If z1,z2,z3 are complex numbers such that ∣z1∣=∣z2∣=∣z3∣=z11+z21+z31=1, then find the value of ∣z1+z2+z3∣. $
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Step-by-Step Solution
Step 1: Use the property of modulus for complex numbers
We are given that the modulus of each complex number z1,z2,z3 is 1. A fundamental property of complex numbers states that if the modulus of a complex number z is 1, then z multiplied by its conjugate zˉ equals 1. This implies that the conjugate of z is equal to its reciprocal, which is a very useful identity for this problem.
Step 2: Substitute reciprocals with conjugates
Using the property derived in the previous step, we can replace each reciprocal term zi1 with its corresponding conjugate ziˉ. This transforms the given expression into the modulus of the sum of the conjugates.
Step 3: Apply the property of conjugate of a sum
The conjugate of a sum of complex numbers is equal to the sum of their conjugates. This property allows us to combine the individual conjugates into the conjugate of the sum z1+z2+z3.
Step 4: Use the property of modulus of a conjugate
Another important property of complex numbers is that the modulus of a complex number is equal to the modulus of its conjugate. Therefore, the modulus of z1+z2+z3 is equal to the modulus of z1+z2+z3.
Step 5: Determine the final value
From the problem statement, we are given that z11+z21+z31=1. By substituting the equivalent expressions from the previous steps, we conclude that ∣z1+z2+z3∣ must also be equal to 1.