# Vectors forming a closed triangle

I've always been taught that vectors that form a closed polygon represent an object being at equilibirium, that is there is no resultant force on the object. However, this has never been intuitive to me. How does one prove, that if the vectors form a closed polygon, the resultant force has to be zero? Furthermore why must the arrows form a closed loop, why is it not enough for the lines themselves to form a closed loop?

If the vectors that represent the forces on a point object can be placed "nose to tail" to form a close loop then the sum of those vectors is zero (since if you treat each vector as a Euclidean vector and follow them around the loop you end up back where you started). Therefore the net force on the point object is zero and it is at equilibrium.

Note that if the object is not a point object then a net force of zero is necessary for equilibrium but not sufficient, since the forces could still create a non-zero moment or couple on the object, even if they net to zero.

If you have to reverse the direction of one of the vectors to make a closed loop then the vectors do not sum to zero, since you have in effect replaced one of the vectors by its negative. So you have, for example, shown that

$$\vec a + \vec b +(-\vec c) = 0$$

but that means that

$$\vec a + \vec b + \vec c = 2\vec c$$

which is not zero (unless $$\vec c=0$$).