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...
Related questions
Question
![### Vector Operations
Given two vectors:
\[
\vec{a} = \begin{pmatrix} 6 \\ 7 \\ -4 \end{pmatrix} \quad \text{and} \quad \vec{b} = \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix}
\]
We need to find the value of the expression:
\[
(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})
\]
To solve this, follow these steps:
1. **Calculate \(\vec{a} - \vec{b}\):**
\[
\vec{a} - \vec{b} = \begin{pmatrix} 6 \\ 7 \\ -4 \end{pmatrix} - \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix} = \begin{pmatrix} 6 - 3 \\ 7 - 2 \\ -4 - 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 5 \\ -6 \end{pmatrix}
\]
2. **Calculate \(\vec{a} + \vec{b}\):**
\[
\vec{a} + \vec{b} = \begin{pmatrix} 6 \\ 7 \\ -4 \end{pmatrix} + \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix} = \begin{pmatrix} 6 + 3 \\ 7 + 2 \\ -4 + 2 \end{pmatrix} = \begin{pmatrix} 9 \\ 9 \\ -2 \end{pmatrix}
\]
3. **Compute the cross product:**
Let \(\vec{u} = \vec{a} - \vec{b} = \begin{pmatrix} 3 \\ 5 \\ -6 \end{pmatrix}\) and \(\vec{v} = \vec{a} + \vec{b} = \begin{pmatrix} 9 \\ 9 \\ -2 \end{pmatrix}\).
The cross product \(\vec{u} \times \vec{v}\) is computed as follows:
\[
\vec{u} \times \vec{v} =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55d3466e-6a31-4aff-972b-e33941523058%2Fa03006ee-d65c-4200-8317-73701475b345%2Fo676kl_processed.png&w=3840&q=75)
Transcribed Image Text:### Vector Operations
Given two vectors:
\[
\vec{a} = \begin{pmatrix} 6 \\ 7 \\ -4 \end{pmatrix} \quad \text{and} \quad \vec{b} = \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix}
\]
We need to find the value of the expression:
\[
(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})
\]
To solve this, follow these steps:
1. **Calculate \(\vec{a} - \vec{b}\):**
\[
\vec{a} - \vec{b} = \begin{pmatrix} 6 \\ 7 \\ -4 \end{pmatrix} - \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix} = \begin{pmatrix} 6 - 3 \\ 7 - 2 \\ -4 - 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 5 \\ -6 \end{pmatrix}
\]
2. **Calculate \(\vec{a} + \vec{b}\):**
\[
\vec{a} + \vec{b} = \begin{pmatrix} 6 \\ 7 \\ -4 \end{pmatrix} + \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix} = \begin{pmatrix} 6 + 3 \\ 7 + 2 \\ -4 + 2 \end{pmatrix} = \begin{pmatrix} 9 \\ 9 \\ -2 \end{pmatrix}
\]
3. **Compute the cross product:**
Let \(\vec{u} = \vec{a} - \vec{b} = \begin{pmatrix} 3 \\ 5 \\ -6 \end{pmatrix}\) and \(\vec{v} = \vec{a} + \vec{b} = \begin{pmatrix} 9 \\ 9 \\ -2 \end{pmatrix}\).
The cross product \(\vec{u} \times \vec{v}\) is computed as follows:
\[
\vec{u} \times \vec{v} =
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning