Question

If magnitude of a complex number $z=4-3i$ is tripled and is rotated anti clockwise by an angle of $\pi$ , then the resulting complex number would be
1. $-12+9i$
2. $12+9i$
3. $7-6i$
4. $7+6i$

Answer 1

To solve this question you must first start by taking the magnitude of the given complex number. Let the complex number be z and its magnitude will be $\left| z \right|$ and the formula to find its magnitude will be $\sqrt{{{a}^{2}}+{{b}^{2}}}$ . Now we need to find the complex number which we will obtain after rotating the complex number z. Let the resulting complex number be ${{z}_{1}}$ and we can find it by using the formula ${{z}_{1}}={{e}^{-i\pi }}$

Complete step by step answer:
Now to start the solution let us assume that
$z=4-3i$
Now to find the magnitude of z we can use it by the formula that
$\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$
Here we know that a is the real part of the complex number and b is the imaginary part of it. Therefore substituting their values inside the formula we get
$\left| z \right|=\sqrt{{{4}^{2}}+{{3}^{2}}}$
Simplifying we get
$\left| z \right|=5$
Now we also know that the complex number is tripled in size therefore the magnitude of required complex number will be
$\left| {{z}_{1}} \right|=5\times 3$
$\left| {{z}_{1}} \right|=15$
Now that we know the magnitude we know that the complex number is then rotated anticlockwise by $\pi$ which means that the new complex number vector will be
${{z}_{1}}={{e}^{-i\pi }}z=(\cos i\pi -\sin i\pi )z$
Therefore simplifying we get
${{z}_{1}}=-4+3i$
Now we know that the new vector is in the direction of the unit vector which is $-\dfrac{4}{5}+\dfrac{3i}{5}$
Now henceforth our required vector we can say will be since its magnitude is $15$
${{z}_{1}}=15\left( -\dfrac{4}{5}+\dfrac{3i}{5} \right)$
${z}_{1}=-12+9i$

So, the correct answer is “Option 1”.

Note: Argand plane is a plane where we can represent any complex number in the easiest way. Here in this plane the x axis which is the horizontal axis represents all real numbers whereas the y axis which is the vertical axis represents all the imaginary numbers .