Suppose `omega` is a cube root of unity with `omega cancel(=)1`. Suppose P and Q are the points on the complex plane defined by ` omega` and `omega^(2)` . If O is the origin, then what is the angle between OP and OQ ?

Options

A. A)`60^(@)`
B. B)`90^(@)`
C. C)`120^(@)`
D. D)`150^(@)`

Answer

To find the angle between the lines OP and OQ where P and Q are the points representing the cube roots of unity \( \omega \) and \( \omega^2 \), we can follow these steps:

### Step 1: Identify the cube roots of unity
The cube roots of unity are the solutions to the equation \( x^3 = 1 \). These roots are:
1. \( 1 \)
2. \( \omega = e^{2\pi i / 3} \)
3. \( \omega^2 = e^{-2\pi i / 3} \)

### Step 2: Represent the points P and Q in the complex plane
In the complex plane:
- Point \( O \) (the origin) is represented as \( 0 \).
- Point \( P \) (representing \( \omega \)) is \( \omega = e^{2\pi i / 3} \).
- Point \( Q \) (representing \( \omega^2 \)) is \( \omega^2 = e^{-2\pi i / 3} \).

### Step 3: Calculate the angle between OP and OQ
The angle \( \theta \) between the two lines OP and OQ can be calculated using the formula for the angle between two complex numbers \( z_1 \) and \( z_2 \):
\[ \theta = \arg(z_2) - \arg(z_1) \]
Here, \( z_1 = \omega \) and \( z_2 = \omega^2 \).

### Step 4: Find the arguments of \( \omega \) and \( \omega^2 \)
- The argument of \( \omega \) is \( \arg(\omega) = \frac{2\pi}{3} \).
- The argument of \( \omega^2 \) is \( \arg(\omega^2) = -\frac{2\pi}{3} \).

### Step 5: Calculate the angle
Now we can calculate the angle:
\[ \theta = \arg(\omega^2) - \arg(\omega) = -\frac{2\pi}{3} - \frac{2\pi}{3} = -\frac{4\pi}{3} \]
To find the positive angle, we can convert this to a positive angle by adding \( 2\pi \):
\[ \theta = -\frac{4\pi}{3} + 2\pi = -\frac{4\pi}{3} + \frac{6\pi}{3} = \frac{2\pi}{3} \]

### Step 6: Convert the angle to degrees
To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):
\[ \theta = \frac{2\pi}{3} \times \frac{180}{\pi} = 120^\circ \]

### Conclusion
The angle between OP and OQ is \( 120^\circ \).