Question

Hi,

Can someone please answer question 1A-15 part b (case n is odd) of the following:

http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/problem-set-1/MIT18_02SC_SupProb1.pdf

Thanks

Answer 1

how about this?
Assume the vertices are in order OP1, OP2, ... OPn
the angle between each vector is 2pi/n
Add OP2 to OP1: tail of OP2 at the head of OP1. OP2 will form an exterior angle of 2pi/n
call the resultant P2
Now add OP3 to P2: OP3 will form an exterior angle of 2pi/n with the "displaced" vector OP2.
Continue. At the end of this process we have n equal sides, with n equal exterior angles of 2pi/n
so we have formed a regular n-gon and have gone 2pi radians back to our starting point.
we conclude P1+P2+...+Pn=0