This is an attempt to solve the system for z1,z2,z3
Using the second equation to compare magnitudes:
8|(z1+z2)||z3|=3|z1||z2|
8|(z1+z2)|1=3*2*2
|z1+z2|=3/2
In polar form let
z1=2(cos(a1)+isin(a1)
z2=2(cos(a2)+isin(a2)
z1+z2=2(cos(a1)+cos(a2)+i(sin(a1)+sin(a2)))
|z1+z2|^2=2((cos(a1)+cos(a2))^2+((sin(a1)+sin(a2))^2)=(3/2)^2
|z1+z2|^2=2(2+2cos(a1)cos(a2)+2sin(a1)sin(a2))=9/4
|z1+z2|^2=4(1+cos(a1)cos(a2)+sin(a1)sin(a2))=9/4
(1+cos(a1-a2)=9/32
cos(a1-a2)=-23/32
but also comparing real parts
cos(a1+a3)+cos(a2+a3)=(3/4)cos(a1+a2)
and imaginary parts
sin(a1+a3)+cos(a2+a3)=(3/4)cos(a1+a2)
Unfortunately, a graph of these three equations does not show any solutions so I'm kind of stuck. I'm not sure I didn't make an algebra mistake somewhere.
Edit: fixed the algebra, but now the diagonal just shows the line on which the real and imaginary cross. Next step: make sense of that.
https://www.desmos.com/calculator/xdsluosbp4
Edited on April 24, 2024, 9:20 am
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Posted by Jer
on 2024-04-24 09:08:27 |
A little start
