Question

Given $z_1+z_2+z_3=A,z_1+z_2\omega +z_3{\omega }^2=B,z_1+z_2{\omega }^3+z_3\omega =C$
Prove : $|A{|}^2+|B{|}^2+|C{|}^2=3(|z_1{|}^2+|z_2{|}^2+|z_3{|}^2)$

Answer 1

We are given
$z_1+z_2+z_3=A$
$z_1+z_2\omega +z_3{\omega }^2=B$
$z_1+z_2{\omega }^2+z_3\omega =C$
We have
$|A{|}^2+|B{|}^2+|C{|}^2=A\overline{A}+B\overline{B}+C\overline{C}$ .....(1)
But $A\overline{A}=(z_1+z_2+z_3)(\overline{z_1}+\overline{z_2}+\overline{z_3})$
$=z_1\overline{z_1}+z_2\overline{z_2}+z_3\overline{z_3}+\overline{z_1}(z_2+z_3)+\overline{z_2}(z_3+z_1)+\overline{z_3}(z_1+z_2)$ ....
$|z_1{|}^2+|z_2{|}^2+|z_3{|}^2+\overline{z_1}(z_2+z_3)+\overline{z_2}(z_3+z_1)+\overline{z_3}(z_1+z_2)$
$B\overline{B}=(z_1+z_2\omega +z_3{\omega }^3)(\overline{z_1}+\overline{z_2}\omega +\overline{z_3}{\omega }^2)$
$=(z_1+z_2\omega +z_3{\omega }^3)(\overline{z_1}+\overline{z_2}{\omega }^2+\overline{z_3}\omega )$
$[\because \overline{\omega ={\omega }^2}and(\omega {)}^2=\omega ]$
$=z_1\overline{z_1}+z_2\overline{z_2}{\omega }^3+z_3\overline{z_3}{\omega }^{.3}+\overline{z_1}(z_2\omega +z_3{\omega }^2)+\overline{z_2}(z_3{\omega }^4+z_1{\omega }^2)+\overline{z_3}(z_1\omega +z_2{\omega }^2)$
$=|z_1{|}^2+|z_2{|}^2+|z_3{|}^2+\overline{z_1}(z_2\omega +z_3{\omega }^2)+\overline{z_2}(z_3\omega +z_1{\omega }^2)+\overline{z_3}(z_1\omega +z_2{\omega }^2)$
Similarly ,
$C\overline{C}=|z_1{|}^2+|z_2{|}^2+|z_3{|}^2+\overline{z_1}(z_2{\omega }^2+z_3\omega )+\overline{z_2}(z_3{\omega }^2+z_1\omega )+\overline{z_3}(z_1{\omega }^2+z_2\omega )$ ....(3)
Adding (1) , (2) and (3) , we get
$A\overline{A}+B\overline{B}+C\overline{C}=3[|z_1{|}^2+|z_2{|}^2+|z_3{|}^2]+\overline{z_1}\left[z_2\right(1+\omega +{\omega }^2)+z_3(1+{\omega }^2+\omega \left)\right]+\overline{z_2}\left[z_3\right(1+\omega +{\omega }^2)+z_1(1+{\omega }^2+\omega \left)\right]+\overline{z_3}\left[z_1\right(1+\omega +{\omega }^2)+z_2(1+{\omega }^2+\omega \left)\right]$
$=3[|z_1{|}^2+|z_2{|}^2+|z_3{|}^2]$
$[\because 1+\omega +{\omega }^2=0]$
$\therefore$ from (1) and (2) , we conclude
$|A{|}^2+|B{|}^2+|C{|}^2=3(|z_1{|}^2+|z_2{|}^2+|z_3{|}^2)$