Think of a vector as being a 'journey' ** from A to B. If you go A to B; then B to C; then C to A; then you end up where you started so your vector for the three journeys is zero.
Here's an example ( I'll use horizontal rather than the usual vertical to save time) :
(1,0) + (0,1) + (-1,0) + (0,-1) take you round the sides of a square and these add to (0,0).
What you describe in your post is a journey around the perimeter of a regular polygon. Once you've gone all the way round, you are back to the start.
http://www.mathsisfun.com/algebra/vectors.html
Bob
** vectors are used for other things than 'journeys', but it's a good place to start.
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