The points representing cube roots of unity

Options

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

Answer

To find the points representing the cube roots of unity, we can follow these steps:

### Step 1: Define the Equation
We start with the equation for the cube roots of unity, which is given by:
\[ z^3 = 1 \]

### Step 2: Rearranging the Equation
We can rearrange this equation to:
\[ z^3 - 1 = 0 \]

### Step 3: Factor the Polynomial
Next, we can factor the polynomial using the difference of cubes:
\[ z^3 - 1 = (z - 1)(z^2 + z + 1) = 0 \]

### Step 4: Find the Roots
From the factorization, we can see that one root is:
\[ z_1 = 1 \]
Now, we need to find the other two roots from the quadratic equation:
\[ z^2 + z + 1 = 0 \]

### Step 5: Apply the Quadratic Formula
Using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 1, c = 1 \):
\[ z = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ z = \frac{-1 \pm \sqrt{1 - 4}}{2} \]
\[ z = \frac{-1 \pm \sqrt{-3}}{2} \]
\[ z = \frac{-1 \pm i\sqrt{3}}{2} \]

### Step 6: Identify the Roots
Thus, the two other roots are:
\[ z_2 = \frac{-1 + i\sqrt{3}}{2} \]
\[ z_3 = \frac{-1 - i\sqrt{3}}{2} \]

### Step 7: Represent the Roots in Complex Form
The three roots of unity are:
1. \( z_1 = 1 \)
2. \( z_2 = \frac{-1 + i\sqrt{3}}{2} \)
3. \( z_3 = \frac{-1 - i\sqrt{3}}{2} \)

### Step 8: Calculate the Modulus
To find the modulus of each root:
- For \( z_1 \):
\[ |z_1| = |1| = 1 \]
- For \( z_2 \):
\[ |z_2| = \sqrt{\left(\frac{-1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]
- For \( z_3 \):
\[ |z_3| = \sqrt{\left(\frac{-1}{2}\right)^2 + \left(\frac{-\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

### Step 9: Conclusion
All three roots lie on the unit circle (radius = 1) in the complex plane. The points are:
- \( z_1 = (1, 0) \)
- \( z_2 = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \)
- \( z_3 = \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) \)

These points form an equilateral triangle, as all distances between the points are equal.