Complex Numbers - Vector question (1 Viewer)

[shongaponga]

Hey can someone please explain to me how to do this vector question from the Arnold textbook. Sorry I don't know how to use Latex yet.

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

Thanks

 

[Carrotsticks]

Hey can someone please explain to me how to do this vector question from the Arnold textbook. Sorry I don't know how to use Latex yet.

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

Thanks

Did you mean:

 

[shongaponga]

No. It's a plus.

 

[deswa1]

Draw out the vectors z1, z2, z1+z2 on the Argand diagram and try using the triangle inequality.

 

[shongaponga]

I was going to try the triangle inequality but then realised that it works for these two conditions : ||z1 + z2|| ≤ |z1| + |z2| and ||z1| - |z2|| ≤ |z1 - z2|, which is different to the equality at hand in this question.

 

[deswa1]

Try taking a more general look at the triangle inequality. ANY two sides are longer than the other one. Look at the triangle that has z1,z2 and z1+z2 as the side lengths and try to work out how to manipulate the inequality to give the desired result.

 

[barbernator]

Hey can someone please explain to me how to do this vector question from the Arnold textbook. Sorry I don't know how to use Latex yet.

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

Thanks

Isn't this true for any conditions? i'm trying to think of an instance where it isn't true but i can't think of one

 

[deswa1]

Isn't this true for any conditions? i'm trying to think of an instance where it isn't true but i can't think of one

The whole inequality is true for all values of z1 and z2 but what the question is asking is under what case is there equality i.e. when is ||z1|-|z2||=|z1+z2|

 

[RealiseNothing]

Isn't this true for any conditions? i'm trying to think of an instance where it isn't true but i can't think of one

You are proving that it is true though? Why do we need to find an instance where it isn't true?

There isn't an instance by the way.

 

[RealiseNothing]

The whole inequality is true for all values of z1 and z2 but what the question is asking is under what case is there equality i.e. when is ||z1|-|z2||=|z1+z2|

Which is at z1 = -z2.

 

[barbernator]

The whole inequality is true for all values of z1 and z2 but what the question is asking is under what case is there equality i.e. when is ||z1|-|z2||=|z1+z2|

oh yeh, i read equality and inequality thanks

 

[barbernator]

Which is at z1 = -z2.

when their arguments are 180 degrees different