Inequations (1 Viewer)

[khorne]

Cambridge asks to prove ||z|-|w|| <= |z+w|

it's in the vector section, so I guess I should use vectors to prove it...

Now, I get how to do it algebraically, but with vectors, I'm not sure how to write a concise proof...

I was thinking:

draw up the parallelogram and acknowledge the condition that is arg(w) - arg(z) is less than 90, that means that the |z+w| diagonal is longer than the ||z|-|w|| diagonal.

Any ideas? As I doubt proof by "obvious"-ness is valid

 

[bachviete]

Cambridge asks to prove ||z|-|w|| <= |z+w|

it's in the vector section, so I guess I should use vectors to prove it...

Now, I get how to do it algebraically, but with vectors, I'm not sure how to write a concise proof...

I was thinking:

draw up the parallelogram and acknowledge the condition that is arg(w) - arg(z) is less than 90, that means that the |z+w| diagonal is longer than the ||z|-|w|| diagonal.

Any ideas? As I doubt proof by "obvious"-ness is valid

I have not touched over complex number vectors in quite some time, but its seam to me that ||z| - |w|| = |z|-|w|, so we must only show



ie

Construct the vectors z and w. Complete the parallelogram to get z+w. From the triangle inequality,



ie

as required. This completes the proof.

 

[pwoh]

Is this exercise 2.3, question 6?
I used the triangle inequality for that (sum of 2 sides > 3rd side)

How did you prove it algebraically though, I couldn't do that... (well it became convoluted so I stopped)

EDIT: Nevermind, bachviete beat me to it lol.

 

[khorne]

Indeed it is:

How did you use the triangle inequality, since you had to make the cross between |z| + |w| => |z+w| => ||z|-|w||

As for the algebraic way:

let |z| = |z+w-w|

i.e |z+w-w| <= |z+w| + |-w|
|z+w| => |z|-|w|

do that for w too, so you get |w|-|z| = -(|z|-|w|)

so since |z+w| => +/- (|z|-|w|), |z+w| => ||z|-|w||

 

[pwoh]

From the green triangle
|w| + |z+w| > |z| (triangle inequality)
Therefore
|z+w| > |z| - |w|

 

[khorne]

Of course, since |z|-|w| is equal to ||z|-|w||, given direction isn't important...

thanks guys