keywords: Math, Plane Side, Point

Method 1

Let ($A*x+B*y+C*z+D=0$) 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 $(A*x+B*y+C*z+D)$) 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:

$$a*x + b*y + c*z = a*x_0 + b*y_0 + c*z_0$$

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 $\overrightarrow{n}$.

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