# Sum of 3 vectors in R3

• February 17th 2013, 01:02 PM
Civy71
Sum of 3 vectors in R3
If the sum of three vectors in R3 is zero, must they lie in the same plane?
• February 17th 2013, 06:53 PM
chiro
Re: Sum of 3 vectors in R3
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.
• February 17th 2013, 06:55 PM
Let the three vectors be $v_1,v_2,v_3$
If the sum of the three vectors is the zero vector, then these vectors are linearly dependent since $v_3 = -v_1 -v_2$ so $v_3$ must lie on the same plane spanned by $v_1$ and $v_2$. Hence all three vectors lie on the same plane