Question

If \( z_1 = (2, -1) \) and \( z_2 = (6, 3) \), then find the amplitude of the expression \[ \text{amp}\left(\frac{z_1 - z_2}{z_1 + z_2}\right) \]

• A. \( \frac{3\pi}{4} - \tan^{-1}\left(\frac{1}{4}\right) \)
• B. \( \frac{\pi}{4} - \tan^{-1}\left(\frac{1}{4}\right) \)
• C. \( \frac{3\pi}{4} + \tan^{-1}\left(\frac{1}{4}\right) \)
• D. \( \frac{\pi}{4} + \tan^{-1}\left(\frac{1}{4}\right) \)

Hint:

To compute the amplitude of a quotient of complex numbers, subtract the arguments of the numerator and denominator.

Answer 1

The Correct Option is A

We are given \( z_1 = (2, -1) \) and \( z_2 = (6, 3) \), and we are asked to find the amplitude of the expression \( \frac{z_1 - z_2}{z_1 + z_2} \).
Step 1: Compute \( z_1 - z_2 \) and \( z_1 + z_2 \). \[ z_1 - z_2 = (2, -1) - (6, 3) = (2 - 6, -1 - 3) = (-4, -4) \] \[ z_1 + z_2 = (2, -1) + (6, 3) = (2 + 6, -1 + 3) = (8, 2) \]
Step 2: Find the amplitude of \( \frac{z_1 - z_2}{z_1 + z_2} \). The amplitude of a complex number \( \frac{z_1 - z_2}{z_1 + z_2} = \frac{(-4, -4)}{(8, 2)} \) is given by the argument of the resulting complex number. The argument of \( \frac{z_1 - z_2}{z_1 + z_2} \) is: \[ \arg\left(\frac{z_1 - z_2}{z_1 + z_2}\right) = \arg(-4 - 4i) - \arg(8 + 2i) \] The argument of \( -4 - 4i \) is \( \frac{5\pi}{4} \), and the argument of \( 8 + 2i \) is \( \tan^{-1}\left(\frac{1}{4}\right) \). Thus, the total amplitude is: \[ \arg\left(\frac{z_1 - z_2}{z_1 + z_2}\right) = \frac{5\pi}{4} - \tan^{-1}\left(\frac{1}{4}\right) \]