Distance Between Two Parallel Planes (Cartesian Form)

Understanding the distance between two parallel planes is an important concept in 3D geometry, particularly in fields such as engineering, physics, and computer graphics. In Cartesian form, planes are typically represented by linear equations. Let's delve into the details of how to calculate the distance between two parallel planes.

Representation of a Plane in Cartesian Form

A plane in three-dimensional space can be represented in Cartesian coordinates by the equation:

$$ ax + by + cz + d = 0 $$

where (a), (b), and (c) are the coefficients that determine the orientation of the plane, and (d) is the constant term that affects the position of the plane with respect to the origin.

Conditions for Parallelism

Two planes are parallel if and only if their normal vectors are parallel (or equivalent). The normal vector of a plane ( ax + by + cz + d = 0 ) is given by ( \vec{n} = (a, b, c) ). Therefore, for two planes ( \pi_1: a_1x + b_1y + c_1z + d_1 = 0 ) and ( \pi_2: a_2x + b_2y + c_2z + d_2 = 0 ) to be parallel, the following condition must be met:

$$ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} $$

Distance Between Two Parallel Planes

The distance (D) between two parallel planes ( \pi_1 ) and ( \pi_2 ) can be found using the formula:

$$ D = \frac{|d_2 - d_1|}{\sqrt{a^2 + b^2 + c^2}} $$

where ( (a, b, c) ) are the coefficients of the normal vector, which are the same for both planes since they are parallel.

Example Calculation

Let's consider two parallel planes given by the equations:

• ( \pi_1: 2x - 3y + 6z - 18 = 0 )
• ( \pi_2: 2x - 3y + 6z + 9 = 0 )

To find the distance between these two planes, we use the formula:

$$ D = \frac{|9 - (-18)|}{\sqrt{2^2 + (-3)^2 + 6^2}} = \frac{27}{\sqrt{4 + 9 + 36}} = \frac{27}{\sqrt{49}} = \frac{27}{7} = 3.857 $$

So the distance between the two planes is approximately 3.857 units.

Table of Differences and Important Points

Aspect	Description
Equation of a Plane	A plane in 3D space is represented by ( ax + by + cz + d = 0 ).
Normal Vector	The normal vector ( \vec{n} ) of a plane is given by the coefficients of the plane equation, ( (a, b, c) ).
Parallel Planes	Two planes are parallel if their normal vectors are parallel, i.e., ( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} ).
Distance Formula	The distance between two parallel planes is given by ( D = \frac{
Example	Given ( \pi_1: 2x - 3y + 6z - 18 = 0 ) and ( \pi_2: 2x - 3y + 6z + 9 = 0 ), the distance is ( D = 3.857 ) units.

Conclusion

The distance between two parallel planes is a straightforward calculation once the equations of the planes are known. It is essential to ensure that the planes are indeed parallel by comparing their normal vectors. The distance formula then provides a simple way to find the shortest distance between the planes. This concept is widely used in various applications, including determining the thickness of a slab, analyzing the spacing between layers in a material, and collision detection in computer graphics.