If the area of the triangle on the complex plane formed by complex numbers `z, omegaz` is `4 sqrt3` square units, then `|z|` is

Options

A. 4
B. 2
C. 6
D. 3

Explanation

To solve the problem, we need to find the modulus of the complex number \( z \) given that the area of the triangle formed by the points \( z \), \( \omega z \), and the origin (0) is \( 4\sqrt{3} \) square units. Here are the steps to find \( |z| \):

### Step 1: Understand the Geometry
The points involved are:
- \( z \) (a complex number)
- \( \omega z \) (where \( \omega \) is a rotation in the complex plane)
- The origin (0)

### Step 2: Identify the Angle
Given that \( \omega \) represents a rotation of \( \frac{2\pi}{3} \) radians, the angle between the vectors \( z \) and \( \omega z \) is \( \frac{2\pi}{3} \).

### Step 3: Area of the Triangle Formula
The area \( A \) of a triangle formed by two vectors \( a \) and \( b \) with an angle \( \theta \) between them is given by:
\[ A = \frac{1}{2} |a| |b| \sin(\theta) \]
In our case, both vectors \( a \) and \( b \) are equal to \( |z| \):
\[ A = \frac{1}{2} |z| |z| \sin\left(\frac{2\pi}{3}\right) = \frac{1}{2} |z|^2 \sin\left(\frac{2\pi}{3}\right) \]

### Step 4: Calculate \( \sin\left(\frac{2\pi}{3}\right) \)
The sine of \( \frac{2\pi}{3} \) is:
\[ \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Step 5: Substitute Values into the Area Formula
Substituting \( \sin\left(\frac{2\pi}{3}\right) \) into the area formula gives:
\[ A = \frac{1}{2} |z|^2 \cdot \frac{\sqrt{3}}{2} \]
This simplifies to:
\[ A = \frac{\sqrt{3}}{4} |z|^2 \]

### Step 6: Set the Area Equal to the Given Value
We know the area is \( 4\sqrt{3} \):
\[ \frac{\sqrt{3}}{4} |z|^2 = 4\sqrt{3} \]

### Step 7: Solve for \( |z|^2 \)
To isolate \( |z|^2 \), multiply both sides by 4:
\[ \sqrt{3} |z|^2 = 16\sqrt{3} \]
Now, divide both sides by \( \sqrt{3} \):
\[ |z|^2 = 16 \]

### Step 8: Find \( |z| \)
Taking the square root of both sides gives:
\[ |z| = 4 \]

### Final Answer
Thus, the modulus of \( z \) is:
\[ |z| = 4 \]

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