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Math Help - Complex Number Question

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    Complex Number Question

    On an Argand diagram, the points P and Q represent z1 and z2 respectively. OPQ is an equilateral triangle. Show that z1 + z2 = z1z2


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  2. #2
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    Quote Originally Posted by acevipa View Post
    On an Argand diagram, the points P and Q represent z1 and z2 respectively. OPQ is an equilateral triangle. Show that z1 + z2 = z1z2

    Since the sides of an equlateral triangle all have the same length sides we know:

    |z1|=|z2|=|z1-z2|

    The internal angles of the triangle are all 60 degress so we also know:

    arg(z1) = arg(z2) + 60

    And

    |z1^2|=|z2^2|

    And

    arg(z1^2)=arg(z2^2) + 120

    So if z1= r\angle\theta then z2= r\angle(\theta+60) and z2^2=r^2\angle(2\theta+2*60)
    z1^2+z2^2=r^2\angle2\theta + r^2\angle(2\theta+120)=r^2(1\angle2\theta+1\angle(  2\theta+120))=r^2(-1\angle(2\theta-120)) =r^2(1\angle(2\theta+60))=r\angle\theta(r\angle(\t  heta+60))=z1z2
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