The vector z=-4+5i is turned counter clockwise through an angle of `180^@` and stretched 1.5 times. The complex number corresponding to the newly obtained vector is

Options

A. `6-15/2i`
B. `-6+15/2`i
C. `6+15/2`i
D. `6+15/2`i

Answer

To solve the problem, we need to perform two operations on the complex number \( z = -4 + 5i \): rotate it counterclockwise by \( 180^\circ \) and stretch it by a factor of \( 1.5 \).

### Step-by-Step Solution:

1. **Identify the Initial Complex Number:**
\[ z = -4 + 5i \]

2. **Convert the Angle to Radians:**
Since we are rotating the vector \( z \) by \( 180^\circ \), we convert this angle to radians:
\[ 180^\circ = \pi \text{ radians} \]

3. **Use Euler's Formula for Rotation:**
According to Euler's formula, rotating a complex number \( z \) by an angle \( \theta \) can be expressed as:
\[ z' = z \cdot e^{i\theta} \]
Here, \( \theta = \pi \):
\[ z' = z \cdot e^{i\pi} \]

4. **Calculate \( e^{i\pi} \):**
From Euler's formula:
\[ e^{i\pi} = \cos(\pi) + i\sin(\pi) = -1 + 0i = -1 \]

5. **Perform the Rotation:**
Substitute \( z \) and \( e^{i\pi} \):
\[ z' = (-4 + 5i) \cdot (-1) \]
Simplifying this gives:
\[ z' = 4 - 5i \]

6. **Stretch the Result by a Factor of 1.5:**
Now, we stretch the vector \( z' \) by \( 1.5 \):
\[ z'' = 1.5 \cdot (4 - 5i) \]
Distributing \( 1.5 \):
\[ z'' = 1.5 \cdot 4 - 1.5 \cdot 5i = 6 - 7.5i \]

7. **Final Result:**
The complex number corresponding to the newly obtained vector is:
\[ z'' = 6 - 7.5i \]

### Summary of the Solution:
The final complex number after rotating \( z = -4 + 5i \) counterclockwise by \( 180^\circ \) and stretching it by \( 1.5 \) is:
\[ \boxed{6 - 7.5i} \]