Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2}\), and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ x+3 y+z=7, x-5 z=0 $$

Step-by-Step Solution

Answer

The given planes are neither parallel nor perpendicular.

Step 1: Identify coefficients

Rewrite both equations in the form \(ax + by + cz = d\). This gives us the following planes: For the first equation, \(a_1 = 1\), \(b_1 = 3\), and \(c_1 = 1\), and for the second equation: \(a_2 = 1\), \(b_2 = 0\), and \(c_2 = -5\).

Step 2: Check if planes are parallel

To check if the planes are parallel, there should exist a constant \(k\) such that \(a_1 = k a_2\), \(b_1 = k b_2\) , and \(c_1 = k c_2\). Comparing coefficients, there is no constant \(k\) for which these equalities hold true. Therefore, these planes are not parallel.

Step 3: Check if planes are perpendicular

Next, check if the planes are perpendicular. If they are, the condition to satisfy is \(a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\). Substituting the coefficients from the planes gives that \(1*1 + 3*0 + 1*(-5) = -4\), which is not equal to 0. Therefore, the planes are not perpendicular.

Key Concepts

Parallel Planes

Parallel planes are like sheets of paper lying on top of each other without touching. They never intersect. For two planes to be considered parallel, their normal vectors must be proportional. In other words, there exists a nonzero constant \(k\) such that:

• \(a_{1} = k \, a_{2}\)
• \(b_{1} = k \, b_{2}\)
• \(c_{1} = k \, c_{2}\)

This means each of the coefficients of \(x\), \(y\), and \(z\) in one plane is some constant multiple of the corresponding coefficients in the other plane. If no such constant exists, the planes are not parallel.
In our exercise, we tried to find such a constant for the planes given, but found it's impossible for the provided equations. This confirms that the planes are not parallel.

Perpendicular Planes

Perpendicular planes meet each other at a right angle, like the pages in an open book making a 90-degree angle. To determine if two planes are perpendicular, we check the dot product of their normal vectors. This is given by the equation:\[a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0\]If this equation holds true, it means the normal vectors of the planes are orthogonal, confirming that the planes are perpendicular. This condition ensures that the planes intersect at a right angle.
By substituting the coefficients of our planes into the dot product formula, the result was \(-4\) instead of \(0\), indicating the planes are not perpendicular. They do not satisfy the condition of forming a right angle at their intersection.

Plane Equations

The equation of a plane in three-dimensional geometry is typically written as \(ax + by + cz = d\). Here, \(a\), \(b\), and \(c\) are the coefficients that describe the orientation of the plane, while \(d\) represents the position of the plane concerning the origin. These coefficients together form the normal vector \((a, b, c)\), which is crucial for understanding the plane's orientation relative to other planes or lines.
When working with plane equations, it is helpful to:

• Identify and compare the coefficients to determine geometric relationships like parallelism or perpendicularity.
• Rearrange equations if needed to standardize the comparison process.

In the exercise, we began by rewriting the plane equations to a comparable form and then analyzed their geometrical relationship by focusing on these coefficients. Recognizing the importance of these coefficients helps in exploring various spatial relationships between planes.