{"id":4989,"date":"2014-03-19T06:05:44","date_gmt":"2014-03-18T21:05:44","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4989"},"modified":"2014-08-01T18:47:10","modified_gmt":"2014-08-01T09:47:10","slug":"triple-products","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4989","title":{"rendered":"Triple products"},"content":{"rendered":"
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Dot and cross multiplication of three vectors \\bold{A}, \\bold{B} and \\bold{C} may produce meaningful products of the form (\\bold{A}\\cdot\\bold{B})\\bold{C}, \\bold{A}\\cdot(\\bold{B}\\times\\bold{C}) and \\bold{A}\\times(\\bold{B}\\times\\bold{C}). The following laws are valid: <\/p>\n

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  1. (\\bold{A}\\cdot\\bold{B})\\bold{C} \\ne \\bold{A}(\\bold{B}\\cdot\\bold{C}) in general<\/li>\n
  2. \\bold{A}\\cdot(\\bold{B}\\times\\bold{C}) = \\bold{B}\\cdot(\\bold{C}\\times\\bold{A}) = \\bold{C}\\cdot(\\bold{A}\\times\\bold{B}) is volume of a parallelepiped having \\bold{A}, \\bold{B}, and \\bold{C} as edges, or the negative of this volume according as \\bold{A}, \\bold{B} and \\bold{C} do or do not form a right-handed system. If \\bold{A} = A_1\\bold{i} + A_2\\bold{j} + A_3\\bold{k}, \\bold{B} = B_1\\bold{i} + B_2\\bold{j} + B_3\\bold{k} and \\bold{C} = C_1\\bold{i} + C_2\\bold{j} + C_3\\bold{k}, then
    \n\\displaystyle \\bold{A}\\cdot(\\bold{B}\\times\\bold{C}) = \\left|\\begin{array}{ccc} A_1 & A_2 & A_3 \\\\ B_1 & B_2 & B_3 \\\\ C_1 & C_2 & C_3 \\end{array}\\right|\\cdots (6)<\/li>\n
  3. \\bold{A} \\times (\\bold{B} \\times \\bold{C}) \\ne (\\bold{A} \\times \\bold{B}) \\times \\bold{C}<\/li>\n
  4. \\bold{A} \\times (\\bold{B} \\times \\bold{C}) = (\\bold{A} \\cdot \\bold{C})\\bold{B} - (\\bold{A} \\cdot \\bold{B})\\bold{C}\\\\ (\\bold{A} \\times \\bold{\\bold{B}}) \\times \\bold{C} = (\\bold{A} \\cdot \\bold{C})\\bold{B} - (\\bold{B} \\cdot \\bold{C})\\bold{A}<\/li>\n<\/ol>\n

    The product \\bold{A} \\cdot (\\bold{B} \\times \\bold{C}) is sometimes called the scalar triple product<\/em> or box product<\/em> and may be denoted by \\left[\\bold{ABC}\\right]. The product \\bold{A} \\times (\\bold{B} \\times \\bold{C}) is called the vector triple product<\/em>. <\/p>\n

    In \\bold{A} \\cdot (\\bold{B} \\times \\bold{C}) parentheses are sometimes omitted and we write \\bold{A} \\cdot \\bold{B} \\times \\bold{C}. However, parentheses must be used in \\bold{A} \\times (\\bold{B} \\times \\bold{C}). Note that \\bold{A} \\cdot (\\bold{B} \\times \\bold{C}) = (\\bold{A} \\times \\bold{B}) \\cdot \\bold{C}. This is often expressed by stating that in a scalar triple product the dot and the cross can be interchanged without affecting the result. <\/p>\n<\/blockquote>\n