{"id":4950,"date":"2014-03-06T06:05:36","date_gmt":"2014-03-05T21:05:36","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4950"},"modified":"2014-08-01T18:48:30","modified_gmt":"2014-08-01T09:48:30","slug":"cross-or-vector-product","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4950","title":{"rendered":"Cross or vector product"},"content":{"rendered":"
\n

The cross or vector product of \\bold{A} and \\bold{B} is a vector \\bold{C} = \\bold{A} \\times \\bold{B} (read \\bold{A} cross \\bold{B}). The magnitude of \\bold{A}\\times\\bold{B} is defined as the product of the magnitudes of \\bold{A} and \\bold{B} and the sine of the angle between them. The direction of the vector \\bold{C} = \\bold{A}\\times\\bold{B} is perpendicular to the plane of \\bold{A} and \\bold{B} and such that \\bold{A}, \\bold{B} and \\bold{C} form a right-handed system. In symbols, <\/p>\n

\\bold{A}\\times\\bold{B} = AB\\sin{\\theta}\\bold{u},\\ 0\\le\\theta\\le\\pi\\cdots(5)<\/p>\n

where \\bold{u} is a unit vector indicating the direction of \\bold{A}\\times\\bold{B}. If \\bold{A} = \\bold{B} or if \\bold{A} is parallel to \\bold{B}, then \\sin\\theta = 0 and we define \\bold{A}\\times\\bold{B} = 0. <\/p>\n

\"\/\/en.wikipedia.org\/wiki\/Vector_product\"<\/a>
\/\/en.wikipedia.org\/wiki\/Vector_product<\/figcaption><\/figure>\n

The following laws are valid: <\/p>\n

    \n
  1. \\bold{A} \\times \\bold{B} = - \\bold{B} \\times \\bold{A} <\/li>\n
  2. \\bold{A} \\times (\\bold{B} + \\bold{C}) = \\bold{A}\\times\\bold{B} + \\bold{A} \\times \\bold{C} <\/li>\n
  3. m( \\bold{A} \\times \\bold{B}) = (m \\bold{A}) \\times \\bold{B} = \\bold{A} \\times (m \\bold{B}) = (\\bold{A} \\times \\bold{B})m <\/li>\n
  4. \\bold{i} \\times \\bold{i} = \\bold{j} \\times \\bold{j} = \\bold{k} \\times \\bold{k} = 0,\\ \\bold{i} \\times \\bold{j} = \\bold{k},\\ \\bold{j} \\times \\bold{k} = \\bold{i},\\ \\bold{k} \\times \\bold{i} = \\bold{j} <\/li>\n
  5. If \\bold{A} = A_1\\bold{i} + A_2\\bold{j} + A_3\\bold{k} and \\bold{B} = B_1\\bold{i} + B_2\\bold{j} + B_3\\bold{k}, then<\/li>\n

    \\displaystyle \\bold{A} \\times \\bold{B} = \\left|\\begin{array}{ccc} \\bold{i} & \\bold{j} & \\bold{k} \\\\ A_1 & A_2 & A_3 \\\\ B_1 & B_2 & B_3 \\end{array}\\right| = \\left|\\begin{array}{cc} A_2 & A_3 \\\\ B_2 & B_3 \\end{array}\\right| \\bold{i} - \\left|\\begin{array}{cc} A_1 & A_3 \\\\ B_1 & B_3 \\end{array}\\right| \\bold{j} + \\left|\\begin{array}{cc} A_1 & A_2 \\\\ B_1 & B_2 \\end{array}\\right| \\bold{k} <\/p>\n

  6. |\\bold{A} \\times \\bold{B}| is the area of a parallelogram with sides \\bold{A} and \\bold{B}. <\/li>\n
  7. If \\bold{A} \\times \\bold{B} = 0 and \\bold{A} and \\bold{B} are not null vectors, then \\bold{A} and \\bold{B} are parallel. <\/li>\n<\/ol>\n

    Note that communicative law for cross products is failed. <\/p>\n<\/blockquote>\n