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

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sdfgsdfgs
 one year ago
Best ResponseYou've already chosen the best response.2for 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.....

dan815
 one year ago
Best ResponseYou've already chosen the best response.2all 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

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2Consider an octagon for a concrete example with even number of vertices, dw:1436576807411:dw How do we use symmetry ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2@dan815 <nitpicking started> do you mean r*e^(\(\color{red}{i}\)2pi*k/n) ? <nitpickign ended />

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2so based on that i think the problem translates to proving \[\large \sum\limits_{k=0}^{n1} e^{i2\pi k/n} = 0\]

sdfgsdfgs
 one year ago
Best ResponseYou've already chosen the best response.2dw:1436577086602:dw

dan815
 one year ago
Best ResponseYou've already chosen the best response.2what if its odd careful there

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2Ahh nice, so we always have opposite vectors for polygons with even number of vertices

sdfgsdfgs
 one year ago
Best ResponseYou've already chosen the best response.2@dan815 it can still be proven by symmetry for odd no. but its less obvious...

dan815
 one year ago
Best ResponseYou've already chosen the best response.2adding 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

dan815
 one year ago
Best ResponseYou've already chosen the best response.2or also like the outer angles of all regular polygon = 360

sdfgsdfgs
 one year ago
Best ResponseYou've already chosen the best response.2dw:1436577289096:dw

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2Building 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 \(x^nr^n\) is \(0\) so i think we're done ?

sdfgsdfgs
 one year ago
Best ResponseYou've already chosen the best response.2@ganeshie8 @dan815 nicely done! :)

dan815
 one year ago
Best ResponseYou've already chosen the best response.2you know kai actually found that equation randomly to generate any n sided regular polygon

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2yeah 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

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2ofcourse wid some extra algebra as we cant multiply vectors like we multiply complex numbers

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2wonder if there is an useful interpretation dot product/cross product of complex numbers

sdfgsdfgs
 one year ago
Best ResponseYou've already chosen the best response.2btw  just realized that proving the ycomponet 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.! :)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2for even no. of vertices the symmetry argument worked like a charm as we found pairs of opposite vectors !

sdfgsdfgs
 one year ago
Best ResponseYou've already chosen the best response.2by lining up 1 of the vertices in the ydirection, the xcomponent of all other vectors (except the 1 pointing in y) in an oddno polygon will equal out as the vectors will be in pair by symmetry. but showing the ycomponent of the vectors will add to zero is much tricky than i though.
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