If the sum of three vectors in R3 is zero, must they lie in the same plane?
Let the three vectors be $\displaystyle v_1,v_2,v_3$
If the sum of the three vectors is the zero vector, then these vectors are linearly dependent since $\displaystyle v_3 = -v_1 -v_2$ so $\displaystyle v_3$ must lie on the same plane spanned by $\displaystyle v_1$ and $\displaystyle v_2$. Hence all three vectors lie on the same plane