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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![### Vector Operations
Given vectors:
\[ \mathbf{a} = \langle 4, 0, -2 \rangle \]
\[ \mathbf{b} = \langle 2, -2, 3 \rangle \]
Compute the following:
1. **Vector Addition:**
\[ \mathbf{a} + \mathbf{b} = ( \_\_ , \_\_ , \_\_ ) \]
2. **Vector Subtraction:**
\[ \mathbf{a} - \mathbf{b} = ( \_\_ , \_\_ , \_\_ ) \]
3. **Scalar Multiplication (2 times vector a):**
\[ 2\mathbf{a} = ( \_\_ , \_\_ , \_\_ ) \]
4. **Linear Combination (3 times vector a plus 4 times vector b):**
\[ 3\mathbf{a} + 4\mathbf{b} = ( \_\_ , \_\_ , \_\_ ) \]
5. **Magnitude of vector a:**
\[ |\mathbf{a}| = \_\_ \]
### Explanation:
- **Vector Addition** \( \mathbf{a} + \mathbf{b} \):
To add two vectors, sum their corresponding components:
\[ \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle \]
- **Vector Subtraction** \( \mathbf{a} - \mathbf{b} \):
To subtract vector b from vector a, subtract their corresponding components:
\[ \mathbf{a} - \mathbf{b} = \langle a_1 - b_1, a_2 - b_2, a_3 - b_3 \rangle \]
- **Scalar Multiplication** \( 2\mathbf{a} \):
To multiply a vector by a scalar, multiply each component of the vector by the scalar:
\[ 2\mathbf{a} = \langle 2 \cdot a_1, 2 \cdot a_2, 2 \cdot a_3 \rangle \]
- **Linear Combination** \( 3\mathbf{a} +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa77ad668-7ef6-46eb-bde1-e7c4f652c232%2F2b019dbd-aede-40fa-8839-3755fbec1232%2Fau8qx89_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Vector Operations
Given vectors:
\[ \mathbf{a} = \langle 4, 0, -2 \rangle \]
\[ \mathbf{b} = \langle 2, -2, 3 \rangle \]
Compute the following:
1. **Vector Addition:**
\[ \mathbf{a} + \mathbf{b} = ( \_\_ , \_\_ , \_\_ ) \]
2. **Vector Subtraction:**
\[ \mathbf{a} - \mathbf{b} = ( \_\_ , \_\_ , \_\_ ) \]
3. **Scalar Multiplication (2 times vector a):**
\[ 2\mathbf{a} = ( \_\_ , \_\_ , \_\_ ) \]
4. **Linear Combination (3 times vector a plus 4 times vector b):**
\[ 3\mathbf{a} + 4\mathbf{b} = ( \_\_ , \_\_ , \_\_ ) \]
5. **Magnitude of vector a:**
\[ |\mathbf{a}| = \_\_ \]
### Explanation:
- **Vector Addition** \( \mathbf{a} + \mathbf{b} \):
To add two vectors, sum their corresponding components:
\[ \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle \]
- **Vector Subtraction** \( \mathbf{a} - \mathbf{b} \):
To subtract vector b from vector a, subtract their corresponding components:
\[ \mathbf{a} - \mathbf{b} = \langle a_1 - b_1, a_2 - b_2, a_3 - b_3 \rangle \]
- **Scalar Multiplication** \( 2\mathbf{a} \):
To multiply a vector by a scalar, multiply each component of the vector by the scalar:
\[ 2\mathbf{a} = \langle 2 \cdot a_1, 2 \cdot a_2, 2 \cdot a_3 \rangle \]
- **Linear Combination** \( 3\mathbf{a} +
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