Question

If $|z_1+z_2|=|z_1-z_2|$ then the difference between the arguments of $z_1$ and $z_2$ is:

• A. $0$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (B)

$\frac{\pi }{2}$

If we consider the geometric representation of addition of complex numbers,
$|z_1+z_2|$ and $|z_1-z_2|$ represent the diagonals of the parallelogram formed having vertices origin ,$z_1$, $z_2$ and $z_1+z_2$.
Now, the lengths of the diagonals are equal. Hence, the parallelogram must be a rectangle.
Hence, the difference in the arguments of $z_1$ and $z_2$ is $\frac{\pi }{2}$
Hence, option B is correct.