Which one of the following is correct in respect of the cube roots of unity?

Options

A. They are collinear
B. They lie on a circle of radius `sqrt3`
C. They form an equilateral triangle
D. None of the above

Explanation

To determine which one of the following statements is correct regarding the cube roots of unity, we will analyze the cube roots of unity and their geometric representation on the Argand plane.

### Step-by-Step Solution:

1. **Identify the Cube Roots of Unity**:
The cube roots of unity are the solutions to the equation \( z^3 = 1 \). The roots are:
- \( z_0 = 1 \)
- \( z_1 = \omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i \)
- \( z_2 = \omega^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \)

2. **Plot the Points on the Argand Plane**:
- The point \( z_0 = 1 \) corresponds to the coordinates \( (1, 0) \).
- The point \( z_1 = \omega \) corresponds to the coordinates \( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \).
- The point \( z_2 = \omega^2 \) corresponds to the coordinates \( \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) \).

3. **Determine the Distances Between the Points**:
We will calculate the distances between the points to check if they form an equilateral triangle.

- **Distance \( AB \)** (between \( z_0 \) and \( z_1 \)):
\[ AB = \sqrt{\left(-\frac{1}{2} - 1\right)^2 + \left(\frac{\sqrt{3}}{2} - 0\right)^2} = \sqrt{\left(-\frac{3}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} \]

- **Distance \( BC \)** (between \( z_1 \) and \( z_2 \)):
\[ BC = \sqrt{\left(-\frac{1}{2} + \frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}\right)^2} = \sqrt{0 + \left(-\sqrt{3}\right)^2} = \sqrt{3} \]

- **Distance \( AC \)** (between \( z_0 \) and \( z_2 \)):
\[ AC = \sqrt{\left(-\frac{1}{2} - 1\right)^2 + \left(-\frac{\sqrt{3}}{2} - 0\right)^2} = \sqrt{\left(-\frac{3}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} \]

4. **Conclusion**:
Since all three distances \( AB \), \( BC \), and \( AC \) are equal to \( \sqrt{3} \), the points \( z_0 \), \( z_1 \), and \( z_2 \) form an equilateral triangle on the Argand plane.

### Final Answer:
The cube roots of unity form an equilateral triangle.

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