# representation of this argument in the complex plane

If $|z_1| =|z_2|$ and $\arg(z_1/z_2)=π$, then $z_1+z_2$ is

My attempt – We can represent $z_1$ and $z_2$ as two complex numbers at the same distances from origin in the Argand plane but I can't seem to represent the argument in the Argand plane. Can someone help me for this?

## 2 Answers

Hint:

The argument of a non-zero complex number corresponds to the polar angle of its affix in polar coordinates; and the argument of a quotient of complex numbers is the difference of their arguments.

• arg(z1/z2)=arg z1-arg z2 ...this i know.But i am having comfusion in representing it in argand plane.Will the complex numbers lie diagonally opposite to each other or something like that. – Hydrous Caperilla Dec 7 '17 at 23:04
• What does ,a rotation of π correspond to, geometrically, in your opinion? – Bernard Dec 7 '17 at 23:06
• for me,it's like the number is rotated π clockwise – Hydrous Caperilla Dec 7 '17 at 23:10
• That's right, but there's a still simpler interpretation. – Bernard Dec 7 '17 at 23:12
• can u tell me about it – Hydrous Caperilla Dec 7 '17 at 23:15

There are multiple ways to solve the problem. One way is to see that $$z_1+z_2 = z_2 \left(1+\frac{z_1}{z_2}\right).$$ Now, $\left|\frac{z_1}{z_2}\right|=1$ and $\arg\left(\frac{z_1}{z_2}\right)=\pi$ implies $\frac{z_1}{z_2}=\cos(\pi)+i \sin(\pi) = ? \implies z_1 + z_2 = ?$

• i got your method but can u help me representing the arguement (z1/z2) in complex plane.I want to solve this question using geometry – Hydrous Caperilla Dec 7 '17 at 23:21