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Math Help - Dot product

  1. #1
    Junior Member
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    Mar 2009
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    Kentucky
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    Dot product

    How is the following equation proved/rationalized?


    A \bullet B=AB\cos\theta
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  2. #2
    Junior Member
    Joined
    Sep 2008
    From
    Wilmington NC
    Posts
    30
    AB = |A| |B| cos(θ)

    this comes from the definition of the inner product (dot product)

    It can be derived using the law of cosines:
    let a, b, c be vectors

    define vector c = a - b

    this forms a triangle, let θ be the angle between a and b, i.e. opposite side c


    from the law of cosines

    c = a + b - 2abcos(θ)

    from another property of the dot product, xx = x

    we replace c, a, and b to get

    cc = aa + bb - 2abcos(θ)

    since c = a - b
    cc = (a -b)(a - b)
    cc = (aa - 2ab + bb)

    plug this in

    (aa - 2ab + bb) = aa + bb - 2abcos(q)

    clean it up by canceling aa, and bb to get:

    -2ab = -2abcos(θ)

    divide by -2 to get the result

    ab = abcos(θ)
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