If ` z ` satisfies the condition arg`(z + i) = (pi)/(4)` . Then the minimum value of ` |z + 1 - i| + |z - 2 + 3 i|` is _______.

Explanation

To solve the problem, we need to find the minimum value of \( |z + 1 - i| + |z - 2 + 3i| \) given that \( \arg(z + i) = \frac{\pi}{4} \).

### Step-by-Step Solution:

1. **Express \( z \) in terms of \( x \) and \( y \)**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

2. **Find the expression for \( z + i \)**:
\[ z + i = x + iy + i = x + i(y + 1) \]

3. **Set up the argument condition**:
Given \( \arg(z + i) = \frac{\pi}{4} \), we know:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Therefore, we have:
\[ \frac{y + 1}{x} = 1 \]

4. **Solve for \( y \)**:
From the equation \( \frac{y + 1}{x} = 1 \), we can rearrange it to find \( y \):
\[ y + 1 = x \implies y = x - 1 \]

5. **Substitute \( y \) into the expression**:
Now substitute \( y = x - 1 \) into the expression \( |z + 1 - i| + |z - 2 + 3i| \):
\[ z + 1 - i = x + (x - 1)i + 1 - i = (x + 1) + (x - 2)i \]
\[ z - 2 + 3i = x + (x - 1)i - 2 + 3i = (x - 2) + (x + 2)i \]

6. **Calculate the magnitudes**:
\[ |z + 1 - i| = \sqrt{(x + 1)^2 + (x - 2)^2} \]
\[ |z - 2 + 3i| = \sqrt{(x - 2)^2 + (x + 2)^2} \]

7. **Simplify the expressions**:
- For \( |z + 1 - i| \):
\[ |z + 1 - i| = \sqrt{(x + 1)^2 + (x - 2)^2} = \sqrt{(x^2 + 2x + 1) + (x^2 - 4x + 4)} = \sqrt{2x^2 - 2x + 5} \]
- For \( |z - 2 + 3i| \):
\[ |z - 2 + 3i| = \sqrt{(x - 2)^2 + (x + 2)^2} = \sqrt{(x^2 - 4x + 4) + (x^2 + 4x + 4)} = \sqrt{2x^2 + 8} \]

8. **Combine the magnitudes**:
Now we need to minimize:
\[ |z + 1 - i| + |z - 2 + 3i| = \sqrt{2x^2 - 2x + 5} + \sqrt{2x^2 + 8} \]

9. **Find the minimum value**:
To find the minimum value of the sum of these two distances, we can analyze the geometric interpretation. The points \( (-1, 2) \) and \( (2, -3) \) are the points we are measuring distances to. The minimum distance occurs when the line connecting these two points intersects the line \( y = x - 1 \).

10. **Calculate the distance between the two points**:
The distance between points \( (-4, 3) \) and \( (4, -3) \) is:
\[ d = \sqrt{(4 - (-4))^2 + (-3 - 3)^2} = \sqrt{(8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Final Answer:
The minimum value of \( |z + 1 - i| + |z - 2 + 3i| \) is \( 10 \).