If not, why not? 1 6 4 -(0·0·0); 3 1 8 6 4. What is span ?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Carpet Ride Problem in a Three-Dimensional World**
In this exercise, you are exploring a three-dimensional world with three modes of transportation, represented by vectors:
\[ \vec{v}_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \vec{v}_2 = \begin{bmatrix} 6 \\ 3 \\ 8 \end{bmatrix}, \quad \vec{v}_3 = \begin{bmatrix} 4 \\ 1 \\ 6 \end{bmatrix} \]
You are allowed to use each mode of transportation only once, either in the forward or backward direction, for a fixed amount of time (\(c_1\) on \(\vec{v}_1\), \(c_2\) on \(\vec{v}_2\), \(c_3\) on \(\vec{v}_3\)).
### Tasks:
1. **Determine Time Amounts:**
Find the amounts of time for each mode of transportation (\(c_1\), \(c_2\), and \(c_3\)) required to make a journey that begins and ends at home, or explain why it cannot be done.
2. **Explore Multiple Journey Options:**
Determine if there is more than one combination of times that allows you to return home. If so, describe the possibilities.
3. **Investigate Hidden Locations:**
Consider if there is any location in this 3D world that cannot be reached using the given modes of transportation. Discuss why or why not.
4. **Determine the Span:**
Identify the span of the set of vectors:
\[
\left\{ \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 6 \\ 3 \\ 8 \end{bmatrix}, \begin{bmatrix} 4 \\ 1 \\ 6 \end{bmatrix} \right\}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a8af915-3f7a-4a41-a3ab-e20554fd1f78%2Fb3e26c7f-3389-4a12-9d22-02809badf83c%2F1nxe35h_processed.png&w=3840&q=75)
Transcribed Image Text:**Carpet Ride Problem in a Three-Dimensional World**
In this exercise, you are exploring a three-dimensional world with three modes of transportation, represented by vectors:
\[ \vec{v}_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \vec{v}_2 = \begin{bmatrix} 6 \\ 3 \\ 8 \end{bmatrix}, \quad \vec{v}_3 = \begin{bmatrix} 4 \\ 1 \\ 6 \end{bmatrix} \]
You are allowed to use each mode of transportation only once, either in the forward or backward direction, for a fixed amount of time (\(c_1\) on \(\vec{v}_1\), \(c_2\) on \(\vec{v}_2\), \(c_3\) on \(\vec{v}_3\)).
### Tasks:
1. **Determine Time Amounts:**
Find the amounts of time for each mode of transportation (\(c_1\), \(c_2\), and \(c_3\)) required to make a journey that begins and ends at home, or explain why it cannot be done.
2. **Explore Multiple Journey Options:**
Determine if there is more than one combination of times that allows you to return home. If so, describe the possibilities.
3. **Investigate Hidden Locations:**
Consider if there is any location in this 3D world that cannot be reached using the given modes of transportation. Discuss why or why not.
4. **Determine the Span:**
Identify the span of the set of vectors:
\[
\left\{ \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 6 \\ 3 \\ 8 \end{bmatrix}, \begin{bmatrix} 4 \\ 1 \\ 6 \end{bmatrix} \right\}
\]
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