Does there exist a dipole field on $S^2$ differing by at most a minus sign between antipodal points? Consider the two-sphere $S^2 \subset \mathbb{R}^3$. By a _dipole field_ on $S^2$, I mean a continuous function $f \colon S^2 \to S^2$ such that (1) $x$ is perpendicular to $f(x)$ for all $x \in S^2$ (this means that $f$ is a continuous _tangent vector field_ on $S^2$), and (2) $f$ vanishes at exactly one point. Question: Does there exist a dipole field on $S^2$ with the property that $f(x) \in \text{span} \left\\{f(-x)\right\\}$ for all $x \in S^2$, except for the point $x$ where $f(x) = 0$?

I think this should work: construct a vector field om $\mathbb R^2-${$(0,0)$} with exactly one zero, and use the stereographic projection to pull it back to $\mathbb S^2$.

Maybe a good point to make here, is that the fact that the stereographic projection is a diffeomorphism allows you to pushforward vector fields; this is not always possible.