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Problem 1.1a from Griffith's E+M

  1. Aug 29, 2012 #1
    1. The problem statement, all variables and given/known data
    Using the definitions 1.1 and 1.4 show that the dot product and cross product are distributive when

    (a) the 3 vectors are coplanar

    2. Relevant equations

    [itex]\vec{A}\cdot\vec{B} = AB\cos(\theta)[/itex]
    [itex]\vec{A}\times\vec{B} = AB\sin(\theta)\hat{n}[/itex]


    3. The attempt at a solution

    See attachment.

    Not exactly sure of where to go from here, or if I'm even going in the right direction. I do want to do this without using components though as the book never used them when introducing the dot product or vectors.

    Haven't attempted cross product yet but I'm assuming it would be using a similar method to the dot product.
     

    Attached Files:

    Last edited: Aug 29, 2012
  2. jcsd
  3. Aug 29, 2012 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    Careful, in the equation [itex]\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = |\mathbf{A}||(\mathbf{B} + \mathbf{C})|\cos\theta[/itex], [itex]\theta[/itex] is the angle between [itex]\mathbf{A}[/itex] and [itex]\mathbf{B} + \mathbf{C}[/itex]; but in your diagram, it looks like you are using it to represent the angle between [itex]\mathbf{B}[/itex] and [itex]\mathbf{C}[/itex].

    Instead, try letting [itex]\theta[/itex] be the angle between [itex]\mathbf{A}[/itex] and [itex]\mathbf{B} + \mathbf{C}[/itex], [itex]\alpha[/itex] be the angle between [itex]\mathbf{A}[/itex] and [itex]\mathbf{B}[/itex] and [itex]\beta[/itex] be the angle between [itex]\mathbf{A}[/itex] and [itex]\mathbf{C}[/itex], then what you want to prove is that

    [tex]|\mathbf{A}||(\mathbf{B} + \mathbf{C})|\cos\theta = |\mathbf{A}||\mathbf{B}|\cos\alpha + |\mathbf{A}||\mathbf{C}|\cos\beta[/tex]
     
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