How to find if the point lies in which half of the tangent plane to a sphere?

Question

Let $S$ be the sphere $$ x^2 + y^2 + z^2 = 14$$

Equation of tangent plane to $S$ at the point $P(1, 2, 3)$ is $$ \quad \quad x+2y+3z-14=0 $$

This plane divides the whole $3-D$ plane in $2$ halves .

Let the two halves be A and B and the origin lies in A half.

I need to find whether the point $Q(3, 2, 1)$ lies in half A or not?

Answer 1

The two halves are the regions where $x+2y+3z-14>0$ and where $x+2y+3z-14<0$. Since the origin is in the latter, the region with $x+2y+3z-14<0$ is A. Finally, plug in the point and see if it is in the same region.

Answer 2

Let $P$ be the tangent plane to the sphere at $(1,2,3)$. for every element $(x,y,z)$ of $R^3$ to know the half of $R^3$ determined by $P$ where $(x,y,z)$ lies firstly compute $(x-1,y-2,z-3)$, and the scalar product $\langle(1,2,3);(x-1,y-2,z-3)\rangle$ its sign determines the half of the plane.