Let $F: S^2 \rightarrow \mathbb{R}^2$ be a continuous function. Does there exist a unite vector $v \in R^2$ and a continuous functions $f(x):S^2\rightarrow \mathbb{R}$ such that $f(x)>0$ on $S^2$ and
$$F(x)\neq f(x)v   \ \ \ \ \forall x\in S^2?$$