If ` | z_(1) + z_(2) | = | z_(1) - z_(2)| ` , the difference in the amplitudes of ` z_(1) and z_(2) ` is

Options

A. ` (pi)/(4)`
B. ` (pi)/(2)`
C. `(pi)/(3)`
D. none

Answer

To solve the problem, we start with the given condition:

**Given:**
\[ |z_1 + z_2| = |z_1 - z_2| \]

### Step 1: Square both sides
We square both sides of the equation to eliminate the modulus:
\[ |z_1 + z_2|^2 = |z_1 - z_2|^2 \]

### Step 2: Expand both sides
Using the property of modulus, we can expand both sides:
\[ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2|z_1||z_2|\cos(\alpha - \beta) \]
\[ |z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2|z_1||z_2|\cos(\alpha - \beta) \]

### Step 3: Set the expanded forms equal
Now, we set the two expanded forms equal to each other:
\[ |z_1|^2 + |z_2|^2 + 2|z_1||z_2|\cos(\alpha - \beta) = |z_1|^2 + |z_2|^2 - 2|z_1||z_2|\cos(\alpha - \beta) \]

### Step 4: Simplify the equation
We can cancel \( |z_1|^2 + |z_2|^2 \) from both sides:
\[ 2|z_1||z_2|\cos(\alpha - \beta) = -2|z_1||z_2|\cos(\alpha - \beta) \]

### Step 5: Combine like terms
Now, we can combine like terms:
\[ 2|z_1||z_2|\cos(\alpha - \beta) + 2|z_1||z_2|\cos(\alpha - \beta) = 0 \]
\[ 4|z_1||z_2|\cos(\alpha - \beta) = 0 \]

### Step 6: Analyze the equation
Since \( |z_1| \) and \( |z_2| \) are magnitudes and cannot be zero (assuming \( z_1 \) and \( z_2 \) are non-zero), we conclude:
\[ \cos(\alpha - \beta) = 0 \]

### Step 7: Find the difference in amplitudes
The condition \( \cos(\alpha - \beta) = 0 \) implies:
\[ \alpha - \beta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]

Thus, the difference in the amplitudes of \( z_1 \) and \( z_2 \) is:
\[ |\alpha - \beta| = \frac{\pi}{2} \]

### Final Answer:
The difference in the amplitudes of \( z_1 \) and \( z_2 \) is \( \frac{\pi}{2} \).

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