Question

Let \(Z_1 = -2 + 2i\) and \(Z_2 = 3i\), then:

• A. \(\text{Arg}(Z_1 Z_2) = \frac{-3\pi}{4}\)
• B. \(\text{Arg}(Z_1 Z_2) = \frac{3\pi}{4}\)
• C. \(\text{Arg}(Z_1 Z_2) = \frac{-\pi}{4}\)
• D. \(\text{Arg}(Z_1 Z_2) = \frac{5\pi}{4}\)

Hint:

Always identify which quadrant your final complex number lies in.
Since both parts of \(-6 - 6i\) are negative, the number is in the 3rd quadrant.
This means its principal argument must be a negative angle between \(-\frac{\pi}{2}\) and \(-\pi\).
Only \(-\frac{3\pi}{4}\) satisfies this condition.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The argument of a complex number is the angle it makes with the positive real axis in the complex plane.
The principal argument, denoted as \(\text{Arg}(Z)\), is restricted to the interval \((-\pi, \pi]\).

Step 2: Key Formula or Approach:
The argument of a product of two complex numbers is the sum of their arguments, adjusted by \(2\pi\) if necessary to keep it in the principal range:
\[ \text{arg}(Z_1 Z_2) = \text{arg}(Z_1) + \text{arg}(Z_2) \]

Step 3: Detailed Explanation:
Let us calculate the argument using two different methods to verify the result.
Method 1: Find the product first
Multiply the complex numbers:
\[ Z_1 Z_2 = (-2 + 2i)(3i) = -6i + 6i^2 \]
Since \(i^2 = -1\):
\[ Z_1 Z_2 = -6 - 6i \]
This complex number lies in the third quadrant of the complex plane because both the real and imaginary parts are negative.
The reference angle \(\theta\) in the first quadrant is:
\[ \tan \theta = \left| \frac{\text{Im}(Z)}{\text{Re}(Z)} \right| = \left| \frac{-6}{-6} \right| = 1 \implies \theta = \frac{\pi}{4} \]
For a complex number in the third quadrant, the principal argument is:
\[ \text{Arg}(Z_1 Z_2) = \theta - \pi = \frac{\pi}{4} - \pi = -\frac{3\pi}{4} \]
Method 2: Use the sum of arguments
- For \(Z_1 = -2 + 2i\) (lies in the second quadrant):
\[ \text{Arg}(Z_1) = \pi - \tan^{-1}\left(\left|\frac{2}{-2}\right|\right) = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]
- For \(Z_2 = 3i\) (lies on the positive imaginary axis):
\[ \text{Arg}(Z_2) = \frac{\pi}{2} \]
Add the arguments:
\[ \text{Arg}(Z_1) + \text{Arg}(Z_2) = \frac{3\pi}{4} + \frac{\pi}{2} = \frac{5\pi}{4} \]
Since \(\frac{5\pi}{4} > \pi\), we subtract \(2\pi\) to get the principal argument:
\[ \text{Arg}(Z_1 Z_2) = \frac{5\pi}{4} - 2\pi = -\frac{3\pi}{4} \]
Both methods yield the same result, which matches Option (A).

Step 4: Final Answer:
Therefore, the correct option is (A).