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Complex numbers HELPP pLEASE (1 Viewer)

pencilspanker

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Hey guys im struggling with a few concepts on complex numbers and just wondering if anyone could clear them up because im legit abt to neck. I know a lot of basic facts but I don't fully understand how they work which is really bugging me... Ok so, for the question:
i) 1.PNG
I get that you go arg(z1-z2) - arg (z1+z2) =90o
But what im struggling to understand is why the angle in between those two diagonals in that parallelogram is going to be 90. Like im trying to manipulate the angles that are forming with the positive x axis but i just dont get it...

ii) Why can we just randomly say that the vector in the form z - (+z0) will always be the vector running from z0 to z?

iii) If i have the vector --> OC = z, is the vector ---> CO = -z???? And if this is true, how does this work because isnt it that when you put a negative out the front you are multiplying by i2 which means you rotate it by 180o. And if that is true do we call that 180 rotation the position vector whilst the vector ---> CO is just the free vector with the exact same magnitude and direction? Here is a retarded diagram to explain exactly what I mean.2.PNG

iv) help with this q plsss 3.PNG

THANK U VERY MUCH TO ANYONE THAT HELPS I REALLLLY FREAKING APPRECIATE IT!!!
 

fan96

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i)





What does this mean?

It means that the vector is perpendicular to . That is basically the condition given to you.

and are the diagonals of the parallelogram with vertices .

But, a parallelogram with perpendicular diagonals is a rhombus.

ii)



iv)

b)



What is ?

(It doesn't matter which point is , although this will be considered later.)

c)

.

But, implies that is always in the clockwise direction of .

.



What are and ?
 
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fan96

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Also the "identity"



isn't actually true because isn't guaranteed to be in the range .

If was instead expressed as the principal value for that angle then I believe it would be true.

So just be careful when using it - if possible find another method.
 

peter ringout

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Hey guys im struggling with a few concepts on complex numbers and just wondering if anyone could clear them up because im legit abt to neck. I know a lot of basic facts but I don't fully understand how they work which is really bugging me... Ok so, for the question:
i) View attachment 34699
I get that you go arg(z1-z2) - arg (z1+z2) =90o
But what im struggling to understand is why the angle in between those two diagonals in that parallelogram is going to be 90. Like im trying to manipulate the angles that are forming with the positive x axis but i just dont get it...

ii) Why can we just randomly say that the vector in the form z - (+z0) will always be the vector running from z0 to z?

iii) If i have the vector --> OC = z, is the vector ---> CO = -z???? And if this is true, how does this work because isnt it that when you put a negative out the front you are multiplying by i2 which means you rotate it by 180o. And if that is true do we call that 180 rotation the position vector whilst the vector ---> CO is just the free vector with the exact same magnitude and direction? Here is a retarded diagram to explain exactly what I mean.View attachment 34700

iv) help with this q plsss View attachment 34701

THANK U VERY MUCH TO ANYONE THAT HELPS I REALLLLY FREAKING APPRECIATE IT!!!

It's always nice to use geometry but it is unfortunate that the question is so ham-fisted. Parts a) and b) are heavy machinery to get to a trivial result. A simple direct algebraic approach for c) is:

c) Divide top and bottom by z1 and let w=z2/z1. Then (1+w)/(1-w)=2i. Just solve for w.

Still you must answer the question on the page!
 
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peter ringout

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Why is that unfortunate?
A poorly written question is always unfortunate. When it comes to writing complex analysis problems I always feel that geometry should be forced to provide elegant proofs for deep results, not difficult proofs for trivial results. Presumably part c) is the point of the question and unfortunately the examiner chose a very tortuous path for the candidates. Once you have c), a) and b) are trivial if you really want them.
 
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pencilspanker

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Forum) do you then translate that green vector to start at w and run to z? So that means it will have the same magnitude and direction but just won’t represent the exact point z-w. It is a free vector?
 

fan96

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Forum) do you then translate that green vector to start at w and run to z? So that means it will have the same magnitude and direction but just won’t represent the exact point z-w. It is a free vector?
The green vector and the vector from to are the same vector. The position doesn't matter.
 

pencilspanker

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The green vector and the vector from to are the same vector. The position doesn't matter.
Right I see. I semi get it now, this might sound retarded, but how can we just magically translate the vectors to wherever we want and say that they represent the same thing?
 

fan96

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Right I see. I semi get it now, this might sound retarded, but how can we just magically translate the vectors to wherever we want and say that they represent the same thing?
Vectors are defined by two things:

- a magnitude e.g.

- a direction e.g.

So if two vectors have the same magnitude and direction then they are equal.
 

HeroWise

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Well it depends on what you are doing. In maths and Physics we see it as same and equal even if they move around but they must have same magnitude and direction. But smthn like in CS, Vectors are rather for storing information, so a translated vector would be different to the original vector
 

Skuxxgolfer

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\left|z+4\right|+2\left|z-3\right|=15

can someone solve this for me please
 

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