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Question

if z1,z2 and z3 are complex numbers such that

|z1|=|z2|=|z3|=1z1+1z2+1z3=1,

then |z1+z2+z3|


A

Equal to 1

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B

Less than 1

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C

Greater than 3

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D

Equal to 3

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Solution

The correct option is A

Equal to 1


1=1z1+1z2+1z3=z1¯z1z1+z2¯z2z2+z3¯z3z3

(hence, |z1|2=1=z1¯¯¯¯¯z1, etc)

|z1+z2+z3|=|¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯z1+z2+z3|= |z1+z2+z3|

(hence,|¯¯¯¯¯z1|=|z1|)


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