Sheaf of Planes

A sheaf of planes with an axis, r, is a set of planes that contain the line r.

Sheaf of Planes

Implicit Equations

The equation of the sheaf of planes with axis, r, is:

Sheaf of Planes Equation

Dividing by λ and making Value of K, the equation becomes:

Sheaf of Planes Equation

Example

Find the equation of plane that passes through the point (3, 2, −3) and belongs to the sheaf of planes with its axis on the following line:

Line Equation

Linear Combination

Operations

Line Solution


Parallel Sheaf of Planes

Parallel Sheaf of Planes

Two planes are parallel if the coefficients x, y, z are proportional to their equations but their independent terms are not.

All planes that are parallel to a given plane admit an equation in the form:

Parallel Sheaf of Planes Equations


Example

Find the equation of the plane that passes through the point (3, −1, 2) and is parallel to Plane Equation Example.

Sheaf of Plane Operations

Sheaf of Plane Solution