## ganeshie8 one year ago show that sum of the vectors drawn from center of a regular polygon to its vertices is 0

1. sdfgsdfgs

for regular polygon w even no. of vertices, it is easy to show the sum of vecto5rs from center to vertices is zero by symmetry... it can be proven by symmetry as well for odd no. of vertices but it is less obvious....hmmmm.....

2. dan815

all the vector sides of an n sided regular polygon centered around the origin can be rewritten was r*e^(2pi*k/n) k is an int from 1 to n and r is the length of each side by this definition we are producing vector from the fact that they are summing to 0

3. ganeshie8

Consider an octagon for a concrete example with even number of vertices, |dw:1436576807411:dw| How do we use symmetry ?

4. dan815

|dw:1436576906626:dw|

5. ganeshie8

@dan815 <nitpicking started> do you mean r*e^($$\color{red}{i}$$2pi*k/n) ? <nitpickign ended />

6. dan815

oh yes sry

7. ganeshie8

so based on that i think the problem translates to proving $\large \sum\limits_{k=0}^{n-1} e^{i2\pi k/n} = 0$

8. dan815

yeah

9. sdfgsdfgs

|dw:1436577086602:dw|

10. dan815

what if its odd careful there

11. ganeshie8

Ahh nice, so we always have opposite vectors for polygons with even number of vertices

12. sdfgsdfgs

@dan815 it can still be proven by symmetry for odd no. but its less obvious...

13. dan815

adding vectors to vectors with the same angle of separation between them and the angles add to 360 has to equal 0 another way to say that complex expression in words

14. geerky42

*

15. dan815

or also like the outer angles of all regular polygon = 360

16. dan815

their sum

17. dan815

|dw:1436577429209:dw|

18. sdfgsdfgs

|dw:1436577289096:dw|

19. ganeshie8

Building on dan's method using complex numbers, the vectors can be viewed as solutions to the equation $\large x^n = r^n$ Clearly the sum of roots of the polynomial $$x^n-r^n$$ is $$0$$ so i think we're done ?

20. sdfgsdfgs

@ganeshie8 @dan815 nicely done! :)

21. dan815

you know kai actually found that equation randomly to generate any n sided regular polygon

22. ganeshie8

yeah earlier i almost forgot we could use complex numbers here, that equation is same as the equation in vectors because we can treat complex numbers literally as 2D vectors

23. dan815

right

24. ganeshie8

ofcourse wid some extra algebra as we cant multiply vectors like we multiply complex numbers

25. ganeshie8

wonder if there is an useful interpretation dot product/cross product of complex numbers

26. sdfgsdfgs

btw - just realized that proving the y-componet all vectors in a regular polygon w odd no. of vertices by symmetry will be much less OBVIOUS than I had initially thought! so good that u guys have proven it by using complex nos.! :)

27. ganeshie8

for even no. of vertices the symmetry argument worked like a charm as we found pairs of opposite vectors !

28. sdfgsdfgs

by lining up 1 of the vertices in the y-direction, the x-component of all other vectors (except the 1 pointing in y) in an odd-no polygon will equal out as the vectors will be in pair by symmetry. but showing the y-component of the vectors will add to zero is much tricky than i though.