Linear Algebra **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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Answered: If not, why not? 1 6 4 -(0·0·0); 3 1 8…

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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

Linear Algebra

**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:
\[
\left\{ \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 6 \\ 3 \\ 8 \end{bmatrix}, \begin{bmatrix} 4 \\ 1 \\ 6 \end{bmatrix} \right\}
\]

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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