How to Calculate the Equation of a Plane

Three Essential Forms of Plane Equations

In mathematics, a plane is a two-dimensional surface that extends infinitely in all directions. It can be defined by its normal vector and a point on the plane. The equation of a plane is a mathematical expression that describes the set of all points that lie on the plane.

General Form

The general form of the equation of a plane is:

``` ax + by + cz + d = 0 ```

where:

* `a`, `b`, and `c` are the coefficients of the variables `x`, `y`, and `z`, respectively * `d` is a constant

Point-Normal Form

The point-normal form of the equation of a plane is:

``` (x - x<sub>0</sub>)a + (y - y<sub>0</sub>)b + (z - z<sub>0</sub>)c = 0 ```

where:

* `(x<sub>0</sub>, y<sub>0</sub>, z<sub>0</sub>)` is a point on the plane * `a`, `b`, and `c` are the components of the normal vector to the plane

Intercept Form

The intercept form of the equation of a plane is:

``` x/a + y/b + z/c = 1 ```

where:

* `a`, `b`, and `c` are the intercepts of the plane with the `x`, `y`, and `z` axes, respectively