Complex NumbersHard
Question
If |z1| = |z2| = |z3| = 1 and z1, z2, z3 are represented by the vertices of an equilateral triangle then
Options
A.z1 + z2 + z3 = 0
B.z1 z2 z3 = 1
C.z1z2 + z2z3 + z3z1 = 0
D.none of these
Solution
All the three vertices lies on circle |z| = 1
take z1 = z1, z2 = z1ω, z3 = z1ω2
so z1 + z2 + z3 = z1(1 + ω + ω2) = 0
z1z2z3 = z13
z1z2 + z2z3 + z3z1 = z12 (ω + ω3 + ω2) = 0
take z1 = z1, z2 = z1ω, z3 = z1ω2
so z1 + z2 + z3 = z1(1 + ω + ω2) = 0
z1z2z3 = z13
z1z2 + z2z3 + z3z1 = z12 (ω + ω3 + ω2) = 0
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