Question

If \[\left| {{z_1} + {z_2}} \right| = \left| {{z_1} - {z_2}} \right|\], then the difference in the amplitudes of \[{z_1}\] and \[{z_2}\] is
A) \[\dfrac{\pi }{4}\]
B) \[\dfrac{\pi }{3}\]
C) \[\dfrac{\pi }{2}\]
D) \[0\]

Answer 1

Hint:
Here, we have to find the difference in the amplitudes of the complex number. First equating the given equality, we will use the inequality to find the difference in the amplitudes of a complex number. The amplitude of a complex number is defined as the angle inclined from the real axis in the direction of the complex number represented on the complex plane.

Formula Used:
We will use the following formula:
i. \[{\left| {{z_1} + {z_2}} \right|^2} = \left( {{z_1} + {z_2}} \right)\left( {\overline {{z_1}} + \overline {{z_2}} } \right)\]
ii. \[{\left| {{z_1} - {z_2}} \right|^2} = \left( {{z_1} - {z_2}} \right)\left( {\overline {{z_1}} - \overline {{z_2}} } \right)\]
iii. \[{z_1}\overline {{z_1}} = {\left| {{z_1}} \right|^2}\] ; \[{z_2}\overline {{z_2}} = {\left| {{z_2}} \right|^2}\]
iv. \[{z_1}\overline {{z_2}} + \overline {{z_1}} {z_2} = 2{\mathop{\rm Re}\nolimits} ({z_1}\overline {{z_2}} )\] where \[{z_1},{z_2}\] are the complex numbers and \[\overline {{z_1}} ,\overline {{z_2}} \] are the complex conjugates

Complete step by step solution:
We are given that \[\left| {{z_1} + {z_2}} \right| = \left| {{z_1} - {z_2}} \right|\]
\[ \Rightarrow \left| {{z_1} + {z_2}} \right| = \left| {{z_1} - {z_2}} \right|\]
Squaring on both the sides, we get
\[ \Rightarrow {\left| {{z_1} + {z_2}} \right|^2} = {\left| {{z_1} - {z_2}} \right|^2}\]
\[{\left| {{z_1} + {z_2}} \right|^2} = \left( {{z_1} + {z_2}} \right)\left( {\overline {{z_1}} + \overline {{z_2}} } \right)\] ; \[{\left| {{z_1} - {z_2}} \right|^2} = \left( {{z_1} - {z_2}} \right)\left( {\overline {{z_1}} - \overline {{z_2}} } \right)\]
By using the above inequality, we get
\[ \Rightarrow \left( {{z_1} + {z_2}} \right)\left( {\overline {{z_1}} + \overline {{z_2}} } \right) = \left( {{z_1} - {z_2}} \right)\left( {\overline {{z_1}} - \overline {{z_2}} } \right)\]
Now, by multiplying, we have
\[ \Rightarrow {z_1}\overline {{z_1}} + {z_2}\overline {{z_2}} + {z_1}\overline {{z_2}} + \overline {{z_1}} {z_2} = {z_1}\overline {{z_1}} + {z_2}\overline {{z_2}} - {z_1}\overline {{z_2}} - \overline {{z_1}} {z_2}\]
Now, by using \[{z_1}\overline {{z_1}} = {\left| {{z_1}} \right|^2}\] ; \[{z_2}\overline {{z_2}} = {\left| {{z_2}} \right|^2}\] and \[{z_1}\overline {{z_2}} + \overline {{z_1}} {z_2} = 2{\mathop{\rm Re}\nolimits} ({z_1}\overline {{z_2}} )\]
\[ \Rightarrow {\left| {{z_1}} \right|^2} + {\left| {{z_2}} \right|^2} + 2{\mathop{\rm Re}\nolimits} \left( {{z_1}\overline {{z_2}} } \right) = {\left| {{z_1}} \right|^2} + {\left| {{z_2}} \right|^2} - 2{\mathop{\rm Re}\nolimits} \left( {{z_1}\overline {{z_2}} } \right)\]
Rewriting the equation, we get
\[ \Rightarrow {\left| {{z_1}} \right|^2} + {\left| {{z_2}} \right|^2} + 2{\mathop{\rm Re}\nolimits} \left( {{z_1}\overline {{z_2}} } \right) - {\left| {{z_1}} \right|^2} - {\left| {{z_2}} \right|^2} + 2{\mathop{\rm Re}\nolimits} \left( {{z_1}\overline {{z_2}} } \right) = 0\]
Adding the terms, we get
\[ \Rightarrow 2{\mathop{\rm Re}\nolimits} \left( {{z_1}\overline {{z_2}} } \right) + 2{\mathop{\rm Re}\nolimits} \left( {{z_1}\overline {{z_2}} } \right) = 0\]
\[ \Rightarrow 4{\mathop{\rm Re}\nolimits} \left( {{z_1}\overline {{z_2}} } \right) = 0\]
\[ \Rightarrow {\mathop{\rm Re}\nolimits} \left( {{z_1}\overline {{z_2}} } \right) = 0\]
So, we get
\[ \Rightarrow \arg \left( {{z_1}\overline {{z_2}} } \right) = \dfrac{\pi }{2}\]
\[ \Rightarrow arg({z_1}) + arg(\overline {{z_2}} ) = \dfrac{\pi }{2}\]
\[ \Rightarrow arg({z_1}) - arg({z_2}) = \dfrac{\pi }{2}\]
Therefore, the difference in the amplitudes of \[{z_1}\] and\[{z_2}\] is \[\dfrac{\pi }{2}\].

Hence, option C is the correct answer.

Additional Information:
The complex number is basically the combination of a real number and an imaginary number. The complex number is of the form \[a + ib\]. The real numbers are the numbers which we usually work on to do the mathematical calculations. But the imaginary numbers are not generally used for calculations but only in the case of imaginary numbers. Let us check the definitions for both the numbers. The result of the multiplication of two complex numbers and its conjugate value should result in a complex number and it should be a positive value.

Note:
We should know that the Argument of Z and Amplitude of Z mean the same thing and are used interchangeably when we talk about complex numbers. When we plot the point of a complex number on a graph, and join it to the origin, the angle it makes with the x-axis is the argument or amplitude of complex number Z. The argument may be negative.