Question

In the Argand plane, the vector $z=4–3i$ is turned in the clockwise sense through $180$ degree and stretched three times. The complex number represented by the new vector is:

• A. $12+9i$
• B. $12-9i$
• C. $-12-9i$
• D. $-12+9i$

Answer 1

The correct option is D

$-12+9i$

Explanation for the correct option:

Finding the complex number:

If $z=a+ib$ then

$\left|z\right|=\sqrt{a^2+b^2}$

Here, $z=4–3i$.

$\Rightarrow a=4;b=-3$

$$\begin{gathered} \therefore \left|z\right|=\sqrt{4^2-{\left(-3\right)}^2} \\ =\sqrt{16+9} \\ =\sqrt{25} \\ =5 \end{gathered}$$

Considering $z_1$ as a new vector obtained by rotating $z$ in the clockwise direction through $180°$

$$\begin{gathered} \therefore z_1=e^{-i\mathrm{\pi }}z \\ =\left(\mathrm{cos\pi }-i\mathrm{sin\pi }\right)z\left[\because e^{-i\theta }=\cos \theta -i\sin \theta \right] \\ =\left(-1\right)z\left[\because \mathrm{cos\pi }=-1\mathrm{and}\mathrm{sin\pi }=0\right] \\ =-4+3i \end{gathered}$$

The unit vector along $z_1$

$$\begin{gathered} \Rightarrow \frac{z_1}{\left|z_1\right|}=\frac{-4+3i}{\sqrt{{\left(-4\right)}^2+3^2}} \\ =\frac{-4+3i}{5} \\ =\frac{-4}{5}+\frac{3}{5}i \end{gathered}$$

Now the required vector if $z=4–3i$ stretched three times

$$\begin{gathered} z_2=3\left|z\right|\left(-\frac{4}{5}+\frac{3}{5}i\right) \\ =15\left(-\frac{4}{5}+\frac{3}{5}i\right)\left[\because \mid z\mid =5\right] \\ =-12+9i \end{gathered}$$
Hence, option (D) is the correct answer.