Let $f$ and $g$ be two $C^{1}$ vector fields in $\mathbb{R}^{2}$ such that $\langle f(x), g(x) \rangle = 0 \,\,\,\forall \,\, x \in \mathbb{R}^{2}$. If $f$ admits a cyclic orbit, prove that $g$ possesses a singularity.
I am pretty sure that the Poincaré-Bendixson Theorem is used here, but I am not sure how.
Let $C \subset \mathbb{R}^2$ be the image of the embedding $S^1 \mapsto \mathbb{R}^2$ which is a cyclic orbit of $f$. By the Schonflies theorem, $C$ is the boundary of an embedded disc $D \subset \mathbb{R}^2$. The vector field $g(x)$ is defined on $D$ and is orthogonal to $D$.
Edited for more detail: The Poincare-Hopf index theorem applies in any situation of a compact manifold $M$ and a vector field on $M$ orthogonal to the boundary, with the conclusion that if $g(x)$ has isolated zeroes then the sum of their indices must be the Euler characteristic of $M$.
Since in this case $M=D$ has Euler characteristic $1$, either $g(x)$ has a non-isolated zero or it has isolated zeroes including one of nonzero index. In either case, $g(x)$ has a zero.