{"id":7630,"date":"2024-02-20T12:05:58","date_gmt":"2024-02-20T03:05:58","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=7630"},"modified":"2024-02-20T12:09:29","modified_gmt":"2024-02-20T03:09:29","slug":"%e5%8f%82%e8%80%83%ef%bc%9asympy-%e3%81%a7%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%ae%e6%bc%94%e7%ae%97","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%ae%e6%bc%94%e7%ae%97\/%e5%8f%82%e8%80%83%ef%bc%9asympy-%e3%81%a7%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%ae%e6%bc%94%e7%ae%97\/","title":{"rendered":"\u53c2\u8003\uff1aSymPy \u3067\u30d9\u30af\u30c8\u30eb\u306e\u6f14\u7b97"},"content":{"rendered":"

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\u5fc5\u8981\u306a\u30e2\u30b8\u30e5\u30fc\u30eb\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n\r\ninit_printing<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SymPy \u306b\u304a\u3051\u308b\u30d9\u30af\u30c8\u30eb\u306e\u8868\u8a18<\/h3>\n
Matrix()<\/code> \u3092\u4f7f\u3063\u3066\u30d9\u30af\u30c8\u30eb\u3092\u8868\u8a18\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# \u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206<\/span>\r\nvar<\/span>(<\/span>'a1:4, b1:4, c1:4'<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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a<\/span> =<\/span> Matrix<\/span>([<\/span>a1<\/span>,<\/span> a2<\/span>,<\/span> a3<\/span>])<\/span>\r\nb<\/span> =<\/span> Matrix<\/span>([<\/span>b1<\/span>,<\/span> b2<\/span>,<\/span> b3<\/span>])<\/span>\r\nc<\/span> =<\/span> Matrix<\/span>([<\/span>c1<\/span>,<\/span> c2<\/span>,<\/span> c3<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u6f14\u7b97<\/h3>\n
\u8db3\u7b97\uff0c\u5f15\u304d\u7b97<\/h4>\n
\u30d9\u30af\u30c8\u30eb\u306e\u8db3\u7b97\u5f15\u304d\u7b97\u3092\u3057\u3066\u307f\u308b\u3068\uff0c\u6210\u5206\u540c\u58eb\u306e\u8db3\u7b97\u5f15\u304d\u7b97\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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a<\/span> +<\/span> b<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{1} + b_{1}\\\\a_{2} + b_{2}\\\\a_{3} + b_{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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a<\/span> -<\/span> b<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{1} – b_{1}\\\\a_{2} – b_{2}\\\\a_{3} – b_{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u6570\u500d<\/h4>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u6570\u500d\u306f\uff0c\u5404\u6210\u5206\u3092\u305d\u308c\u305e\u308c\u5b9a\u6570\u500d\u3057\u305f\u3082\u306e\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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C<\/span> *<\/span> a<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}C a_{1}\\\\C a_{2}\\\\C a_{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d<\/h3>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f $\\boldsymbol{a}\\cdot\\boldsymbol{b} = $a.dot(b)<\/code> \u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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a<\/span>.<\/span>dot<\/span>(<\/span>b<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
$\\displaystyle a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055<\/h4>\n
\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}$ \u306e\u5927\u304d\u3055 $|\\boldsymbol{a}|$ \u306f\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u306e\u5e73\u65b9\u6839\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002
\n$$|\\boldsymbol{a}| \\equiv \\sqrt{\\boldsymbol{a}\\cdot\\boldsymbol{a}}$$<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u3092\u8fd4\u3059\u95a2\u6570 norm()<\/code> \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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def<\/span> norm<\/span>(<\/span>v<\/span>):<\/span>\r\n    return<\/span> sqrt<\/span>(<\/span>v<\/span>.<\/span>dot<\/span>(<\/span>v<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4eca\u5b9a\u7fa9\u3057\u305f norm()<\/code> \u3092\u4f7f\u3063\u3066\uff0c\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}$ \u306e\u5927\u304d\u3055 $|\\boldsymbol{a}|$ \u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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norm<\/span>(<\/span>a<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle \\sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d<\/h3>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u306f $\\boldsymbol{a}\\times\\boldsymbol{b} = $a.cross(b)<\/code> \u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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a<\/span>.<\/span>cross<\/span>(<\/span>b<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{2} b_{3} – a_{3} b_{2}\\\\- a_{1} b_{3} + a_{3} b_{1}\\\\a_{1} b_{2} – a_{2} b_{1}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5916\u7a4d\u306f\u300c\u53cd\u4ea4\u63db\u300d<\/h4>\n
$\\boldsymbol{a}\\times\\boldsymbol{b} {\\color{red}{\\neq}} \\boldsymbol{b}\\times\\boldsymbol{a}$ \u3067\u3042\u308b\u3053\u3068\uff0c\u5177\u4f53\u7684\u306b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u300c\u53cd\u4ea4\u63db\u300d\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002
\n$$\\boldsymbol{a}\\times\\boldsymbol{b} = – \\boldsymbol{b}\\times\\boldsymbol{a}$$<\/p>\n
$\\boldsymbol{a}\\times\\boldsymbol{b} + \\boldsymbol{b}\\times\\boldsymbol{a} = \\boldsymbol{0}$ \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3044\u3044\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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a<\/span>.<\/span>cross<\/span>(<\/span>b<\/span>)<\/span> +<\/span> b<\/span>.<\/span>cross<\/span>(<\/span>a<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u4e09\u91cd\u7a4d<\/h3>\n
\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d<\/h4>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}$ \u306b\u3088\u308b\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306f $\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c})$ \u3067\u3042\u308a\uff0c\u3053\u308c\u306f\u7d50\u5c40\u306f\u30b9\u30ab\u30e9\u30fc\u3068\u306a\u308a\u307e\u3059\u3002\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u516c\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n
$$\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c})
\n= \\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a})
\n= \\boldsymbol{c} \\cdot (\\boldsymbol{a}\\times\\boldsymbol{b}) $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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$\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c})
\n= \\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a})$ \u3064\u307e\u308a\uff0c
\n$\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c}) -\\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a}) = 0$ \u3092\u793a\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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a<\/span>.<\/span>dot<\/span>(<\/span>b<\/span>.<\/span>cross<\/span>(<\/span>c<\/span>))<\/span> -<\/span> b<\/span>.<\/span>dot<\/span>(<\/span>c<\/span>.<\/span>cross<\/span>(<\/span>a<\/span>))<\/span>\r\n# \u30ab\u30c3\u30b3\u3092\u306f\u305a\u3057\u3066\u5c55\u958b\u3059\u308b<\/span>\r\nexpand<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u540c\u69d8\u306b\u3057\u3066\uff0c$\\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a}) -\\boldsymbol{c} \\cdot (\\boldsymbol{a}\\times\\boldsymbol{b}) = 0$ \u3092\u793a\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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b<\/span>.<\/span>dot<\/span>(<\/span>c<\/span>.<\/span>cross<\/span>(<\/span>a<\/span>))<\/span> -<\/span> c<\/span>.<\/span>dot<\/span>(<\/span>a<\/span>.<\/span>cross<\/span>(<\/span>b<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
$\\displaystyle b_{1} \\left(- a_{2} c_{3} + a_{3} c_{2}\\right) + b_{2} \\left(a_{1} c_{3} – a_{3} c_{1}\\right) + b_{3} \\left(- a_{1} c_{2} + a_{2} c_{1}\\right) – c_{1} \\left(a_{2} b_{3} – a_{3} b_{2}\\right) – c_{2} \\left(- a_{1} b_{3} + a_{3} b_{1}\\right) – c_{3} \\left(a_{1} b_{2} – a_{2} b_{1}\\right)$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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expand<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
$\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d<\/h4>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}$ \u306b\u3088\u308b\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306f $\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c})$ \u3067\u3042\u308a\uff0c\u3053\u308c\u306f\u7d50\u5c40\u306f\u30d9\u30af\u30c8\u30eb\u3092\u3064\u304f\u308a\u307e\u3059\u3002\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u516c\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n
$$\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c})
\n= (\\boldsymbol{a}\\cdot\\boldsymbol{c})\\,\\boldsymbol{b}
\n-(\\boldsymbol{a}\\cdot\\boldsymbol{b})\\,\\boldsymbol{c} $$<\/p>\n
\u4e00\u6319\u306b\u8a3c\u660e\u3059\u308b\u306b\u306f\uff0c<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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a<\/span>.<\/span>cross<\/span>(<\/span>b<\/span>.<\/span>cross<\/span>(<\/span>c<\/span>))<\/span> -<\/span> (<\/span>a<\/span>.<\/span>dot<\/span>(<\/span>c<\/span>)<\/span> *<\/span> b<\/span> -<\/span> a<\/span>.<\/span>dot<\/span>(<\/span>b<\/span>)<\/span> *<\/span> c<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[15]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{2} \\left(b_{1} c_{2} – b_{2} c_{1}\\right) – a_{3} \\left(- b_{1} c_{3} + b_{3} c_{1}\\right) – b_{1} \\left(a_{1} c_{1} + a_{2} c_{2} + a_{3} c_{3}\\right) + c_{1} \\left(a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}\\right)\\\\- a_{1} \\left(b_{1} c_{2} – b_{2} c_{1}\\right) + a_{3} \\left(b_{2} c_{3} – b_{3} c_{2}\\right) – b_{2} \\left(a_{1} c_{1} + a_{2} c_{2} + a_{3} c_{3}\\right) + c_{2} \\left(a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}\\right)\\\\a_{1} \\left(- b_{1} c_{3} + b_{3} c_{1}\\right) – a_{2} \\left(b_{2} c_{3} – b_{3} c_{2}\\right) – b_{3} \\left(a_{1} c_{1} + a_{2} c_{2} + a_{3} c_{3}\\right) + c_{3} \\left(a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}\\right)\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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expand<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2592,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-7630","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7630","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=7630"}],"version-history":[{"count":3,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7630\/revisions"}],"predecessor-version":[{"id":7633,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7630\/revisions\/7633"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2592"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=7630"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}