Ebenengleichung Normalform

Normal Form for Plane Equations

Definition

The normal form of a plane equation is a representation of a plane in Cartesian coordinates using a normal vector and a distance from the origin.

Equation

The equation of a plane in normal form is given by: ``` n · (x - p) = d ``` where * **n** is the normal vector to the plane, with components (a, b, c). * **p** is a point on the plane, with coordinates (x₀, y₀, z₀). * **d** is the directed distance from the origin to the plane, which is positive if **n** points away from the origin and negative if **n** points towards the origin.

Applications

The normal form of a plane equation is useful for various applications, including: * Determining whether a point lies on a plane * Finding the intersection of planes * Computing the distance between a point and a plane * Determining the direction of a plane's normal vector

Conclusion

The normal form of a plane equation provides a concise and geometrically intuitive way to represent planes in Cartesian coordinates. Its simplicity and versatility make it a valuable tool for a wide range of mathematical and engineering applications. By understanding the concept of the normal form, individuals can effectively analyze and manipulate planes in three-dimensional space.