keywords: Math, Plane Side, Point

Method 1

Let $$Ax+By+C*z+D=0$$ be the equation determining your plane.

Substitute the $$(X, Y, Z)$$ coordinates of a point into the left hand side of the equation (I mean the $$Ax+By+C*z+D$$) and look at the sign of the result.

The points having the same sign are on the same side of the plane.

Origin:
Check which side of a plane points are on
https://stackoverflow.com/questions/15688232/check-which-side-of-a-plane-points-are-on

Plane Equation:
If point $$P_0 = (x_0, y_0, z_0)$$, and vector $$n = (a, b, c)$$, Plane Equation is:

$$ax + by + cz = ax_0 + by_0 + cz_0$$

Method 2

The plane passing through $$P_1(X_1, Y_1, Z_1)$$, $$P_2(X_2, Y_2, Z_2)$$, and $$P_3(X_3, Y_3, Z_3)$$, the point to check is $$A(X_a, Y_a, Z_a)$$.
So we can get the normal vector of plane using cross product:

$$\overrightarrow{n}=\overrightarrow{P_1P_2}\times\overrightarrow{P_1P_3}$$

then calculate the dot product of $$\overrightarrow{n}$$ and $$\overrightarrow{P_1A}$$:

$$ a=\overrightarrow{n}\cdot\overrightarrow{P_1A}$$

Point $$A$$ is on one side if $$a$$ > 0, otherwise it’s on the another side.

Nobody deserves your tears, but whoever deserves them will not make you cry. ― Gabriel García Márquez