x orthogonal to nonzero vectors u and v and x $\in$ Span{u,v} show x = 0

Question

I can't quite figure this one out, I know that x $\cdot$ u = 0 and x $\cdot$ v = 0 and x = au + bv, but I don't know how to go from there to show x = 0

Answer 1

You've written $\mathbf{x} = a \mathbf{u} + b \mathbf{v}$. Good so far. What happens when you dot product both sides of this equation with $\mathbf{x}$?

Answer 2

If you have a matrix $\left(\begin{matrix}u & v\end{matrix}\right)$ with x both in the column space and the left null space, what does that say about x?

Answer 3

The numeric solution was already pointed at, but imo the geometric solution is way more intuitive and easier.

$Span(u,v)$ is either a plane or a line, which $u$ and $v$ lie on. Further we know $x \perp u,v$ which means $x$ can't lie on that plane/line, unless it is zero.