Question

If z1 & z2 are two complex numbers and |z1 +z2| = |z1| + |z2| . Then proved that arg z1 = arg z

Please solve it......

Answer 1

|z1 + z2| = |z1| + |z2|
On squaring both sides of this equation we get
|z1|2 + |z2|2 + 2|z1| |z2| cos (arg z1 – arg z2) = |z1|2 + |z2|2 + 2|z1| |z2|
This gives 2|z1| |z2| cos (arg z1 – arg z2) = 2|z1| |z2|
Hence, cos (arg z1 – arg z2) = 1
Hence, arg (z1) – arg (z2) = 0.
Therefore, arg(z1) = arg(z2)