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

Options

A. A)They are collinear
B. B)They lie on a circle of radius `sqrt( 3)`
C. C)They form an equilateral triangle
D. D)None of these

Answer

To determine which statement is correct regarding the cube roots of unity, we will first find the cube roots of unity and analyze their properties step by step.

### Step-by-Step Solution:

1. **Understanding the Cube Roots of Unity**:
The cube roots of unity are the solutions to the equation \(x^3 = 1\). This can be rewritten as:
\[ x^3 - 1 = 0 \]
This factors as:
\[ (x - 1)(x^2 + x + 1) = 0 \]
From this, we can see that one root is \(x = 1\).

2. **Finding the Other Roots**:
To find the other roots, we need to solve the quadratic equation \(x^2 + x + 1 = 0\). Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = 1\), and \(c = 1\):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2} \]
Thus, the two complex roots are:
\[ x_2 = \frac{-1 + i\sqrt{3}}{2}, \quad x_3 = \frac{-1 - i\sqrt{3}}{2} \]

3. **Identifying the Roots**:
The three cube roots of unity are:
\[ x_1 = 1, \quad x_2 = \frac{-1 + i\sqrt{3}}{2}, \quad x_3 = \frac{-1 - i\sqrt{3}}{2} \]

4. **Plotting the Roots**:
We can plot these roots on the complex plane:
- \(x_1 = 1\) is at the point \((1, 0)\).
- \(x_2 = \frac{-1 + i\sqrt{3}}{2}\) is at the point \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\).
- \(x_3 = \frac{-1 - i\sqrt{3}}{2}\) is at the point \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).

5. **Analyzing the Geometry**:
The three points form an equilateral triangle in the complex plane. The distance from the origin to each of these points is 1, indicating that they lie on a circle of radius 1 centered at the origin.

6. **Conclusion**:
Among the options provided, the correct statement is that the cube roots of unity form an equilateral triangle.

### Final Answer:
The correct option is: **They form an equilateral triangle.**