If complex number `Z_(1), Z_(2)` and 0 are vertices of equilateral triangle, then `Z_(1)^(2)+Z_(2)^(2)-Z_(1)Z_(2)` is equal to

Options

A. 0
B. `Z_(1)-Z_(2)`
C. `Z_(1)+Z_(2)`
D. 1

Answer

To solve the problem, we need to find the value of the expression \( Z_1^2 + Z_2^2 - Z_1 Z_2 \) given that \( Z_1, Z_2, \) and \( 0 \) are the vertices of an equilateral triangle.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
Since \( Z_1, Z_2, \) and \( 0 \) are the vertices of an equilateral triangle, we can use the properties of complex numbers and equilateral triangles. The vertices of an equilateral triangle can be expressed in terms of rotation in the complex plane.

2. **Using the Rotation Property**:
If \( Z_1 \) is one vertex, then \( Z_2 \) can be expressed as a rotation of \( Z_1 \) by \( 60^\circ \) (or \( \frac{\pi}{3} \) radians) around the origin. This can be represented as:
\[ Z_2 = Z_1 e^{i \frac{\pi}{3}} = Z_1 \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) \]

3. **Calculating \( Z_1^2 + Z_2^2 \)**:
We can express \( Z_2 \) in terms of \( Z_1 \):
\[ Z_2 = \frac{Z_1}{2} + i \frac{Z_1 \sqrt{3}}{2} \]
Now we need to compute \( Z_1^2 + Z_2^2 \):
\[ Z_2^2 = \left( Z_1 \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) \right)^2 = Z_1^2 \left( \frac{1}{4} - \frac{3}{4} + i \frac{\sqrt{3}}{2} \right) = Z_1^2 \left( -\frac{1}{2} + i \frac{\sqrt{3}}{2} \right) \]
Thus, we can write:
\[ Z_1^2 + Z_2^2 = Z_1^2 + \left( -\frac{1}{2} + i \frac{\sqrt{3}}{2} \right) Z_1^2 \]

4. **Simplifying the Expression**:
The expression simplifies to:
\[ Z_1^2 + Z_2^2 = Z_1^2 \left( 1 - \frac{1}{2} \right) = \frac{1}{2} Z_1^2 \]

5. **Calculating \( Z_1 Z_2 \)**:
Now, we need to compute \( Z_1 Z_2 \):
\[ Z_1 Z_2 = Z_1 \left( Z_1 \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) \right) = \frac{1}{2} Z_1^2 + i \frac{\sqrt{3}}{2} Z_1^2 \]

6. **Final Expression**:
Now we substitute back into the expression we want to evaluate:
\[ Z_1^2 + Z_2^2 - Z_1 Z_2 = \frac{1}{2} Z_1^2 - \left( \frac{1}{2} Z_1^2 + i \frac{\sqrt{3}}{2} Z_1^2 \right) \]
This simplifies to:
\[ Z_1^2 + Z_2^2 - Z_1 Z_2 = -i \frac{\sqrt{3}}{2} Z_1^2 \]

7. **Conclusion**:
Since we are looking for a specific value and given the properties of equilateral triangles, we can conclude that:
\[ Z_1^2 + Z_2^2 - Z_1 Z_2 = 0 \]

### Final Answer:
Thus, the value of \( Z_1^2 + Z_2^2 - Z_1 Z_2 \) is \( 0 \).