Question

If \( |z_1| = |z_2| = |z_3| = 1 \) and \( z_1 + z_2 + z_3 = \sqrt{2} + i \), then the number \( z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1 \) is:

• A. a positive real number
• B. a negative real number
• C. always zero
• D. a purely imaginary number

Hint:

When dealing with complex numbers on the unit circle, their conjugates are simply their reciprocals. This property can help simplify expressions involving complex conjugates.

Answer 1

The Correct Option is C

Step 1: Use the given conditions.
We are given that \( |z_1| = |z_2| = |z_3| = 1 \), which means that \( z_1, z_2, z_3 \) lie on the unit circle in the complex plane. We are also given that \( z_1 + z_2 + z_3 = \sqrt{2} + i \). Let’s use these conditions in the equation \( z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1 \).
Step 2: Analyze the expression.
To simplify \( z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1 \), note that for any complex number \( z \) on the unit circle, \( \bar{z} = \frac{1}{z} \). Using this, we rewrite the expression as: [ z₁ \frac1z₂ + z₂ \frac1z₃ + z₃ \frac1z₁ ] This is a cyclic sum of complex conjugates. Given the symmetry of the unit circle and the constraints, the sum simplifies to zero.
Step 3: Conclusion.
Since the sum simplifies to zero, we conclude that the value of the expression is always zero. Final Answer: always zero.