If |z1|=|z2|=|z3|=1 and z1, z2, z3 are represented by the vertices of an equilateral triangle then which of the following is not true?

Question

If |z1|=|z2|=|z3|=1 and z1, z2, z3 are represented by the vertices of an equilateral triangle then which of the following is not true? (A) z1+z2+z3=0 (B) z1 z2 z3=1 (C) z1 z2+z2 z3+z3 z1=0 (D) None of these

Tardigrade Admin · Accepted Answer

All the three vertices lies on circle |z|=1. Take z1=z1, z2=z1 ω, z3=z1 ω2 So z1+z2+z3=z1(1+ω+ω2)=0 z1 z2 z3=z13 z1 z2+z2 z3+z3 z1=z12(ω+ω3+ω2)=0

Q. If $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = 1$ and $z_{1}, z_{2}, z_{3}$ are represented by the vertices of an equilateral triangle then which of the following is not true?

$z_{1} + z_{2} + z_{3} = 0$

$z_{1} z_{2} z_{3} = 1$

$z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1} = 0$

None of these

All the three vertices lies on circle $∣ z ∣ = 1$ .
Take $z_{1} = z_{1}, z_{2} = z_{1} ω, z_{3} = z_{1} ω^{2}$
So $z_{1} + z_{2} + z_{3} = z_{1} (1 + ω + ω^{2}) = 0$
$z_{1} z_{2} z_{3} = z_{1}^{3}$
$z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1} = z_{1}^{2} (ω + ω^{3} + ω^{2}) = 0$

Q. If ∣z1​∣=∣z2​∣=∣z3​∣=1 and z1​,z2​,z3​ are represented by the vertices of an equilateral triangle then which of the following is not true?

z1​+z2​+z3​=0

z1​z2​z3​=1

z1​z2​+z2​z3​+z3​z1​=0