SOLUTION: If z1 , z2 are two complex numbers , |z1|=|z2| , and Arg(z1 + z2) = pi/4 ,find Arg(z1×z2)

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Question 1207479: If z1 , z2 are two complex numbers , |z1|=|z2| , and Arg(z1 + z2) = pi/4 ,find Arg(z1×z2)
Answer by ikleyn(52795) About Me  (Show Source):
You can put this solution on YOUR website!
.

arg(z1*z2) is always equal to arg(z1) + arg(z2)    (by the modulo of 2pi).



From the other side, if  |z1| = |z2|,  then arg(z1+z2) = %28arg%28z1%29%2Barg%282%29%29%2F2 

according to the parallelogram rule of adding vectors (= complex numbers), which

is in this case the "rhombus" rule of adding vectors.



From this, we conclude that, under given conditions,

    arg(z1*z2) = arg(z1) + arg(z2) = 2%2A%28%28arg%28z1%29+%2B+arg%28z2%29%29%2F2%29 = 2*arg(z1+z2) = 2%2A%28pi%2F4%29 = pi%2F2.    ANSWER

Solved.