# Thread: Vector Span and Orthogonality

1. ## Vector Span and Orthogonality

Prove thath if v, w, and x are in R2 or R3 an v is orthogonal to both w and x, then v is orthogonal to every vector span in {w, x}.

I'm totally lost on this problem. So far what I've got is that
v . w = 0
v . x = 0

v1w1 + .... + vnwn = 0
v1x1 + .... + vnxn = 0
v1x1 + .... + v1x1 = v1w1 + .... + vnwn
v1(x1 - w1) + .... + vn (xn - wn) = 0
(I'm not even sure if this part is useful or not)

Let's say there is a y that is a vector in the span of {w, x}. How would I prove that v . y = 0?

Thanks

2. Originally Posted by Brokescholar
Prove thath if v, w, and x are in R2 or R3 an v is orthogonal to both w and x, then v is orthogonal to every vector span in {w, x}.
Any vector in the spam of $\displaystyle \{ \bold{w},\bold{x} \}$ can be written in the form $\displaystyle a\bold{w}+b\bold{x}$.

Then $\displaystyle \bold{v}\cdot (a\bold{w}+b\bold{x}) = a \bold{v}\cdot \bold{w} + b \bold{v}\cdot \bold{x} = a\bold{0}+b\bold{0} = \bold{0}$.