Determine the angle between each of the following pairs of planes. (a) 3x + 2y - 7z = 4 and (2, 4,-5) (x-2, y - 1, z +9) = 0. (b) 4r + 2z+ y = 1 and -2z+x - 7y = 3.
Determine the angle between each of the following pairs of planes. (a) 3x + 2y - 7z = 4 and (2, 4,-5) (x-2, y - 1, z +9) = 0. (b) 4r + 2z+ y = 1 and -2z+x - 7y = 3.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Can you please help me with this problem and can you do a step by step on how to solve both of these problems thank u
![**Title: Determining the Angle Between Pairs of Planes**
When working with planes in a three-dimensional space, it is often useful to determine the angle between them. This lesson will guide you through how to find the angle between two pairs of planes using their respective equations.
**Problem Statement**:
Determine the angle between each of the following pairs of planes.
**(a) Pair of Planes:**
1. Plane 1: \( 3x + 2y - 7z = 4 \)
2. Plane 2: \( \langle 2, 4, -5 \rangle \cdot \langle x - 2, y - 1, z + 9 \rangle = 0 \)
**(b) Pair of Planes:**
1. Plane 1: \( 4x + 2z + y = 1 \)
2. Plane 2: \( -2z + x - 7y = 3 \)
**Method**:
To determine the angle between two planes, we need to find the angle between their normal vectors. The normal vector to a plane of the form \( ax + by + cz = d \) is \( \langle a, b, c \rangle \).
1. Identify the normal vectors from the given plane equations.
2. Use the dot product formula to find the cosine of the angle between the normal vectors:
\[
\cos \theta = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|}
\]
where \( \mathbf{n}_1 \) and \( \mathbf{n}_2 \) are the normal vectors of the planes, and \( \| \mathbf{n} \| \) denotes the magnitude of vector \( \mathbf{n} \).
3. Compute the angle \( \theta \) using:
\[
\theta = \arccos \left( \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|} \right)
\]
Let's apply this method to each pair of planes.
**Detailed Solutions**:
**(a) Pair of Planes:**
1. For the plane \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c5a2bc3-e88c-4d18-8e0f-eea96fe50ab5%2Fbba7018f-b8ee-414f-a007-ce10a5c0da8c%2Fzvoi4gc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Determining the Angle Between Pairs of Planes**
When working with planes in a three-dimensional space, it is often useful to determine the angle between them. This lesson will guide you through how to find the angle between two pairs of planes using their respective equations.
**Problem Statement**:
Determine the angle between each of the following pairs of planes.
**(a) Pair of Planes:**
1. Plane 1: \( 3x + 2y - 7z = 4 \)
2. Plane 2: \( \langle 2, 4, -5 \rangle \cdot \langle x - 2, y - 1, z + 9 \rangle = 0 \)
**(b) Pair of Planes:**
1. Plane 1: \( 4x + 2z + y = 1 \)
2. Plane 2: \( -2z + x - 7y = 3 \)
**Method**:
To determine the angle between two planes, we need to find the angle between their normal vectors. The normal vector to a plane of the form \( ax + by + cz = d \) is \( \langle a, b, c \rangle \).
1. Identify the normal vectors from the given plane equations.
2. Use the dot product formula to find the cosine of the angle between the normal vectors:
\[
\cos \theta = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|}
\]
where \( \mathbf{n}_1 \) and \( \mathbf{n}_2 \) are the normal vectors of the planes, and \( \| \mathbf{n} \| \) denotes the magnitude of vector \( \mathbf{n} \).
3. Compute the angle \( \theta \) using:
\[
\theta = \arccos \left( \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|} \right)
\]
Let's apply this method to each pair of planes.
**Detailed Solutions**:
**(a) Pair of Planes:**
1. For the plane \(
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