Determine Whether The Planes Are Parallel Perpendicular Or Neither

Determining Whether Planes are Parallel, Perpendicular, or Neither

Determining the relationship between two planes in three-dimensional space—whether they are parallel, perpendicular, or neither—is a fundamental concept in linear algebra and vector calculus. Understanding this relationship is crucial in various fields, including computer graphics, physics, and engineering. This comprehensive guide will explore the methods and underlying principles involved in determining the relative orientation of two planes.

Understanding Plane Equations

Before we delve into the methods, let's refresh our understanding of plane equations. A plane in three-dimensional space can be represented by a linear equation of the form:

Ax + By + Cz + D = 0

where A, B, and C are the components of a vector n = <A, B, C> which is normal to the plane, and D is a constant. The normal vector is perpendicular to every vector lying in the plane.

Different forms of the plane equation exist, including the point-normal form and the intercept form. However, the general form (Ax + By + Cz + D = 0) is the most convenient for comparing two planes.

Method 1: Comparing Normal Vectors

The most straightforward method for determining the relationship between two planes is by analyzing their normal vectors. Let's consider two planes:

• Plane 1: A₁x + B₁y + C₁z + D₁ = 0 (with normal vector n₁ = <A₁, B₁, C₁>)
• Plane 2: A₂x + B₂y + C₂z + D₂ = 0 (with normal vector n₂ = <A₂, B₂, C₂>)

Parallel Planes

Two planes are parallel if and only if their normal vectors are parallel. This means that one normal vector is a scalar multiple of the other:

n₁ = k * n₂ where k is a non-zero scalar.

In other words:

• A₁ = kA₂
• B₁ = kB₂
• C₁ = kC₂

If this condition holds true, the planes are parallel. If, additionally, D₁/k ≠ D₂, the planes are parallel and distinct. If D₁/k = D₂, the planes are coincident (they are essentially the same plane).

Perpendicular Planes

Two planes are perpendicular if and only if their normal vectors are perpendicular. The dot product of two perpendicular vectors is zero. Therefore, the planes are perpendicular if:

n₁ • n₂ = 0

This translates to:

A₁A₂ + B₁B₂ + C₁C₂ = 0

If this condition is met, the planes are perpendicular.

Neither Parallel Nor Perpendicular

If neither of the above conditions is satisfied (the normal vectors are neither parallel nor perpendicular), then the planes are neither parallel nor perpendicular. They intersect at an angle other than 0° or 90°.

Method 2: Using the Angle Between Planes

A more general approach involves calculating the angle between the two planes. The angle θ between two planes is defined as the angle between their normal vectors. We can use the dot product formula to find this angle:

cos θ = (n₁ • n₂) / (||n₁|| ||n₂||)

where:

• n₁ • n₂ is the dot product of the normal vectors.
• ||n₁|| and ||n₂|| are the magnitudes (lengths) of the normal vectors.

The magnitude of a vector n = <A, B, C> is calculated as:

||n|| = √(A² + B² + C²)

Based on the value of θ:

• θ = 0°: The planes are parallel.
• θ = 90°: The planes are perpendicular.
• 0° < θ < 90°: The planes are neither parallel nor perpendicular.

Example Problems

Let's illustrate these methods with some examples:

Example 1:

• Plane 1: 2x + 3y - z + 5 = 0 (n₁ = <2, 3, -1>)
• Plane 2: 4x + 6y - 2z - 10 = 0 (n₂ = <4, 6, -2>)

Notice that n₂ = 2n₁. Therefore, the planes are parallel. Furthermore, since -10/2 ≠ 5, they are distinct parallel planes.

Example 2:

• Plane 1: x - 2y + z + 3 = 0 (n₁ = <1, -2, 1>)
• Plane 2: 2x + y - z - 1 = 0 (n₂ = <2, 1, -1>)

Let's calculate the dot product: n₁ • n₂ = (1)(2) + (-2)(1) + (1)(-1) = -1 ≠ 0. The planes are not perpendicular.

Let's check for parallelism: There's no scalar k such that <1, -2, 1> = k<2, 1, -1>. Therefore, the planes are neither parallel nor perpendicular.

Example 3:

• Plane 1: 3x + y + 2z - 4 = 0 (n₁ = <3, 1, 2>)
• Plane 2: -6x - 2y - 4z + 1 = 0 (n₂ = <-6, -2, -4>)

Notice that n₂ = -2n₁. Thus, the planes are parallel. Since 1/(-2) ≠ -4, they are distinct parallel planes.

Example 4:

• Plane 1: x + 2y - z = 0 (n₁ = <1, 2, -1>)
• Plane 2: 2x - y + z = 0 (n₂ = <2, -1, 1>)

n₁ • n₂ = (1)(2) + (2)(-1) + (-1)(1) = -1. The planes are not perpendicular.

There's no scalar k satisfying the condition for parallelism.

Let's calculate the angle:

cos θ = (-1) / (√6 * √6) = -1/6

θ = arccos(-1/6) ≈ 99.59°

The angle between the planes is approximately 99.59°. This indicates that the planes are neither parallel nor perpendicular.

Conclusion

Determining whether two planes are parallel, perpendicular, or neither is a crucial skill in various mathematical and scientific disciplines. By analyzing their normal vectors using the methods described above—comparing normal vectors directly or calculating the angle between them—we can confidently classify the relationship between any two given planes. Remember to carefully consider both the direction and magnitude of the normal vectors for accurate results. Practicing with diverse examples will solidify your understanding and enable you to solve similar problems efficiently.