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 :

Options

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

Answer

To solve the problem, we need to find the value of \( z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1 \) given that \( |z_1| = |z_2| = |z_3| = 1 \) and \( z_1 + z_2 + z_3 = \sqrt{2} + i \).

### Step-by-Step Solution:

1. **Understanding the Magnitudes**:
Since \( |z_1| = |z_2| = |z_3| = 1 \), we can express each \( z_i \) in the form \( z_i = e^{i\theta_i} \) for some angle \( \theta_i \).

2. **Using the Given Sum**:
We know that:
\[ z_1 + z_2 + z_3 = \sqrt{2} + i \]
This means the sum of these complex numbers results in a complex number with a real part of \( \sqrt{2} \) and an imaginary part of \( 1 \).

3. **Finding the Conjugate**:
The conjugate of the sum is:
\[ \overline{z_1 + z_2 + z_3} = \overline{\sqrt{2} + i} = \sqrt{2} - i \]
According to the properties of complex numbers, we have:
\[ \overline{z_1 + z_2 + z_3} = \overline{z_1} + \overline{z_2} + \overline{z_3} \]

4. **Multiplying the Sums**:
Now, we can multiply the sum by its conjugate:
\[ (z_1 + z_2 + z_3)(\overline{z_1} + \overline{z_2} + \overline{z_3}) = (\sqrt{2} + i)(\sqrt{2} - i) \]
Simplifying the right side:
\[ (\sqrt{2})^2 - (i)^2 = 2 + 1 = 3 \]

5. **Expanding the Left Side**:
The left side expands to:
\[ z_1 \overline{z_1} + z_1 \overline{z_2} + z_1 \overline{z_3} + z_2 \overline{z_1} + z_2 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1} + z_3 \overline{z_2} + z_3 \overline{z_3} \]
Since \( |z_i| = 1 \), we have \( z_i \overline{z_i} = 1 \). Therefore:
\[ 1 + 1 + 1 + (z_1 \overline{z_2} + z_2 \overline{z_1} + z_2 \overline{z_3} + z_3 \overline{z_2} + z_3 \overline{z_1} + z_1 \overline{z_3}) = 3 \]
This simplifies to:
\[ 3 + (z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) + (z_2 \overline{z_1} + z_3 \overline{z_2} + z_1 \overline{z_3}) = 3 \]

6. **Finding the Desired Expression**:
Let \( \omega = z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1} \). The equation becomes:
\[ 3 + \omega + \overline{\omega} = 3 \]
Thus:
\[ \omega + \overline{\omega} = 0 \]
This implies that \( \omega \) is purely imaginary.

7. **Conclusion**:
Therefore, the value of \( z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1 \) is purely imaginary, and since we are not given specific values, we conclude that it can take the form \( ki \) where \( k \) is a real number.