Question

If $z = 3 - 4i$ is turned ${90^0}$ in anti clock direction, then new position of $z$ is
A.$3 - 4i$
B.$4 - 3i$
C.$4 + 3i$
D.$3 + 4i$

Answer 1

Hint: If a point on the complex plane is rotated by a some angle $\theta $ then the new position of the point is calculated as follows
$z' = z{e^{i\theta }}$ Where $z'\;{\text{and}}\;z$ are the new position and the original position respectively, and $\theta $ is the angle by which the point is rotated in an anti clockwise direction with respect to the imaginary axis.
${e^{i\theta }}$ can be calculated with the help of the Euler’s formula as follows:
${e^{i\theta }} = \cos \theta + i\sin \theta $

Complete step by step solution:
In order to find the new position of $z = 3 - 4i$ which is being rotated by ${90^0}$ in anti clock direction, we know that when a point in a complex plane is rotated with an angle $\theta $ then the formula for new position of the point is given as
$z' = z{e^{i\theta }},\;{\text{where}}\;\;z'\;{\text{and}}\;z$ are the new position and the original position respectively, whereas ${e^{i\theta }}$ is the Euler’s expression which is calculated as ${e^{i\theta }} = \cos \theta + i\sin \theta $
Now, coming to the question, we can write the new position of $z$ as
$
   \Rightarrow z' = z{e^{i\theta }} \\
   \Rightarrow z' = (3 - 4i){e^{i{{90}^0}}} \\
 $
Solving it further with help of the Euler’s formula,
$
   \Rightarrow z' = (3 - 4i)(\cos {90^0} + i\sin {90^0}) \\
   \Rightarrow z' = (3 - 4i)(0 + i) \\
   \Rightarrow z' = (3 - 4i)i \\
   \Rightarrow z' = (3i - 4{i^2}) \\
 $
Now we know that value of ${i^2} = - 1$
$
   \Rightarrow z' = \left( {3i - ( - 4)} \right) \\
   \Rightarrow z' = 4 + 3i \\
 $

Therefore option C is the correct option.

Note:
When finding the new position take care of the rotation of the given angle, is it clockwise or anticlockwise and also the reference axis or line of from which the angle is being measured, generally when only angle is given then we consider the ideal condition that is we take the rotation in anti clock direction and the reference line to be x-axis or the imaginary axis in complex plane.