triangle inequality (complex number) (1 Viewer)

[lugana]

Hi guys,

tell me how to solve these two questions please?

1.
Show that ||z₁|+|z₂||<or=|z₁+z₂|. State the condition for equality to hold.


2.
Show that |z₁+z₂+...+z_n|<or=|z₁|+|z₂|+...+|z_n|



Seems I'm in stuck with those types of questions...

 

[Carrotsticks]

Hi guys,

tell me how to solve these two questions please?

1.
Show that ||z₁|+|z₂||<or=|z₁+z₂|. State the condition for equality to hold.


2.
Show that |z₁+z₂+...+z_n|<or=|z₁|+|z₂|+...+|z_n|



Seems I'm in stuck with those types of questions...

There is nothing to solve there. You just gave us a whole bunch of letters and numbers.

 

[Nooblet94]

Q1.


Q2.

Judging from the thread title I think that's what you mean?


Q1 - Consider the triangle formed by the origin, Z1 and Z1+Z2. We know that in any triangle the sum of any two side lengths is greater than the length of the third side. Hence, OB is less than (or equal to) OA+AB, from which you obtain the required inequality. The inequality holds when Z1=Z2


Q2 - By definition, a straight line is the path with the shortest distance between two points (in this case, the modulus of the sum of numbers), and therefore any other path (the sum of the moduli) will be longer. Hence, the inequality.

 

[lolcakes52]

Q1.


Q2.

Judging from the thread title I think that's what you mean?


Q1 - Consider the triangle formed by the origin, Z1 and Z1+Z2. We know that in any triangle the sum of any two side lengths is greater than the length of the third side. Hence, OB is less than (or equal to) OA+AB, from which you obtain the required inequality. The inequality holds when Z1=Z2
View attachment 24125

Q2 - By definition, a straight line is the path with the shortest distance between two points (in this case, the modulus of the sum of numbers), and therefore any other path (the sum of the moduli) will be longer. Hence, the inequality.

Didn't you just prove the triangle inequality using the triangle inequality?

 

[Carrotsticks]

Didn't you just prove the triangle inequality using the triangle inequality?

Not quite sure why, but I found this comment to be very funny.

But in all seriousness, that does seem to be the case. He utilised the fact that "We know that in any triangle the sum of any two side lengths is greater than the length of the third side", which is said triangle inequality.

 

[seanieg89]

And the equality conditions stated are incorrect.

 

[Carrotsticks]

Equality occurs when the angle between Z_1 and Z_2 is equal to pi.

Geometrically, this means equality occurs in the degenerate case where the 'triangle' is a line.

 

[seanieg89]

Equality occurs when the angle between Z_1 and Z_2 is equal to pi.

Geometrically, this means equality occurs in the degenerate case where the 'triangle' is a line.

You mean an angle of 0 right?

 

[Carrotsticks]

You mean an angle of 0 right?

I was pertaining to the geometric definition of the three vectors as the sides of a triangle Z1, Z2, and Z3.

 

[largarithmic]

I was pertaining to the geometric definition of the three vectors as the sides of a triangle Z1, Z2, and Z3.

No, that isnt a situation that corresponds with the actual question. The reason is in giving the sides vectors you havent given them directions. For the angles between Z1 and Z2 to actually be alpha, you need Z3 = Z1-Z2 or Z3 = Z2-Z1. If you want Z3=Z1+Z2 (so that it actually corresponds to the question), you need Z1 and Z2 sorta going 'in the same direction' if you think about it, in which case Z1 and Z2 are gonna have the same argument and thus angle between them 0.

 

[Carrotsticks]

No, that isnt a situation that corresponds with the actual question. The reason is in giving the sides vectors you havent given them directions. For the angles between Z1 and Z2 to actually be alpha, you need Z3 = Z1-Z2 or Z3 = Z2-Z1. If you want Z3=Z1+Z2 (so that it actually corresponds to the question), you need Z1 and Z2 sorta going 'in the same direction' if you think about it, in which case Z1 and Z2 are gonna have the same argument and thus angle between them 0.

Yes, if we consider Z_1 + Z_2 = Z3 (thus giving them directions), then equality occurs when Arg (Z_1) - Arg (Z_2) = 0 ---> Arg (Z_1) = Arg (Z_2) , as you said.

You are correct in saying that my diagram did not correspond to the original question directly.