If for complex numbers `z_(1)` and `z_(2)`, arg `z_(1)-"arg"(z_(2))=0` then `|z_(1)-z_(2)|` is equal to

Options

A. `|z_(1)|+|z_(2)|`
B. `|z_(1)|-|z_(2)|`
C. `||z_(1)|-|z_(2)|`
D. 0

Answer

To solve the problem, we need to analyze the given condition:

**Given:**
For complex numbers \( z_1 \) and \( z_2 \), \( \arg(z_1) - \arg(z_2) = 0 \).

This implies that \( \arg(z_1) = \arg(z_2) \). Therefore, both complex numbers \( z_1 \) and \( z_2 \) have the same argument, meaning they lie on the same line from the origin in the complex plane.

### Step-by-Step Solution:

1. **Understanding the Condition:**
Since \( \arg(z_1) = \arg(z_2) \), we can express \( z_1 \) and \( z_2 \) in terms of their magnitudes and the common argument \( \theta \):
\[ z_1 = r_1 e^{i\theta}, \quad z_2 = r_2 e^{i\theta} \]
where \( r_1 = |z_1| \) and \( r_2 = |z_2| \).

2. **Finding the Difference:**
Now, we need to find \( |z_1 - z_2| \):
\[ z_1 - z_2 = r_1 e^{i\theta} - r_2 e^{i\theta} = (r_1 - r_2)e^{i\theta} \]

3. **Calculating the Modulus:**
The modulus of the difference is given by:
\[ |z_1 - z_2| = |(r_1 - r_2)e^{i\theta}| = |r_1 - r_2| \cdot |e^{i\theta}| = |r_1 - r_2| \]
Since \( |e^{i\theta}| = 1 \).

4. **Conclusion:**
Therefore, we conclude that:
\[ |z_1 - z_2| = | |z_1| - |z_2| | \]

### Final Result:
Thus, the value of \( |z_1 - z_2| \) is equal to \( ||z_1| - |z_2|| \).

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