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Law of Vectors

  1. Mar 3, 2008 #1
    1. The problem statement, all variables and given/known data

    The identity below is significant because it relates 3 different kinds of products: a cross product and a dot product of 2 vectors on the left side, and the product of 2 real numbers on the right side. Prove the identity below.

    | a × b |² + (a • b)² = |a|²|b|²

    2. Relevant equations

    | a × b | = |a||b|sinθ
    (a • b) = |a||b|cosθ

    3. The attempt at a solution

    My work, LSH:

    = | a × b |² + (a • b)²

    = (|a||b|sinθ)(|a||b|sinθ) + (|a||b|cosθ)(|a||b|cosθ)

    = (|a|²)(|a||b|)(|a|sinθ)(|a||b|)(|b|²)(|b|sinθ)(|a| sinθ)(|b|sinθ)(sin²θ) + (|a|²)(|a||b|)(|a|cosθ)(|a||b|)(|b|²)(|b|cosθ)(|a| cosθ)(|b|cosθ)(cos²θ)

    = (|a|²)(|a||b|)²(|a|sinθ)²(|b|²)(|b|sinθ)²(sin²θ) + (|a|²)(|a||b|)²(|a|cosθ)²(|b|²)(|b|cosθ)²(cos²θ)

    = (|a|²|b|²(|a||b|)²) [(|a|sinθ)²(|b|sinθ)²(sin²θ) + (|a|cosθ)²(|b|²)(|b|cosθ)²(cos²θ)]

    = (|a|²|b|²(|a||b|)²) [(|a|²)(sin²θ)(|b|²)(sin²θ)(sin²θ) + (|a|²)(cos²θ)(|b|²)(|b|²)(cos²θ)(cos²θ)]

    = (|a|²|b|²(|a||b|)²) [(sin²θ)(sin²θ)(sin²θ) + (cos²θ)(cos²θ)(cos²θ)]

    And now I don't know what else to do! Please help. Did I mess up somewhere in my steps? Or is it possible to common factor still?
  2. jcsd
  3. Mar 3, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Stop with this. (|a||b|sinθ)(|a||b|sinθ) + (|a||b|cosθ)(|a||b|cosθ). That's |a|^2*|b|^2*(sin^2(theta)+cos^2(theta)). Now what? I don't know what you are doing on the following lines.
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