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Anyone have a clear definition of how to do a cross product?

  1. Nov 7, 2004 #1
    anyone have a clear definition of how to do a cross product?
     
  2. jcsd
  3. Nov 7, 2004 #2

    Galileo

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    The cross-product [itex]\vec C[/itex] of two vectors [itex]\vec A[/itex] and [itex]\vec B[/itex] is most conveniently defined by:
    [itex]|\vec C|=|\vec A||\vec B|\sin(\theta)[/itex]
    where [itex]\theta[/itex] is the angle between [itex]\vec A[/itex] and [itex]\vec B[/itex].
    This gives the magnitude of [itex]\vec C[/itex]. The direction is given by the right-hand rule.

    To calculate the cross product when you know the components,
    it's usually easiest to form the symbolic 3X3 determinant:
    [tex]\vec A=(A_x,A_y,A_z)[/tex]
    [tex]\vec B=(B_x,B_y,B_z)[/tex]
    [tex]\vec C = \left|
    \begin{array}{ccc}
    \hat x & \hat y &\hat z \\
    A_x & A_y & A_z\\
    B_x & B_y & B_z
    \end{array}\right|
    [/tex]
    This follows from [itex]\hat x \times \hat y = \hat z[/itex] (and the other possible product combinations of these unit vectors) and the distributivity of the cross product (which is tedious to prove IIRC).
     
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