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Math Help - Another complex number

  1. #1
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    Another complex number

    In an Argand diagram O is the origin, P represents the number 7 i and Q represent the number 12 + 4i.

    Calculate the size of angle OPQ.


    Ive done 1st part but how can I calculate angle OPQ in this case?
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by geton View Post
    In an Argand diagram O is the origin, P represents the number 7 i and Q represent the number 12 + 4i.

    Calculate the size of angle OPQ.


    Ive done 1st part but how can I calculate angle OPQ in this case?
    What's the first part?

    I'd call the point P(7, -1) and Q(12, 4) and use them as vectors based at the origin. Then use the dot product to get the angle.

    The other way is a bit messier: put each point into the form r~e^{i~\theta}. The find the difference in the arguments.

    -Dan
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  3. #3
    Member SengNee's Avatar
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    Quote Originally Posted by geton View Post
    In an Argand diagram O is the origin, P represents the number 7 i and Q represent the number 12 + 4i.

    Calculate the size of angle OPQ.


    Ive done 1st part but how can I calculate angle OPQ in this case?
    You can find out the modulus and arg of P and Q.
    After that,you have to find the angle POQ.
    Then, use the cosine formula to calculate length PQ.
    Last step is use sine formula to get the angle OPQ.
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  4. #4
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    Fro any two non-zero complex numbers z & w the angle between them can be found by: \Theta  = \arccos \left( {\frac{{{\mathop{\rm Re}\nolimits} (z){\mathop{\rm Re}\nolimits} (w) + {\mathop{\rm Im}\nolimits} (z){\mathop{\rm Im}\nolimits} (w)}}{{\left| z \right|\left| w \right|}}} \right)
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