Hello, I had a physics test two weeks ago and I was thinking about an exercise that I solved without any mathematical proof.
Basically I need to prove that if I consider the vectors from the center of a circle to the vertices of any regular polygon, than the sum of all vectors is zero.
So it reduces to prove that:
for any n greater or equal to 2.
Can anyone help me?
I think you can solve it by doing the complex expression in this way sum( r^n) = (1-r^(n+1))/(1-r).
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I thought about this for quite a while and the only thing I could come up with involves complex numbers. i.e. the complex nth roots of 1 - there are n of them. This is what has been suggested above by Jack. I used DeMoivre's Theorem and the polar form of a complex number, whereas Jack is using the exponential form. Both amount to the same thing in the end.
I don't know what experience you have in working with complex numbers or in finding things like: the 2 complex square roots, the 3 complex cube roots, the 4 complex 4th roots, etc..... for any complex number. If you know about these things and are interested I can show you my step by step process. It all relates to summing sin(2pi/n) and cos(2pi/n) and shows that both sums are 0.
It seems that you are a physics student so I think I know where you are coming from. You have n vectors - all of the same magnitude and originating from the same initial point and all with equal angles between them - 2pi/n. The tips of these vectors form the regular n-gon and the sides of the n-gon are all the "delta" v's. You can easily see that when you add up all the delta v's you get 0, but our problem is the the v's are not lined up tip to tail so we need to show that if you do line up the v's tip to tail you get back to where you started from. This is what the complex number thing does only it uses components - just as your original question asks. You are really showing that all the vertical components add to 0 and all the horizontal components add to 0.
Let me know if you want to see more detail.
maybe you can try to write it as a geometric series in x using
sin(x) = (exp(ix)-exp(-ix)/(2i) etc similar for cos(x).
then just simplify the complex expressions.
sum( r^n) = (1-r^(n+1))/(1-r)