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

Concept: For a complex number \( z = x + iy \), the argument \( \arg(z) \) is the angle made with the positive real axis. For a purely imaginary number:
• If \( y > 0 \), then \( \arg(z) = \frac{\pi}{2} \)
• If \( y < 0 \), then \( \arg(z) = -\frac{\pi}{2} \)

Step 1: Express the complex number.
Given: \[ z = \frac{3i}{2} = 0 + \frac{3}{2}i \]

Step 2: Identify its position in the complex plane.
Since the real part is \(0\) and the imaginary part is positive, the number lies on the positive imaginary axis.

Step 3: Determine the argument.
For any positive purely imaginary number: \[ \arg(z) = \frac{\pi}{2} \]