Problem 25 - Calculus Early Transcendentals

Question

Determine if the following pairs of planes are parallel, orthogonal, or neither parallel nor orthogonal. $$x+y+4 z=10 \text { and }-x-3 y+z=10$$

Step-by-Step Solution

Answer

Answer: The planes are orthogonal.

Step 1: Identify the normal vectors

We start by identifying the normal vectors of the two given planes: For the first plane, \(x+y+4z=10\), the normal vector is \(\mathbf{n}_1 = \begin{pmatrix}1\\1\\4\end{pmatrix}\). For the second plane, \(-x-3y+z=10\), the normal vector is \(\mathbf{n}_2 = \begin{pmatrix}-1\\-3\\1\end{pmatrix}\).

Step 2: Check if the normal vectors are parallel

To check if these normal vectors are parallel, we need to see if one is a scalar multiple of the other. This means there exists a scalar \(k\) such that \(\mathbf{n}_1 = k\mathbf{n}_2\). Dividing the components of \(\mathbf{n}_1\) by the corresponding components of \(\mathbf{n}_2\), we obtain: \(\frac{1}{-1} = -1\), \(\frac{1}{-3} = -\frac{1}{3}\), and \(\frac{4}{1} = 4\) Since these values are not equal, the vectors are not parallel.

Step 3: Check if the normal vectors are orthogonal

To check if these normal vectors are orthogonal, we need to find the dot product between the two vectors: \(\mathbf{n}_1 \cdot \mathbf{n}_2 = \begin{pmatrix}1\\1\\4\end{pmatrix} \cdot \begin{pmatrix}-1\\-3\\1\end{pmatrix} = (1)(-1) + (1)(-3) + (4)(1) = -1 -3 +4 = 0\) Since the dot product is 0, the vectors are orthogonal. This means that the two given planes are orthogonal.

Key Concepts

Normal Vectors

In vector analysis, normal vectors play a critical role in understanding the orientation of planes in three-dimensional space. A normal vector is a vector that is perpendicular to the surface at a given point. For a plane defined by the equation \(ax + by + cz = d\), the normal vector is \( \mathbf{n} = \begin{pmatrix}a \ b \ c\end{pmatrix} \). This vector tells us the direction that is perpendicular to the plane.

• The coefficients \(a\), \(b\), and \(c\) in a plane equation directly form the components of the normal vector.
• Given two planes, knowing their normal vectors can help determine the spatial relationship between the planes.

Comprehending the concept of normal vectors allows you to analyze and interpret the orientation of complex geometrical structures in space.

Parallel Planes

Parallel planes are planes that never intersect. This occurs when their normal vectors are proportional, meaning one is a scalar multiple of the other. If \(\mathbf{n}_1 = k\mathbf{n}_2\), the planes are parallel, where \(k\) is a non-zero scalar.

• Checking each component of the normal vectors to see if they maintain the same ratio is critical to identifying parallelism.
• If the ratios are all the same, the vectors (and hence the planes) are parallel.

In the exercise provided, you can see that the vectors \(\begin{pmatrix}1\1\4\end{pmatrix}\) and \(\begin{pmatrix}-1\-3\1\end{pmatrix}\) do not maintain consistent ratios across their components, indicating that these planes are not parallel.

Orthogonal Planes

Orthogonal planes are planes that intersect each other at right angles. Two planes are orthogonal if their normal vectors are orthogonal, which means their dot product is zero. In mathematical terms, two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal if \(\mathbf{a} \cdot \mathbf{b} = 0\).

• The dot product involves multiplying corresponding components of vectors and adding the results.
• When the dot product of two normal vectors is zero, the planes they describe meet at a right angle.

In the original problem, the normal vectors \(\begin{pmatrix}1\1\4\end{pmatrix}\) and \(\begin{pmatrix}-1\-3\1\end{pmatrix}\) were shown to have a dot product of zero. Therefore, the planes are orthogonal.