Question

Given $z_1+z_2+z_3=A,z_1+z_2\omega +z_3{\omega }^2=B,z_1+z_2{\omega }^3+z_3\omega =C$
Express$z_1+z_2+z_3$ in terms of A, B, C

Answer 1

We are given
$z_1+z_2+z_3=A$ .....(1)
$z_1+z_2\omega +z_3{\omega }^2=B$ ....(2)
$z_1+z_2{\omega }^2+z_3\omega =C$ ....(3)
Adding (1), (2) and (3) , we get
$3z_1+z_2(1+\omega +{\omega }^2)+z_3(1+{\omega }^2+\omega )=A+B+C$
or $z_1=\frac{A+B+C}{3}$ $[\because (1+\omega +{\omega }^2)=0]$
now multiplying (1,) (2) and (3) by 1, ${\omega }^2$ , $\omega$ respectively and adding , we get
$z_1(1+{\omega }^2+\omega )+z_2(1+{\omega }^3+{\omega }^3)+z_3(1+{\omega }^4+{\omega }^2)$
$=A+B{\omega }^2+C\omega$
or $z_2=\frac{A+B{\omega }^2+C\omega }{3}$
[$\because$ 1 + ${\omega }^4+{\omega }^2=1+\omega +{\omega }^2=0$ and ${\omega }^3$ = 1]
Similarly , $z_3=\frac{A+B\omega +C{\omega }^2}{3}$