If `z in C`, then minimum value of `|z - 2 + 3i| + |z - 1 + i|` is

Options

A. `sqrt(5)`
B. `2 sqrt(5)`
C. `sqrt(13) - sqrt(2)`
D. 0

Answer

To find the minimum value of the expression \( |z - (2 - 3i)| + |z - (1 - i)| \), we can interpret the terms geometrically. The expression represents the sum of distances from the complex number \( z \) to the points \( 2 - 3i \) and \( 1 - i \) in the complex plane.

### Step-by-step Solution:

1. **Identify the Points**:
Let \( A = 2 - 3i \) and \( B = 1 - i \). We need to minimize the expression \( |z - A| + |z - B| \).

2. **Use the Triangle Inequality**:
According to the triangle inequality, for any point \( z \) in the complex plane, the following holds:
\[ |z - A| + |z - B| \geq |A - B| \]
This means that the minimum value of the sum of distances occurs when \( z \) lies on the line segment joining points \( A \) and \( B \).

3. **Calculate the Distance \( |A - B| \)**:
We find \( A - B \):
\[ A - B = (2 - 3i) - (1 - i) = 2 - 3i - 1 + i = 1 - 2i \]
Now, calculate the modulus:
\[ |A - B| = |1 - 2i| = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

4. **Conclusion**:
The minimum value of \( |z - (2 - 3i)| + |z - (1 - i)| \) is \( |A - B| = \sqrt{5} \).

### Final Answer:
The minimum value is \( \sqrt{5} \).

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