The vector z = - 4 + 5 i is turned counterclockwise through an angle of ` 180^(@)` and stretched ` 1 (1)/(2)` times. The complex number corresponding to newly obtained vector is

Options

A. `6 - (15)/(2)i`
B. `- 6 + (15)/(2) i`
C. `6 + (15)/(2)i`
D. none of these

Answer

To solve the problem, we need to perform two operations on the complex number \( z = -4 + 5i \): rotating it counterclockwise by \( 180^\circ \) and then stretching it by a factor of \( \frac{3}{2} \).

### Step-by-Step Solution:

1. **Identify the complex number**:
\[ z = -4 + 5i \]

2. **Rotate the complex number by \( 180^\circ \)**:
- To rotate a complex number by \( 180^\circ \), we can multiply it by \( -1 \) (which is equivalent to multiplying by \( e^{i\pi} \)).
\[ z' = z \cdot (-1) = (-4 + 5i) \cdot (-1) = 4 - 5i \]

3. **Stretch the vector by a factor of \( \frac{3}{2} \)**:
- To stretch the complex number, we multiply it by \( \frac{3}{2} \).
\[ z'' = z' \cdot \frac{3}{2} = (4 - 5i) \cdot \frac{3}{2} = \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 5i = 6 - \frac{15}{2}i \]

4. **Final Result**:
- The newly obtained complex number after rotation and stretching is:
\[ z'' = 6 - \frac{15}{2}i \]

### Conclusion:
The complex number corresponding to the newly obtained vector is \( 6 - \frac{15}{2}i \).

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