How to Find the Equation of a 3D Plane Passing Through Three Points

Understanding how to find the equation of a plane passing through three given points is a fundamental concept in both mathematics and computer science. This article will explore two primary methods to achieve this: using determinants and vectors. We will also discuss the significance of the derived equation in real-world applications.

Using Determinants to Find the Plane Equation

The process of finding the equation of a plane in three-dimensional space that passes through three given points involves using determinants and matrices. This method is based on the fact that the equation of the plane can be derived using the points directly.

Method 1: Using Determinants

Given three non-collinear points ( 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 plane passing through these points can be described as the set of all points ((x, y, z)) that satisfy the following determinant equation:

[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

Alternatively, the plane can also be given by the equation:

[begin{vmatrix}x-x_{1}  y-y_{1}  z-z_{1}x-x_{2}  y-y_{2}  z-z_{2}x-x_{3}  y-y_{3}  z-z_{3}end{vmatrix}]  0

Method 2: Using Equations and Matrix Manipulation

Another approach is to solve a system of linear equations. Given the points ( p_1, p_2, p_3 ), the plane can be described by an equation of the form ( ax by cz d 0 ).

To find ( a, b, c, d ), we solve the following system of equations:

[a x_{1}   b y_{1}   c z_{1}   d  0] [a x_{2}   b y_{2}   c z_{2}   d  0] [a x_{3}   b y_{3}   c z_{3}   d  0]

This system can be solved using Cramer's rule and basic matrix manipulations. Let ( D ) be the determinant of the matrix formed by the coordinates of the points:

[D  begin{vmatrix}x_{1}  y_{1}  z_{1}x_{2}  y_{2}  z_{2}x_{3}  y_{3}  z_{3}end{vmatrix}]

If ( D eq 0 ), the values for ( a, b, c, d ) can be calculated as follows:

[a  frac{D_{yz}}{D}, quad b  frac{D_{xz}}{D}, quad c  frac{D_{xy}}{D}, quad d  - (ax_{1}   by_{1}   cz_{1})]

Here, ( D_{yz}, D_{xz}, D_{xy} ) are the determinants of the submatrices obtained by removing the first row and the corresponding column.

Conceptual Understanding and Application

The equation of the plane can also be found using a vector approach. If three points ( p_1, p_2, p_3 ) define a plane, we can use the cross-product of two vectors lying in the plane to find the normal vector. Let ( v_1 p_2 - p_1 ) and ( v_2 p_3 - p_1 ) be two vectors lying in the plane. The normal vector ( vec{n} ) to the plane is given by the cross-product ( vec{n} v_1 times v_2 ).

Using the point-normal form of the plane equation ( vec{n} cdot (vec{r} - vec{r_0}) 0 ), where ( vec{r} ) is any point on the plane and ( vec{r_0} ) is a known point on the plane, we get the equation of the plane:

[vec{n} cdot (vec{r} - vec{r_0})  0]

This equation can be expanded into the standard form ( ax by cz d 0 ).

Conclusion

Understanding how to find the equation of a plane passing through three points is crucial in many fields, including computer graphics, robotics, and engineering. The methods outlined above provide a robust way to determine the plane equation, either using determinants or vectors. By mastering these techniques, one can effectively solve a wide range of problems involving 3D geometric objects.

Keywords

3D plane equation three points vector