Root of unity- help plz (1 Viewer)

HeroicPandas · Dec 28, 2012

$$a) Let the points A_1, A_2, ..., A_n$ $ represent the nth roots of unity, w_1, w_2,..., w_n,$ $ and suppose P represents any complex number z such that \left | z \right |=1. \\ \\ $(i)$ Prove that w_1 + w_2 + ... + w_n = 0 \\ \\ $(ii)$ Show that PA_i^2 = (z-w_i)(\bar{z}-\bar{w_i})$ $ for i = 1, 2, ..., n \\ \\$ $(iii) Prove that \sum_{i=1}^{n}$ $ PA_i^2 = 2n$

Someone help me plz!!

I need help with part ii, its IMPOSSIBRU!

(If there are any mistakes, this is 1993 question 8a)

twinklegal19 · Dec 29, 2012

Sorry for the messiness, might have missed out some lines or something

edit: new solution

Sy123 · Dec 29, 2012

twinklegal19 said:
I remember a tutor going through this question once.

Sorry for the messiness, might have missed out some lines or something

edit: oops forgot about w_n on the diagram

Was there any additional information missing in the question?
At the moment the question doesn't make any sense. Because you got PA^2 = AP^2
And we can only derive that from some additional information..

twinklegal19 · Dec 29, 2012

Sy123 said:
Was there any additional information missing in the question?
At the moment the question doesn't make any sense. Because you got PA^2 = AP^2
And we can only derive that from some additional information..

It's the distance between P and A_i all squared so it doesn't matter the order? It just looks nicer but not necessary I guess. I'll redo this again since I rushed it

Sy123 · Dec 29, 2012

Oh I completely stuffed up that interpretation..............
I took them as complex numbers... ಠ_ಠ

HeroicPandas · Dec 29, 2012

i dont get why $A_i P^2 = |\vec{A_i P}|^2$

OMG NVM, same problem as Sy123

asianese · Jan 2, 2013

Hahaha I remember doing this just before my trial...I did the solution but have since burned the book in which I did it.

It involved Pythagoras' theorem I'm pretty sure....

Root of unity- help plz (1 Viewer)

HeroicPandas

Heroic!

twinklegal19

.

Sy123

This too shall pass

twinklegal19

.

Sy123

This too shall pass

HeroicPandas

Heroic!

asianese

Σ

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