I have to show the following:
If $v, w \in \mathbb{R}^3 \setminus \{(0,0,0)\}$ such that the set of vectors orthogonal to both of them is a plane through the origin, then each is a scalar multiple of the other.
I've proved that if $u^\perp = \{x \in \mathbb{R}^3 : x \perp u\}=\{x \in \mathbb{R}^3 : \langle x , u \rangle=0\}$, the set of all vectors which are ortogonal to $u$, then $v^\perp=w^\perp$ [since $v^\perp \cap w^\perp$ is a plane through origin (given) then dim$(v^\perp \cap w^\perp)$=dim$(v^\perp)$=dim$(w^\perp)$=2 and $v^\perp \cap w^\perp \subseteq v^\perp$,$v^\perp \cap w^\perp \subseteq w^\perp$]. Will this be of any help? I can't go any further. Many thanks.