Suppose we have 3 complex numbers , such that $$|z_1|=|z_2|=|z_3|=1$$ and they form equilateral triangle then will condition $$z_1.z_2.z_3=1$$ always be true? I know cube roots of unity , that is $w,w^2,1$ satisfy here but is this always true?

  • $\begingroup$ Why should the product be 1? Try with any complex number, say $2+i$. $\endgroup$ – ghosts_in_the_code Dec 10 '14 at 17:28
  • $\begingroup$ @user45195 Its mod is not one. $\endgroup$ – Tesla Dec 10 '14 at 17:30
  • $\begingroup$ @user45195 which number should be $2+i$? $\endgroup$ – mrf Dec 10 '14 at 17:31
  • $\begingroup$ I mean to say the condition needn't necessarily be true. $\endgroup$ – ghosts_in_the_code Dec 11 '14 at 9:35

No. Rotate the three point by multiplying with $e^{i\theta}$. Then the product is multiplied by $e^{3i\theta}$.

  • $\begingroup$ Hmm so will they always satisfy z1 + z2 + z3 = 0 ?? $\endgroup$ – Tesla Dec 10 '14 at 17:36

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