I'm looking for a general way to parametrise the distribution of of vectors that have a non-negative dot-product with a known vector. I believe this is a more general version of the question asked in all vectors making $ \text{negative dot product} $ with the vector $(1,1,1)$..
The motivation of the problem is to parametrise the distribution of vectors that point away from the origin of a 3-D (Cartesian) space at a location $\vec{r}$. This is required to find which possible vectors point away from the centre of the volume to provide gradient constraints to a Gaussian Process.
If I understand the related question correctly, my first step should be to find the plane that is normal to $\vec{r}$ (which I remember being taught how to do once upon a time, but have since forgotten - I'm sure I can figure it out again, but pointers would be helpful). However, once I have identified that plane, I am struggling to understand how I determine the distribution of vectors that meet my needs. Ideally I would like to sample from the distribution of x, y and z components for these vectors. I can place some constraints on the range of projections in the direction of $\vec{r}$ to limit the length of these vectors.
Any help would be appreciated, and I will happily try to clarify anything that is unclear!
The answer is
$$\vec{v}(\alpha, \beta, \gamma) = \alpha \vec{e}_r + \beta \vec{e}_b + \gamma \vec{e}_c$$
Where $\alpha, \beta, \gamma$ are real numbers, and $\vec{e}_r, \vec{e}_b, \vec{e}_c$ are 3 orthonormal vectors. So $\vec{e}_r$ points in the same direction as your $\vec{r}$, ant the other two vectors are orthogonal to it. Additional constraint that you require is $\alpha \geq 0$, whereas the other two numbers can be arbitrary.
The only remaining thing to do is to find the vectors $\vec{e}_b, \vec{e}_c$. The procedure to do this is called Gram-Schmidt process.