{"id":4945,"date":"2014-03-05T06:05:45","date_gmt":"2014-03-04T21:05:45","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4945"},"modified":"2014-08-01T18:49:17","modified_gmt":"2014-08-01T09:49:17","slug":"dot-or-scalar-product","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4945","title":{"rendered":"Dot or scalar product"},"content":{"rendered":"
\n

The dot or scalar product of two vectors \\bold{A} and \\bold{B}, denoted by \\bold{A}\\cdot\\bold{B} (read \\bold{A} dot \\bold{B}) is defined as the product of the magnitude of \\bold{A} and \\bold{B} and the cosine of the angle between them. In symbols, <\/p>\n

\\bold{A}\\cdot\\bold{B} = AB\\cos\\theta,\\ 0\\le\\theta\\le\\pi\\cdots(4)<\/p>\n

Note that \\bold{A}\\cdot\\bold{B} is a scalar and not a vector. <\/p>\n

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

    \n
  1. \\bold{A}\\cdot\\bold{B} = \\bold{B}\\cdot\\bold{A}<\/li>\n

    Communicative Law for Dot Products<\/p>\n

  2. \\bold{A}\\cdot(\\bold{B} + \\bold{C}) = \\bold{A}\\cdot\\bold{B} + \\bold{A}\\cdot\\bold{C}<\/li>\n

    Distributive Law<\/p>\n

  3. m(\\bold{A}\\cdot\\bold{B}) = (m\\bold{A})\\cdot\\bold{B} = \\bold{A}\\cdot(m\\bold{\\bold{B}}) = (\\bold{A}\\cdot\\bold{B})m<\/li>\n

    where m is a scalar.<\/p>\n

  4. \\bold{i}\\cdot\\bold{i} = \\bold{j}\\cdot\\bold{j} = \\bold{k}\\cdot\\bold{k} = 1,\\ \\bold{i}\\cdot\\bold{j} = \\bold{j}\\cdot\\bold{k} = \\bold{k}\\cdot\\bold{i} = 0<\/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
      \n
    • \\bold{A}\\cdot\\bold{B} = A_1B_1 + A_2B_2 + A_3B_3<\/li>\n
    • \\bold{A}\\cdot\\bold{A} = A^2 = A_1^2 + A_2^2 + A_3^2<\/li>\n
    • \\bold{B}\\cdot\\bold{B} = B^2 = B_1^2 + B_2^2 + B_3^2<\/li>\n<\/ul>\n
    • If \\bold{A}\\cdot\\bold{B} = 0 and \\bold{A} and \\bold{B} are not null vectors, then \\bold{A} and \\bold{B} are perpendicular. <\/li>\n<\/ol>\n<\/blockquote>\n