Complex Number Question (1 Viewer)

[Sp3ctre]

I'm not sure how to go about this question and I can't understand the worked out solutions either. All help is appreciated, thanks in advance.

Show that ||z₁|-|z₂|| ≤ |z₁+z₂|. State the condition for eqality to hold.

 

[Sp3ctre]

Nevermind figured it out, was easier than I thought

 

[pikachu975]

Nevermind figured it out, was easier than I thought

Do you mind showing the answer so we can see?

 

[trecex1]

Do you mind showing the answer so we can see?

|z1| |z2| |z1+z2| are length of three sides of a triangle.
The length of one side of a triangle is greater than or equal to the difference of the other two. Equality when the points 0,z1,z2 are collinear.

 

[calamebe]

And if anyone wants to know why that works, just use the cosine formula and note that the cosine of any angle is less than or equal to 1.

 

[Mahan1]

Do you mind showing the answer so we can see?

By triangle inequality:
















that's why we use absolute value.

 

[Sp3ctre]

Do you mind showing the answer so we can see?

It might be a bit hard to understand without the diagram but I showed it to my teacher and he said it was correct.

|z1|-|z2| ≤ |z1+z2|
If |z1| > |z2|
|z1| ≤ |z1+z2| + |z2|

Statement only holds true for equality when
arg(z1)-arg(z2)=pi or arg(z1) = pi + arg(z2)
OR
z1 = -kz2