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

Hint: 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 .