Question

can u teach me, how i can solve this problem...
Triangle inequality for complex numbers is |Z1 + Z2|<= |Z1| + |Z2| . Write down at least eight different conditions for which |Z1 + Z2|= |Z1| + |Z2| .

Answer 1

Let the angle between the two complex position be @
So the length of the Z1+Z2 complex point or in other words |Z1+Z2| is =
\[\sqrt(Z _{1}^2\sin^2@+Z _{2}^2+Z _{1}^2\cos^2@+Z _{1}Z _{2}\cos@)\]

Continued ........................

Answer 2

Let name that expression as A

Answer 3

Now when |Z1+Z2|=|Z1|+|Z2|
Z1+Z2=A

Answer 4

Squaring both sides and simplifying we get @=0

Answer 5

So you can get the equation only when the amplitude of the complex number is the same. I hope you have understood

Please note that here the symbol Z1 and Z2 is the magnitude of the actual Z1 and Z2

Answer 6

i got what u had explained...thank you so much...this open stdy really halp me to think out of the box..

Answer 7

As you asked on through chat how to calculate amplitude of a complex number, so here is the reply

Lets take a complex number x+iy
So its amplitude is y/x
Lets take another example z=3+7i
So its amplitude would be 7/3

According the reply I gave you, the equation |z1|+|z2|=|z1+z2| is valid only when the amplitude of two complex number is the same. So for an example

The complex number z1= 2+i4
and the complex number z2=6+i12
holds the relationship |z1|+|z2|=|z1+z2|

Hope, I have made myself clear

Answer 8

So silly of me!
All this time, I have been mentioning a wrong term.
I am extremely sorry for that.

Please note that in the above answer everywhere I have used the word ""amplitude"", it must be instead ""argument""