If `|z_(1)|=|z_(2)|=|z_(3)|=1` and `z_(1)+z_(2)+z_(3)=sqrt(2)+i`, then the complex number `z_(2)barz_(3)+z_(3)barz_(1)+z_(1)barz_(2)`, is

Options

A. purely real
B. purely imaginary
C. a positive real number
D. none of these

Explanation

To solve the problem, we start with the given conditions:

1. \( |z_1| = |z_2| = |z_3| = 1 \)
2. \( z_1 + z_2 + z_3 = \sqrt{2} + i \)

We need to find the value of \( z_2 \overline{z_3} + z_3 \overline{z_1} + z_1 \overline{z_2} \).

### Step 1: Modulus of the Sum
First, we take the modulus of both sides of the equation \( z_1 + z_2 + z_3 = \sqrt{2} + i \).

\[ |z_1 + z_2 + z_3| = |\sqrt{2} + i| \]

### Step 2: Calculate the Modulus of the Right Side
Now, calculate the modulus of the right side:

\[ |\sqrt{2} + i| = \sqrt{(\sqrt{2})^2 + 1^2} = \sqrt{2 + 1} = \sqrt{3} \]

### Step 3: Square Both Sides
Next, we square both sides:

\[ |z_1 + z_2 + z_3|^2 = |\sqrt{3}|^2 \]

This gives us:

\[ |z_1 + z_2 + z_3|^2 = 3 \]

### Step 4: Expand the Left Side
Now, we expand the left side using the properties of modulus:

\[ |z_1 + z_2 + z_3|^2 = |z_1|^2 + |z_2|^2 + |z_3|^2 + 2 \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \]

Since \( |z_1| = |z_2| = |z_3| = 1 \), we have:

\[ |z_1|^2 = |z_2|^2 = |z_3|^2 = 1 \]

Thus,

\[ 1 + 1 + 1 + 2 \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) = 3 \]

### Step 5: Simplify the Equation
This simplifies to:

\[ 3 + 2 \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) = 3 \]

### Step 6: Solve for the Real Part
Subtracting 3 from both sides gives:

\[ 2 \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) = 0 \]

Dividing by 2:

\[ \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) = 0 \]

### Step 7: Find the Desired Expression
We need to find \( z_2 \overline{z_3} + z_3 \overline{z_1} + z_1 \overline{z_2} \). Since the real part of this expression is 0, it implies that this expression is purely imaginary.

### Conclusion
Thus, the complex number \( z_2 \overline{z_3} + z_3 \overline{z_1} + z_1 \overline{z_2} \) is purely imaginary.