Question

If \( z_1 = -1 + i \) and \( z_2 = 2i \), where \( i = \sqrt{-1} \), then \( \text{Arg}(z_1 / z_2) \) is:

• A. \( \frac{3\pi}{4} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{\pi}{3} \)

Hint:

To find the argument of a complex number division, subtract the arguments of the numerator and denominator. Always express the result in the correct range of principal arguments.

Answer 1

The Correct Option is B

Step 1: Represent \( z_1 \) and \( z_2 \) in polar form. For \( z_1 = -1 + i \): \[ |z_1| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}, \quad \text{and} \quad \text{Arg}(z_1) = \tan^{-1}\left(\frac{1}{-1}\right) + \pi = \frac{3\pi}{4}. \] For \( z_2 = 2i \): \[ |z_2| = |2i| = 2, \quad \text{and} \quad \text{Arg}(z_2) = \frac{\pi}{2}. \] Step 2: Division of \( z_1 \) and \( z_2 \). The modulus of \( z_1 / z_2 \) is: \[ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} = \frac{\sqrt{2}}{2}. \] The argument of \( z_1 / z_2 \) is: \[ \text{Arg}\left(\frac{z_1}{z_2}\right) = \text{Arg}(z_1) - \text{Arg}(z_2) = \frac{3\pi}{4} - \frac{\pi}{2}. \] Simplify: \[ \text{Arg}\left(\frac{z_1}{z_2}\right) = \frac{3\pi}{4} - \frac{2\pi}{4} = \frac{\pi}{4}. \] Step 3: Conclusion. The argument of \( z_1 / z_2 \) is \( \frac{\pi}{4} \).