Can I form a non-trivial loop from these vectors?

Question

Suppose that I am given $M>2$ (non-zero) vectors $v_i$ in $\rm I\!R^2$. How can I tell if a non-negative linear combination of these vectors form a loop; i.e. is there a solution to $0 = \sum_i a_i v_i$ with $a_i>=0$ for all $i$ and $1=\sum_i a_i$?