How to check if a surface normal faces inwards or outwards an ellipsoid?

Question

I have a half ellipsoid parametrised by $a,b,c,\theta,\phi$ and centered at origin. I have an optimization problem where I estimate the three components of a 3D direction vector $\hat{d}=(x,y,z)$ where, $-1\leq x,y,z\leq 1$. Assume the estimated vector is normalized. Is it possible to know if this estimated direction vector points towards the inside or outside the ellipsoid with only this information? If yes, how? The origin of the direction vector always lies on the surface of the ellipsoid.

Answer 1

As answered by @intelligenti pauca, if the origin of the direction vector on the ellipsoid's surface is $P$, then $\overrightarrow{OP} \cdot \hat{d} > 0$ if $\hat{d}$ points outwards.