta, 0) \\end{eqnarray} \u5916\u7a4d\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{eqnarray} \\left(\\boldsymbol{a}\\times\\boldsymbol{b} \\right)_x &=& a_y b_z -a_z b_y = 0 \\\\
\\left(\\boldsymbol{a}\\times\\boldsymbol{b} \\right)_y &=& a_z b_x -a_x b_z = 0 \\\\
\\left(\\boldsymbol{a}\\times\\boldsymbol{b} \\right)_z &=& a_x b_y -a_y b_x = |\\boldsymbol{a}|\\,|\\boldsymbol{b}| \\sin\\theta \\\\
\\therefore \\boldsymbol{a}\\times\\boldsymbol{b} &=& |\\boldsymbol{a}|\\,|\\boldsymbol{b}| \\sin\\theta \\,\\boldsymbol{k}
\n\\equiv
\n|\\boldsymbol{a}|\\,|\\boldsymbol{b}| \\sin\\theta \\,\\boldsymbol{n}
\n\\end{eqnarray}
\uff08\u3053\u3053\u3067\u306f \\(\\boldsymbol{n}\\) \u3059\u306a\u308f\u3061 \\(\\boldsymbol{k}\\) \u304c \\(\\boldsymbol{a}\\) \u306b\u3082 \\(\\boldsymbol{b}\\) \u306b\u3082\u76f4\u4ea4\u3059\u308b\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3002\uff09\u3044\u3063\u305f\u3093\u8a3c\u660e\u3055\u308c\u308b\u3068\uff0c\u3053\u306e\u5f0f\u306f\u5ea7\u6a19\u7cfb\u306e\u53d6\u308a\u65b9\u306b\u3088\u3089\u305a\u306b\u6210\u308a\u7acb\u3064\u306e\u3067\uff0c\\( \\boldsymbol{a}, \\boldsymbol{b} \\) \u304c \\(xy\\) \u5e73\u9762\u4e0a\u306b\u306a\u3044\u3088\u3046\u306a\u4e00\u822c\u306e\u5834\u5408\u306b\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n\\(| \\boldsymbol{a} \\times \\boldsymbol{b} |\\) \u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{a}, \\, \\boldsymbol{b} \\) \u3067\u4f5c\u3089\u308c\u308b\u5e73\u884c\u56db\u8fba\u5f62\u306e\u9762\u7a4d\u3092\u8868\u3059\u3053\u3068\u306f\u4ee5\u4e0b\u306e\u56f3\u304b\u3089\u308f\u304b\u308b\u3002<\/p>\n
<\/div>\n\u307e\u305f\uff0c\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5916\u7a4d\u306f\u30bc\u30ed\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3053\u3068\u306f\u3059\u3067\u306b\u8ff0\u3079\u305f\u304c\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304c\u5e73\u884c\u3067\u3042\u308c\u3070\uff0c\u5916\u7a4d\u306f\u30bc\u30ed\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002\u8a3c\u660e\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n
\\(\\boldsymbol{b}\\) \u304c \\(\\boldsymbol{a}\\) \u306b\u5e73\u884c\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u3042\u308b\u5b9a\u6570 \\(k\\) \u3092\u4f7f\u3063\u3066\uff0c\\(\\boldsymbol{b}= k \\boldsymbol{a}\\) \u3068\u66f8\u3051\u308b\u3068\u3044\u3046\u3053\u3068\u3002\u3057\u305f\u304c\u3063\u3066\uff0c
\n$$\\boldsymbol{a}\\times\\boldsymbol{b} = \\boldsymbol{a}\\times(k \\boldsymbol{a} ) = k \\boldsymbol{a}\\times\\boldsymbol{a} =\\boldsymbol{0}$$<\/p>\n\u30d9\u30af\u30c8\u30eb\u306e\u4e09\u91cd\u7a4d<\/h3>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306b\u3088\u308b\u300c\u639b\u3051\u7b97\u300d<\/p>\n
\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d<\/h4>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u300c\u639b\u3051\u7b97\u300d\u306b\u3088\u3063\u3066\u6700\u7d42\u7684\u306b\u30b9\u30ab\u30e9\u30fc\u3092\u3064\u304f\u308b\u6f14\u7b97\u306a\u306e\u3067\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u300c\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u300d\u3068\u547c\u3076\uff08\u307e\u304e\u3089\u308f\u3057\u3044\uff09\u3002
\n$$\\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c})$$ \u3053\u308c\u304c\u30b9\u30ab\u30e9\u30fc\u3092\u3064\u304f\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u308f\u304b\u308b\u3002<\/p>\n- \n
- \u5148\u306b\u304b\u3063\u3053\u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u3092\u8a08\u7b97\u3059\u308b\u306e\u3067 \\( (\\boldsymbol{b} \\times \\boldsymbol{c}) \\) \u306f\u5916\u7a4d\u306a\u306e\u3067\u30d9\u30af\u30c8\u30eb\u3092\u4f5c\u308b\u3002\u3053\u308c\u3092\u305f\u3068\u3048\u3070 \\(\\boldsymbol{d}\\) \u3068\u3057\u3088\u3046\u3002$$\\boldsymbol{b} \\times \\boldsymbol{c} = \\boldsymbol{d}$$<\/li>\n
- \u6b21\u306b\uff0c\\(\\boldsymbol{a}\\cdot\\boldsymbol{d}\\) \u3092\u8a08\u7b97\u3059\u308b\u304c\uff0c\u3053\u308c\u306f\u5185\u7a4d\u306a\u306e\u3067\u6700\u7d42\u7684\u306b\u7b54\u3048\u306f\u30b9\u30ab\u30e9\u30fc\u3002<\/li>\n<\/ol>\n\n
\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306e\u91cd\u8981\u306a\u516c\u5f0f<\/h4>\n<\/div>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u9806\u756a\u3092\u30b5\u30a4\u30af\u30ea\u30c3\u30af\u306b\u5909\u3048\u3066\u3082\u5024\u306f\u5909\u308f\u3089\u306a\u3044\u3002\u3064\u307e\u308a\uff0c
\n$$ \\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c}) = \\boldsymbol{b}\\cdot (\\boldsymbol{c} \\times \\boldsymbol{a}) = \\boldsymbol{c}\\cdot (\\boldsymbol{a} \\times \\boldsymbol{b}) $$<\/div>\n<\/div>\n\u76f4\u63a5\u8a08\u7b97\u3057\u3066\uff0c\u305f\u3068\u3048\u3070 \\( \\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c}) = \\boldsymbol{b}\\cdot (\\boldsymbol{c} \\times \\boldsymbol{a})\\) \u3092\u793a\u3057\u3066\u307f\u308b\u3002<\/div>\n\\begin{eqnarray}
\n\\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c}) &=&
\na_x (\\boldsymbol{b} \\times \\boldsymbol{c})_x + a_y (\\boldsymbol{b} \\times \\boldsymbol{c})_y + a_z (\\boldsymbol{b} \\times \\boldsymbol{c})_z \\\\
\n&=& a_x ({\\color{blue}{b_y}} c_z -{\\color{green}{b_z}} c_y) + a_y ({\\color{green}{b_z}} c_x -{\\color{red}{b_x}} c_z) + a_z ({\\color{red}{b_x}} c_y -{\\color{blue}{b_y}} c_x) \\\\
\n&=& {\\color{red}{b_x}} (c_y a_z -c_z a_y) + {\\color{blue}{b_y}} (c_z a_x -c_x a_z) + {\\color{green}{b_z}} (c_x a_y -c_y a_x) \\\\
\n&=&\u00a0 b_x (\\boldsymbol{c}\\times\\boldsymbol{a})_x + b_y (\\boldsymbol{c}\\times\\boldsymbol{a})_y + b_z (\\boldsymbol{c}\\times\\boldsymbol{a})_z \\\\
\n&=& \\boldsymbol{b}\\cdot (\\boldsymbol{c}\\times\\boldsymbol{a})
\n\\end{eqnarray}<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\u4ee5\u4e0b\u306e\u56f3\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306f\uff0c3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u4f5c\u3089\u308c\u308b\u5e73\u884c\u516d\u9762\u4f53\u306e\u4f53\u7a4d $V$ \u306b\u306a\u308b\u3002<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n\n\u53c2\u8003\uff1a<\/h5>\n<\/div>\n
\\(\\boldsymbol{a}\\times\\boldsymbol{b}\\) \u306f \\(\\boldsymbol{a}\\) \u306b\u3082 \\(\\boldsymbol{b}\\) \u306b\u3082\u76f4\u4ea4\u3059\u308b\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3053\u3068\u3092\uff0c\u30b9\u30ab\u30e9\u30fc3\u91cd\u7a4d\u306e\u6027\u8cea\u3092\u4f7f\u3063\u3066\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/div>\n$$ {\\color{red}{\\boldsymbol{a}}} \\cdot(\\boldsymbol{a}\\times\\boldsymbol{b}) = \\boldsymbol{b}\\cdot ({\\color{red}{\\boldsymbol{a}}} \\times\\boldsymbol{a}) = 0, \\quad \\because \\boldsymbol{a} \\times\u00a0 \\boldsymbol{a} = \\boldsymbol{0}$$$$\\therefore\\ \\ \\boldsymbol{a} \\perp \\boldsymbol{a}\\times\\boldsymbol{b}$$<\/div>\n\u540c\u69d8\u306b\u3057\u3066 \\(\\boldsymbol{b}\\perp \\boldsymbol{a}\\times\\boldsymbol{b}\\) \u3082\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/div>\n\n\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d<\/h4>\n<\/div>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u300c\u639b\u3051\u7b97\u300d\u306b\u3088\u3063\u3066\u6700\u7d42\u7684\u306b\u30d9\u30af\u30c8\u30eb\u3092\u3064\u304f\u308b\u6f14\u7b97\u306a\u306e\u3067\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u300c\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u300d\u3068\u547c\u3076\uff08\u3084\u3084\u3053\u3057\u3044\uff09\u3002$$ \\boldsymbol{a}\\times(\\boldsymbol{b}\\times\\boldsymbol{c})$$<\/div>\n\u3053\u308c\u304c\u30d9\u30af\u30c8\u30eb\u3092\u3064\u304f\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u308f\u304b\u308b\u3002<\/div>\n\n- \n
- \u5148\u306b\u304b\u3063\u3053\u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u8a08\u7b97\u3002\\( (\\boldsymbol{b}\\times\\boldsymbol{c}) \\) \u306f\u5916\u7a4d\u306a\u306e\u3067\u30d9\u30af\u30c8\u30eb\u3092\u4f5c\u308b\u3002\u3053\u308c\u3092\u305f\u3068\u3048\u3070 \\(\\boldsymbol{d}\\) \u3068\u3057\u3088\u3046\u3002<\/li>\n
- \u6b21\u306b\uff0c\\( \\boldsymbol{a}\\times\\boldsymbol{d} \\) \u3092\u8a08\u7b97\u3059\u308b\u304c\uff0c\u3053\u308c\u306f\u5916\u7a4d\u306a\u306e\u3067\u6700\u7d42\u7684\u306b\u7b54\u3048\u306f\u30d9\u30af\u30c8\u30eb\u3002<\/li>\n<\/ol>\n
<\/h5>\n
\u5916\u7a4d\u306e\u300c\u7d50\u5408\u5247\u300d\u306f\u6210\u308a\u7acb\u305f\u306a\u3044\u3002<\/b><\/u><\/span>\u3064\u307e\u308a\uff0c$$\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c}) {\\color{red}{\\neq}} (\\boldsymbol{a}\\times\\boldsymbol{b}) \\times \\boldsymbol{c}$$<\/p>\n
<\/p>\n
\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306e\u91cd\u8981\u306a\u516c\u5f0f<\/h4>\n
$$\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c}) = (\\boldsymbol{a}\\cdot\\boldsymbol{c}) \\boldsymbol{b} -(\\boldsymbol{a}\\cdot\\boldsymbol{b}) \\boldsymbol{c} $$<\/p>\n
\u6210\u5206\u3092\u76f4\u63a5\u8a08\u7b97\u3057\u3066\u3053\u306e\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\u4f8b\u3068\u3057\u3066 $x$ \u6210\u5206\u306e\u8a08\u7b97\uff1a<\/p>\n
\\begin{eqnarray}
\n\\bigl\\{\\boldsymbol{a}\\times(\\boldsymbol{b}\\times\\boldsymbol{c})\\bigr\\}_x &=& a_y (\\boldsymbol{b}\\times\\boldsymbol{c})_z -a_z (\\boldsymbol{b}\\times\\boldsymbol{c})_y \\\\
\n&=& a_y ({\\color{blue}{b_x}} c_y -b_y {\\color{green}{c_x}}) -a_z (b_z {\\color{green}{c_x}} -{\\color{blue}{b_x}} c_z) \\\\
\n&=& ({\\color{white}{a_x c_x}} {\\color{white}{+}\\ \\, } a_y c_y + a_z c_z) {\\color{blue}{b_x}}\u00a0 -({\\color{white}{a_x b_x}\\\u00a0 }{\\color{white}{+}} a_y b_y + a_z b_x) {\\color{green}{c_x}} \\\\
\n&=& (a_x c_x + a_y c_y + a_z c_z) {\\color{blue}{b_x} } -(a_x b_x + a_y b_y + a_z b_x) {\\color{green}{c_x}} \\\\
\n&=& \\bigl\\{(\\boldsymbol{a}\\cdot\\boldsymbol{c}) \\boldsymbol{b} -(\\boldsymbol{a}\\cdot\\boldsymbol{b}) \\boldsymbol{c} \\bigr\\}_x
\n\\end{eqnarray}<\/p>\n\u30d9\u30af\u30c8\u30eb\u306e\u6f14\u7b97\u306e\u307e\u3068\u3081<\/h3>\n
\u4ee5\u4e0a\u306e\u7d50\u679c\u3092\u307e\u3068\u3081\u308b\u3068…<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u8868\u3057\u65b9<\/h4>\n
$$ \\boldsymbol{a} = a_x \\,\\boldsymbol{i} + a_y \\,\\boldsymbol{j} + a_z \\,\\boldsymbol{k}$$ \u307e\u305f\u306f $$ \\boldsymbol{a} = (a_x, \\ a_y, \\ a_z) $$<\/tt><\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d<\/h4>\n
$$\\boldsymbol{a}\\cdot\\boldsymbol{b} = a_x b_x + a_y b_y + a_z b_z = |\\boldsymbol{a} | |\\boldsymbol{b} | \\cos\\theta$$<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055<\/h4>\n
$$ |\\boldsymbol{a} | \\equiv \\sqrt{\\boldsymbol{a}\\cdot\\boldsymbol{a}} = \\sqrt{a_x^2 + a_y^2 + a_z^2} $$<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d<\/h4>\n
\\begin{eqnarray}
\n\\boldsymbol{a}\\times\\boldsymbol{b} &=& (a_y b_z -a_z b_y) \\,\\boldsymbol{i} + (a_z b_x -a_x b_z) \\,\\boldsymbol{j} +(a_x b_y -a_y b_x) \\,\\boldsymbol{k} \\\\
\n&=& |\\boldsymbol{a}| |\\boldsymbol{b}| \\sin\\theta\\ \\boldsymbol{n}
\n\\end{eqnarray}
\n\\(\\boldsymbol{n}\\) \u306f \\(\\boldsymbol{a}\\) \u306b\u3082 \\(\\boldsymbol{b}\\) \u306b\u3082\u76f4\u4ea4\u3057\uff0c\u5411\u304d\u304c \\(\\boldsymbol{a}\\) \u304b\u3089 \\(\\boldsymbol{b}\\) \u306b\u53f3\u306d\u3058\u3092\u56de\u3057\u305f\u3068\u304d\u306e\u9032\u884c\u65b9\u5411\u3067\u3042\u308b\u3088\u3046\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3002<\/p>\n\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306e\u516c\u5f0f<\/h4>\n
$$ \\boldsymbol{a}\\cdot (\\boldsymbol{b} \\times \\boldsymbol{c}) = \\boldsymbol{b}\\cdot (\\boldsymbol{c} \\times \\boldsymbol{a}) = \\boldsymbol{c}\\cdot (\\boldsymbol{a} \\times \\boldsymbol{b}) $$<\/p>\n
<\/p>\n
\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306e\u516c\u5f0f<\/h4>\n
$$\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c}) = (\\boldsymbol{a}\\cdot\\boldsymbol{c}) \\boldsymbol{b} -(\\boldsymbol{a}\\cdot\\boldsymbol{b}) \\boldsymbol{c} $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u30d9\u30af\u30c8\u30eb\u306e\u8db3\u3057\u7b97\uff0c\u5f15\u304d\u7b97\uff0c\u5185\u7a4d\uff0c\u5916\u7a4d\uff0c\u4e09\u91cd\u7a4d\u3002<\/p>