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Thread: Law of Cosines

  1. #1
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    Law of Cosines

    If $\displaystyle \bold{C} = \bold{A} + \bold{B} $

    Then $\displaystyle \bold{C} \cdot \bold{C} = (\bold{A} + \bold{B}) \cdot (\bold{A} + \bold{B}) $

    $\displaystyle |\bold{C}|^{2} = |\bold{A}^{2}| + |\bold{B}^{2}| + 2|\bold{A}||\bold{B}| \cos \theta $ where $\displaystyle \theta $ is the angle between the extension of $\displaystyle \bold{A} $ and $\displaystyle \bold{B} $.

    Then this is usually expressed as $\displaystyle C^{2} = A^{2} + B^{2} - 2AB \cos \phi $ where $\displaystyle \phi $ is the angle between $\displaystyle \bold{A} $ and $\displaystyle \bold{B} $.

    But why isn't $\displaystyle \bold{C} \cdot \bold{C} = |\bold{C}| $?
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  2. #2
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    Because say $\displaystyle \bold{C} = (x,y)$ then $\displaystyle |\bold{C}| = \sqrt{x^2+y^2}$ and $\displaystyle \bold{C}\cdot \bold{C} = x^2+y^2$. So you are off by a square root.
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