Sum of 3 vectors in R3

**Civy71** · February 17th 2013, 01:02 PM

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

**chiro** · February 17th 2013, 06:53 PM

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.

**MacstersUndead** · 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

**Civy71** · February 18th 2013, 05:45 AM

Thanks for the help MacstersUndead. Cool Avatar picture.

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