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Law of Vectors (Cross Product)

  1. Mar 4, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove:
    (au) × v + (bu) × v = [(a + b)u] × v

    2. Relevant equations

    http://en.wikipedia.org/wiki/Cross_product

    3. The attempt at a solution

    u = ( x , y , z )
    v = ( x₂, y₂,z₂)

    LHS:
    = (au) × v + (bu) × v
    = [ ay z₂- y₂az , - (axz₂- x₂az) , axy₂- x₂ay ] + [ by z₂- y₂bz , - (bxz₂- x₂bz) , bxy₂- x₂by ]


    RHS:
    = [(a + b)u] × v
    = [ (a + b)x , (a + b)y , (a + b)z ] × ( x₂, y₂,z₂)
    = [ (a + b)y z₂- y₂(a + b)z , -((a + b)x z₂- x₂(a + b)z) , (a + b)x y₂- x₂(a + b)y]

    This is how far I got to prove left side equals right side. . .now I don't know what to do.
     
    Last edited: Mar 4, 2008
  2. jcsd
  3. Mar 4, 2008 #2
    Remember that vectors add component-wise (that is, the x-components add when adding two vectors), and that two vectors are equal if their components are equal.

    So, you're almost there! Simplify the left hand side a bit more by combining it into one vector instead of the sum of two vectors and compare the components!
     
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