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3D Vectors Problem - Find Cross Product

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Given |[itex]\vec{a}[/itex]| = 8, |[itex]\vec{b}[/itex]| = 9 and the angle between vector [itex]\vec{a}[/itex] and [itex]\vec{b}[/itex] is 48° find the cross product, [itex]\vec{a}[/itex] X [itex]\vec{b}[/itex].

    2. Relevant equations

    Let θ = angle between [itex]\vec{a}[/itex] and [itex]\vec{b}[/itex].

    [itex]\vec{a}[/itex] . [itex]\vec{b}[/itex] = ( [itex]\vec{x}[/itex]1 * [itex]\vec{x}[/itex]2 ) + ( [itex]\vec{y}[/itex]1 * [itex]\vec{y}[/itex]2 ) + ( [itex]\vec{z}[/itex]1 * [itex]\vec{z}[/itex]2 ) = |[itex]\vec{a}[/itex]| * |[itex]\vec{b}[/itex]| * Cosθ

    |[itex]\vec{a}[/itex] X [itex]\vec{b}[/itex]| = |[itex]\vec{a}[/itex]| * |[itex]\vec{b}[/itex]| * Sinθ

    [itex]\vec{a}[/itex] X [itex]\vec{b}[/itex] = [ ( [itex]\vec{y}[/itex]1 * [itex]\vec{z}[/itex]2 ) - ( [itex]\vec{y}[/itex]2 * [itex]\vec{z}[/itex]1 ) , ( [itex]\vec{z}[/itex]1 * [itex]\vec{x}[/itex]2 ) - ( [itex]\vec{z}[/itex]2 * [itex]\vec{x}[/itex]1 ) , ( [itex]\vec{x}[/itex]1 * [itex]\vec{y}[/itex]2 ) - ( [itex]\vec{x}[/itex]2 * [itex]\vec{y}[/itex]1 ) ]

    3. The attempt at a solution

    Honestly have no idea how to work this out, the only thing I thought of was assuming the coordinates of one of the vectors. Such as [itex]\vec{a}[/itex] = [0,8,0]. With that use it to solve for the coordinates of [itex]\vec{b}[/itex] with the dot product formula then find the cross product between [itex]\vec{a}[/itex] and [itex]\vec{b}[/itex]. Probably not the right way to do the question though, there might be a formula or method I am not aware.
     
  2. jcsd
  3. Mar 20, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The data given is sufficient to find the length of [itex]\vec{a}\times\vec{b}[/itex] but not to find [itex]\vec{a}\times\vec{b}[/itex]. To see that just imagine rotating the vectors [itex]\vec{a}[/itex] and [itex]vec{b}[/itex] while maintaining the same lengths and angle between them. Clearly the resultant vector will shift direction as you do that.
     
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