Thread: Sum of 3 vectors in R3

If the sum of three vectors in R3 is zero, must they lie in the same plane?

Hey Civy71.

They could lie all on the same line or the same plane. They all lie on the same point if all vectors are the zero vector.

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

Thanks for the help MacstersUndead. Cool Avatar picture.