Question

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

Answer 1

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.....

Answer 2

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

Answer 3

Consider an octagon for a concrete example with even number of vertices,

How do we use symmetry ?

Answer 4

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

Answer 5

oh yes sry

Answer 6

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

Answer 7

yeah

Answer 8

what if its odd careful there

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

*

Answer 13

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

Answer 14

their sum