In 3D geometry, a plane is a flat, two-dimensional surface that extends infinitely in space. It can be uniquely defined by:
\[ Ax + By + Cz + D = 0 \] This is the most commonly used equation of a plane, where \( A, B, C \) define the normal vector to the plane.
\[ \begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0 \] This form uses the determinant to ensure the three vectors lie in the same plane.
Expanded Form:
\[ \left| \begin{array}{cc} y_2 - y_1 & z_2 - z_1 \\ y_3 - y_1 & z_3 - z_1 \end{array} \right| (x - x_1) \]
\[ + \left| \begin{array}{cc} z_2 - z_1 & x_2 - x_1 \\ z_3 - z_1 & x_3 - x_1 \end{array} \right| (y - y_1) \]
\[ + \left| \begin{array}{cc} x_2 - x_1 & y_2 - y_1 \\ x_3 - x_1 & y_3 - y_1 \end{array} \right| (z - z_1) = 0 \]
If the plane cuts the x, y, and z axes at \( a, b, c \), then the equation is:
\[
\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1
\]
The normal form uses angles \( \alpha, \beta, \gamma \) made by the perpendicular from the origin: \[ x \cos \alpha + y \cos \beta + z \cos \gamma = p \] where \( p \) is the distance from origin to the plane.
The perpendicular distance from point \( M(x_0, y_0, z_0) \) to the plane is: \[ d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
