Question

If \( z = \frac{3i}{2} \), what is the value of \( \arg(z) \)?

• A. \(0 \)
• B. \( \frac{\pi}{2} \)
• C. \( \pi \)
• D. \( \frac{3\pi}{2} \)

Hint:

If a complex number lies on: - Positive real axis \( \Rightarrow \arg = 0 \) - Positive imaginary axis \( \Rightarrow \arg = \frac{\pi}{2} \) - Negative real axis \( \Rightarrow \arg = \pi \) - Negative imaginary axis \( \Rightarrow \arg = -\frac{\pi}{2} \)

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the principal argument of a given complex number \( z \).
The argument represents the angle that the vector representing the complex number makes with the positive real axis in the Argand plane.
Step 2: Key Formula or Approach:
For a purely imaginary complex number \( z = iy \):
1. If \( y>0 \), the number lies on the positive imaginary axis, and \( \arg(z) = \frac{\pi}{2} \).
2. If \( y<0 \), the number lies on the negative imaginary axis, and \( \arg(z) = -\frac{\pi}{2} \) or \( \frac{3\pi}{2} \).
Step 3: Detailed Explanation:
Given complex number is \( z = \frac{3}{2}i \).
We can write this in the form \( x + iy \) as \( z = 0 + \frac{3}{2}i \).
Here, the real part \( x = 0 \) and the imaginary part \( y = \frac{3}{2} \).
Since \( y = \frac{3}{2}>0 \), the point \( (0, \frac{3}{2}) \) is located on the positive Y-axis (Imaginary axis).
The angle formed with the positive X-axis (Real axis) is exactly \( 90^\circ \) or \( \frac{\pi}{2} \) radians.
Step 4: Final Answer:
The value of \( \arg(z) \) is \( \frac{\pi}{2} \).