If `z=omega,omega^2w h e r eomega` is a non-real complex cube root of unity, are two vertices of an equilateral triangle in the Argand plane, then the third vertex may be represented by `z=1` b. `z=0` c. `z=-2` d. `z=-1`

Options

A. `z=1`
B. `z=0`
C. `z= -2`
D. `z = - 1`

Explanation

To find the third vertex of an equilateral triangle in the Argand plane, given two vertices \( z = \omega \) and \( z = \omega^2 \) where \( \omega \) is a non-real complex cube root of unity, we can follow these steps:

### Step 1: Understand the properties of cube roots of unity
The complex cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3} = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \]
\[ \omega^2 = e^{-2\pi i / 3} = -\frac{1}{2} - i \frac{\sqrt{3}}{2} \]
These roots satisfy the equation \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

### Step 2: Set up the vertices of the triangle
Let the vertices of the equilateral triangle be:
- Vertex A: \( A = \omega \)
- Vertex B: \( B = \omega^2 \)
- Vertex C: \( C = x \) (the third vertex we need to find)

### Step 3: Use the property of equal distances
In an equilateral triangle, the distances between all pairs of vertices are equal:
\[ |A - B| = |B - C| = |C - A| \]

### Step 4: Calculate the distance \( |A - B| \)
Calculate the distance between \( A \) and \( B \):
\[ |A - B| = |\omega - \omega^2| = |(-\frac{1}{2} + i \frac{\sqrt{3}}{2}) - (-\frac{1}{2} - i \frac{\sqrt{3}}{2})| \]
This simplifies to:
\[ |A - B| = |i \sqrt{3}| = \sqrt{3} \]

### Step 5: Set up the equations for the distances
Now, we need to set up the equations for \( |B - C| \) and \( |C - A| \):
1. \( |B - C| = |x - \omega^2| \)
2. \( |C - A| = |x - \omega| \)

### Step 6: Set the equations equal to \( |A - B| \)
We have:
\[ |x - \omega^2| = \sqrt{3} \]
\[ |x - \omega| = \sqrt{3} \]

### Step 7: Solve for \( x \)
From \( |x - \omega^2| = \sqrt{3} \) and \( |x - \omega| = \sqrt{3} \), we can express these in terms of the coordinates of \( \omega \) and \( \omega^2 \):
- \( \omega = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \)
- \( \omega^2 = -\frac{1}{2} - i \frac{\sqrt{3}}{2} \)

Using the distance formula:
1. For \( |x - \omega^2| = \sqrt{3} \):
\[ |x + \frac{1}{2} + i \frac{\sqrt{3}}{2}| = \sqrt{3} \]
2. For \( |x - \omega| = \sqrt{3} \):
\[ |x + \frac{1}{2} - i \frac{\sqrt{3}}{2}| = \sqrt{3} \]

### Step 8: Solve the equations
This leads to two equations:
1. \( (x + \frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2 = 3 \)
2. \( (x + \frac{1}{2})^2 + (-\frac{\sqrt{3}}{2})^2 = 3 \)

Both equations simplify to:
\[ (x + \frac{1}{2})^2 + \frac{3}{4} = 3 \]
\[ (x + \frac{1}{2})^2 = \frac{9}{4} \]
Taking the square root gives:
\[ x + \frac{1}{2} = \pm \frac{3}{2} \]
Thus, we have:
1. \( x + \frac{1}{2} = \frac{3}{2} \) → \( x = 1 \)
2. \( x + \frac{1}{2} = -\frac{3}{2} \) → \( x = -2 \)

### Conclusion
The third vertex \( C \) can be represented by \( z = 1 \) or \( z = -2 \).

### Final Answer
The third vertex may be represented by:
- \( z = 1 \) (Option a)
- \( z = -2 \) (Option c)