If `z_(1),z_(2)` and `z_(3)` be unimodular complex numbers, then the maximum value of `|z_(1)-z_(2)|^(2)+|z_(2)-z_(3)|^(2)+|z_(3)-z_(1)|^(2)`, is

Options

A. 6
B. 9
C. 12
D. 3

Explanation

To find the maximum value of the expression \( |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 \) where \( z_1, z_2, z_3 \) are unimodular complex numbers (i.e., \( |z_1| = |z_2| = |z_3| = 1 \)), we can follow these steps:

### Step 1: Understanding Unimodular Complex Numbers
Unimodular complex numbers lie on the unit circle in the complex plane. Therefore, we can express them as:
- \( z_1 = e^{i\theta_1} \)
- \( z_2 = e^{i\theta_2} \)
- \( z_3 = e^{i\theta_3} \)

where \( \theta_1, \theta_2, \theta_3 \) are angles in radians.

### Step 2: Expressing the Distances
We need to compute the distances:
- \( |z_1 - z_2|^2 = |e^{i\theta_1} - e^{i\theta_2}|^2 \)
- \( |z_2 - z_3|^2 = |e^{i\theta_2} - e^{i\theta_3}|^2 \)
- \( |z_3 - z_1|^2 = |e^{i\theta_3} - e^{i\theta_1}|^2 \)

Using the formula \( |a - b|^2 = (a - b)(\overline{a - b}) \), we can rewrite these distances:
\[ |z_1 - z_2|^2 = |e^{i\theta_1} - e^{i\theta_2}|^2 = 2 - 2\cos(\theta_1 - \theta_2) \]
\[ |z_2 - z_3|^2 = |e^{i\theta_2} - e^{i\theta_3}|^2 = 2 - 2\cos(\theta_2 - \theta_3) \]
\[ |z_3 - z_1|^2 = |e^{i\theta_3} - e^{i\theta_1}|^2 = 2 - 2\cos(\theta_3 - \theta_1) \]

### Step 3: Summing the Distances
Now we can sum these expressions:
\[ |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 = (2 - 2\cos(\theta_1 - \theta_2)) + (2 - 2\cos(\theta_2 - \theta_3)) + (2 - 2\cos(\theta_3 - \theta_1)) \]
This simplifies to:
\[ = 6 - 2(\cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1)) \]

### Step 4: Maximizing the Expression
To maximize the expression \( 6 - 2(\cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1)) \), we need to minimize \( \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) \).

The minimum value of \( \cos(x) \) is -1. Therefore, the minimum value of the sum can be achieved when the angles are spaced evenly around the unit circle, leading to:
\[ \cos(\theta_1 - \theta_2) = \cos(\theta_2 - \theta_3) = \cos(\theta_3 - \theta_1) = -\frac{1}{2} \]
This gives:
\[ \cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_3 - \theta_1) = -\frac{3}{2} \]

### Step 5: Final Calculation
Substituting back, we find:
\[ 6 - 2\left(-\frac{3}{2}\right) = 6 + 3 = 9 \]

### Conclusion
Thus, the maximum value of \( |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 \) is \( \boxed{9} \).