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Chapters

Finding the Equation of the Sheaf of Planes
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Parallel Sheaf of Planes
Example

In mathematics, a sheaf of planes refers to the set of all planes that contain the same common line. It is also referred to as a pencil or fan of planes. To comprehend the sheaf of planes, consider the concept of the line at infinity where a set of parallel planes can be viewed as a sheaf of planes intersecting in a line at infinity. In other words, we can say that a sheaf of plane is the set of all planes through a line. This line is also known as the axis of the sheaf, and the sheaf itself is sometimes referred to as a pencil.

The following figure shows a sheaf of planes with an axis, r. You can see here that a set of planes contain the common line r.

Figure 1 - Sheaf of Planes

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Finding the Equation of the Sheaf of Planes

If the line is:

$\text{[math]}$

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

$\text{[math]}$

Dividing by λ and making $\text{[math]}$ , the equation becomes:

$\text{[math]}$

In the next section, we will be solving a couple of examples in which we will find the equation of the plane that passes through certain points and belongs to the sheaf of planes with an axis on the line.

Example 1

Find the equation of plane that passes through the point (1, 2, 4) and belongs to the sheaf of planes with its axis on the following line:

$\text{[math]}$

Solution

As we know that the equation of the sheaf of planes with axis, r, is:

$\text{[math]}$

When we divide it by λ and substitute $\text{[math]}$ , the equation becomes:

$\text{[math]}$

Plug values if x, y, and z in the above equation to find the value of k:

$\text{[math]}$

This shows that k = 1

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

So, the equation of the plane passing through the points $\text{[math]}$ is $\text{[math]}$ .

Example 2

Find the equation of plane that passes through the point (2, -1, 3) and belongs to the sheaf of planes with its axis on the following line:

$\text{[math]}$

Solution

As we know that the equation of the sheaf of planes with axis, r, is:

$\text{[math]}$

When we divide it by λ and substitute $\text{[math]}$ , the equation becomes:

$\text{[math]}$

Plug values if x, y, and z in the above equation to find the value of k:

$\text{[math]}$

This shows that k = 1

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

So, the equation of the plane passing through the points $\text{[math]}$ is $\text{[math]}$ .

Example 3

Find the equation of plane that passes through the point (1, -2, 3) and belongs to the sheaf of planes with its axis on the following line:

$\text{[math]}$

Solution

As we know that the equation of the sheaf of planes with axis, r, is:

$\text{[math]}$

When we divide it by λ and substitute $\text{[math]}$ , the equation becomes:

$\text{[math]}$

Plug values if x, y, and z in the above equation to find the value of k:

$\text{[math]}$

This shows that k = 1

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

So, the equation of the plane passing through the points $\text{[math]}$ is $\text{[math]}$ .

Example 4

Find the equation of plane that passes through the point (-2, 4, 1) and belongs to the sheaf of planes with its axis on the following line:

$\text{[math]}$

Solution

As we know that the equation of the sheaf of planes with axis, r, is:

$\text{[math]}$

When we divide it by λ and substitute $\text{[math]}$ , the equation becomes:

$\text{[math]}$

Plug values if x, y, and z in the above equation to find the value of k:

$\text{[math]}$

This shows that k = 1

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

So, the equation of the plane passing through the points $\text{[math]}$ is $\text{[math]}$ .

Example 5

Find the equation of plane that passes through the point (-1, -5, 2) and belongs to the sheaf of planes with its axis on the following line:

$\text{[math]}$

Solution

As we know that the equation of the sheaf of planes with axis, r, is:

$\text{[math]}$

When we divide it by λ and substitute $\text{[math]}$ , the equation becomes:

$\text{[math]}$

Plug values if x, y, and z in the above equation to find the value of k:

$\text{[math]}$

This shows that k = 1

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

So, the equation of the plane passing through the points $\text{[math]}$ is $\text{[math]}$ .

Example 6

Find the equation of plane that passes through the point (2, 4, 6) and belongs to the sheaf of planes with its axis on the following line:

$\text{[math]}$

Solution

As we know that the equation of the sheaf of planes with axis, r, is:

$\text{[math]}$

When we divide it by λ and substitute $\text{[math]}$ , the equation becomes:

$\text{[math]}$

Plug values if x, y, and z in the above equation to find the value of k:

$\text{[math]}$

This shows that k = 1

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

So, the equation of the plane passing through the points $\text{[math]}$ is $\text{[math]}$ .

Parallel Sheaf of Planes

Figure 2 - 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:

$\text{[math]}$ , where $\text{[math]}$

Example

Find the equation of the plane that passes through the point (3, −1, 2) and is parallel to $\text{[math]}$ .

Solution

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

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Sheaf of Planes

Finding the Equation of the Sheaf of Planes

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Example 5

Solution

Example 6

Solution

Parallel Sheaf of Planes

Example

Solution

Theory

Irregular Polygons

Regular Pentagons

Points, Lines and Planes

Quadrilaterals and Regular Polygons

Rhombus

Triangle

Diagonals of a Polygon

Concentric Circles

Area and Perimeter of a Triangle

Circular Sectors

Regular Polygons

Similar Triangles

Angles of a Polygon

Parallelograms

Orthocenter, Centroid, Circumcenter and Incenter of a Triangle

Trapezoids

Circumference

Equilateral Triangles

Circles

Quadrilaterals

Circular Segments

Angle Bisectors

Coplanar

Angles of the Triangle

Angles

Apothems

Intersection of Three Planes

Height of a Polygon

Polygon

Plane Equation

Line Segments

Points

Medians of a Triangle

Star Polygons

Inscribed Polygons

Sheaf of Planes

Lune of Hippocrates

Area and Perimeter of Polygons

Circumscribed Polygons

Polygons

Rectangle

Right Triangle

Lines

Regular Hexagons

Plane

Triangles

Rhomboid

Sides of a Polygon

Perpendicular Bisectors

Intersection of Two Planes

Squares

Formulas

Circle Formulas

Plane Formulas

Geometry Formulas

Triangle Formulas

Area and Perimeter Formulas

Area Formulas

Perimeter Formulas

Exercises

Rectangle Problems

Trapezoid Problems