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Homework Help: Proving distributivity of Dot/Cross product

  1. Feb 5, 2012 #1
    Using the definitions in equations 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive;
    (a) when the three vectors are coplanar;
    (b) in the general case.

    Eq. 1.1) A dot B = ABcosθ

    Eq. 1.4) A cross B = ABsinθN

    This is exactly how my book puts the formulas.

    I know how the definition of the dot product is derived, and that it's distributive over vector addition, but I don't understand why they're asking why the three vectors are coplanar. I don't see where the third vector comes into play. I haven't even tried solving this on the cross product side because I know if I don't conceptually grasp the dot product part of it the cross product will only frustrate me.

    Here's my attempt at this proof:

    Part A: Stared at it for a while trying to figure it out and eventually gave up.

    Part B: Broke out the comfort food. Cried a little.

    1.2 Is the cross product associative?

    (Vector A cross Vector B) cross vector C (equals?) Vector A cross (Vector B cross Vector C)

    I know the cross product isn't associate because the order of the cross product determines the direction of the resultant vector, but I feel like there's more to it.

    Thank you all so much for your help!
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Feb 5, 2012 #2

    Ray Vickson

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    The question is asking you to prove [itex] \vec{A}\cdot(\vec{B} + \vec{C}) = \vec{A}\cdot \vec{B} + \vec{A} \cdot \vec{C}, [/itex] and the same type of result using [itex] \times [/itex] instead of [itex] \cdot[/itex] . So, of course there have to be three vectors involved.

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