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Math Help - Angle of Parallelogram diagonals

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    Angle of Parallelogram diagonals

    Hello. I need help on this problem


    Calculate the angle between parallelogram diagonals builded onto vectors a = 2p+q and b=p-3q, if |p|=5, |q|=2, and the angle (p, q)=\frac{\pi}{3}
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  2. #2
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    Quote Originally Posted by faktor-g View Post
    Calculate the angle between parallelogram diagonals builded onto vectors a = 2p+q and b=p-3q, if |p|=5, |q|=2, and the angle (p, q)=\frac{\pi}{3}
    This is a nightmare of a problem in so far as calculation goes.
    I will not do it for you. But I gladly give you these hints.
    The diagonals of this parallelogram are a + b\;\& \,a - b.
    \left( {a + b} \right) \cdot \left( {a - b} \right) = a \cdot a - b \cdot b = \left\| a \right\|^2  - \left\| b \right\|^2 .
    \frac{1}{2} = \cos \left( {\frac{\pi }{3}} \right) = \frac{{p \cdot q}}{{\left\| p \right\|\left\| q \right\|}}.

    Now you can put this all together. Maybe someone here who does other people’s work can understand this question and give you a complete solution. But if you really want to understand you will do it for yourself.
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  3. #3
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    Quote Originally Posted by faktor-g View Post
    Hello. I need help on this problem


    Calculate the angle between parallelogram diagonals builded onto vectors a = 2p+q and b=p-3q, if |p|=5, |q|=2, and the angle (p, q)=\frac{\pi}{3}
    1. I've made an exact drawing of the parallelogram. So you have the possibility to control your results.

    2. Maybe you can use the cosine rule to calculate the angle in question:

    Let \mu denote the angle included by the diagonals of the parallelogram. Then

    (|a|)^2 = (\frac12 |a+b|)^2 + (\frac12|a-b|)^2 - 2 \cdot \frac12 |a+b| \cdot \frac12 |a-b| \cdot \cos(\mu)

    Solve for \cos(\mu) and afterwards determine the value of \mu.
    Attached Thumbnails Attached Thumbnails Angle of Parallelogram diagonals-winkel_parallelodiagonal.png  
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