Given two vectors $a$ and $b$ and a hyper-surface $S$, I would like to know whether the two vectors point in the same direction regarding the surface.
Example 1
Here they point in different directions:
$$ S = x \cdot (0,1) + 0 \\ a = (1,0)\\ b = (-1,0) $$
Example 2
But here they also point in different directions:
$$ S = x \cdot (0,1) + 0 \\ a = (0.1,10)\\ b = (-0.1,10) $$
Example 3
Here they point in the same direction:
$$ S = x \cdot (1,0,0) + y \cdot (0,1,0) + 5\\ a = (1,1,1)\\ b = (-1,-1,1)\\ $$
Example 4
Here they point in different directions:
$$ S = x \cdot (1,0,0) + y \cdot (0,1,0) + 5\\ a = (1,1,1)\\ b = (1,1,-1)\\ $$