Question

If magnitude of a complex number $4-3i$ is tripled and is rotated anticlockwise by an angle $\mathrm{\pi }$ , then resulting complex number would be:

• A. $-12+9i$
• B. $12+9i$
• C. $7-9i$
• D. $7+6i$

Answer 1

The correct option is A

$-12+9i$

Explanation for the correct answer:

We know that the magnitude of a complex number $z=xi+y$ is given by $\sqrt{x^2+y^2}$

Given $4-3i$ will be equal to $\sqrt{4^2+{(-3)}^2}=5$

Now when a vector is rotated at an angle of $\mathrm{\pi }$ , it will be given by $e^{-i\mathrm{\pi }}$

So, rotating $4-3i$ at $\mathrm{\pi }$ and tripling the magnitude, we get,

$$\begin{gathered} =e^{i\mathrm{\pi }}\times 3\times 5 \\ =e^{i\mathrm{\pi }}\times 3\times \left(4-3i\right)\mathbf{[}\mathbf{\because }\mathbf{5}\mathbf{=}\mathbf{4}\mathbf{-}\mathbf{3}\mathbf{i}\mathbf{]} \\ =\left(\mathrm{cos\pi }+i\mathrm{sin\pi }\right)\times 3\times \left(4-3i\right)\mathbf{}\mathbf{[}\mathbf{\because }{\mathbf{e}}^{\mathbf{i}\mathbf{\pi }}\mathbf{=}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\pi }\mathbf{+}\mathbf{i}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\pi }\mathbf{]} \\ =(-1+0)\times (12-9i) \\ =-12+9i \end{gathered}$$

Therefore, option (A) is the correct answer.