1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook