Complex Numbers - Vectors (1 Viewer)

[McSo]

Hi, I'm having some major problems with vectors. I've been recently trying to revise the complex numbers chapters and I've come to realise that I suck at anything vector related... I think I can do the basic things but when it comes to anything a bit more complicated I get stuck.. For example in Excercise 2.3 from the Arnold 4u Math's Book, I can only do a few questions...
The ones I can't do are:

5) On an Argnd diagram the points P and Q represent the numbers z1 and z2 respectively. OPQ is an equilateral triangle. Show that z1^2 + z2^2 = z1z2

6) Show that ||z1| - |z2|| <= |z1 + z2|. State the condition for equality to hold.

8) Show that |z1 + z2 ... + zn| <= |z1| + |z2| + ... |zn|

9) On an Argand diagram the points A and B represent the numbers z1 and z2 respectively. I is the point (1,0). D is the point such that triangle oID is similar to triangle OBA. Show that D represents z1/z2.

Can anyone help me out with those questions, and does anyone know of any good sites/resources that I could use that would make understanding these concepts easier? Thanks.

 

[hyparzero]

5) If OPQ is an equilateral triangle, then |z₁| = |z₂| = |z₁ - z₂|

and just go on from there....

6) In any triangle, the sum of any two sides is always greater than the third
In the triangle OPQ;
|OP| = |z₁| and |OQ| = |z₂|

Let |OR| be |OP + OQ| = |z₁ + z₂| .... where R = z₁ + z₂

But |OQ| + |OQ| > |OR| and
|PQ| < |OP| + |OQ|

Thus |PQ| < |OR|

.'. ||z₁| - |z₂|| < |z₁ + z₂|

But the inequality
||z₁| - |z₂|| <= |z₁ + z₂|
will only hold true iff OPQR is a square:
ie: Arg(z₁/z₂) = pi/2

and so on..

 

[McSo]

5) If OPQ is an equilateral triangle, then |z₁| = |z₂| = |z₁ - z₂|

and just go on from there....

6) In any triangle, the sum of any two sides is always greater than the third
In the triangle OPQ;
|OP| = |z₁| and |OQ| = |z₂|

Let |OR| be |OP + OQ| = |z₁ + z₂| .... where R = z₁ + z₂

But |OQ| + |OQ| > |OR| and
|PQ| < |OQ| + |OQ|

Thus |PQ| < |OR|

.'. ||z₁| - |z₂|| < |z₁ + z₂|

But the inequality
||z₁| - |z₂|| <= |z₁ + z₂|
will only hold true iff OPQR is a square:
ie: Arg(z₁/z₂) = pi/2

and so on..

Thanks... But with question 5, where do I go on from there?

Oh also... whats the actual reason that |PQ| < |OQ| + |OQ| ? Like if you draw it, its obvious, but yeah...

 

[hyparzero]

Oh also... whats the actual reason that |PQ| < |OP| + |OQ| ? Like if you draw it, its obvious, but yeah...

|PQ| = |OQ|

therefore
|PQ| < |OP| + |OQ|

 

[McSo]

Ah right, thanks.

 

[Shady01]

tagging on Cis(pi/5) can that be put into cartesian and plotted?

 

[hyparzero]

tagging on Cis(pi/5) can that be put into cartesian and plotted?

why not?

 

[echelon4]

Here's the solution for q5. It's a bit messy. Donno if there's an easier way though.....

Sorry bout the size of the picture, i donno how to scan it properly...