Let -HHI 4 u1 = , u2 uz = 3 3 Compute each of the following. (A graphing calculator is recommended.) (a) u2 · u3 32 (b) ||u1|| V 39 (c) ||3u1 + 2u3|| V955 (d) ||5u1 - 4u2 - u3| V1479
Let -HHI 4 u1 = , u2 uz = 3 3 Compute each of the following. (A graphing calculator is recommended.) (a) u2 · u3 32 (b) ||u1|| V 39 (c) ||3u1 + 2u3|| V955 (d) ||5u1 - 4u2 - u3| V1479
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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![### Vector Operations
Let the vectors be defined as follows:
\[ \mathbf{u_1} = \begin{bmatrix} 5 \\ 0 \\ -1 \\ 3 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} 2 \\ 8 \\ 4 \\ 0 \end{bmatrix}, \quad \mathbf{u_3} = \begin{bmatrix} -4 \\ 0 \\ 3 \\ -3 \end{bmatrix} \]
Compute each of the following. (A graphing calculator is recommended.)
#### (a) Compute \( \mathbf{u_2} \cdot \mathbf{u_3} \)
- **Solution:** \( 32 \) ✔️
#### (b) Compute \( \| \mathbf{u_1} \| \)
- **Solution:** \( \sqrt{39} \) ✔️
#### (c) Compute \( \| 3 \mathbf{u_1} + 2 \mathbf{u_3} \| \)
- **Solution Attempt:** \( \sqrt{955} \) ❌
#### (d) Compute \( \| 5 \mathbf{u_1} - 4 \mathbf{u_2} - \mathbf{u_3} \| \)
- **Solution Attempt:** \( \sqrt{1479} \) ❌
**Explanation of Calculations:**
- **Dot Product:** The dot product is calculated as the sum of the products of corresponding entries of the two sequences of numbers.
- **Magnitude (Norm):** The magnitude of vector \( \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix} \) is calculated by \( \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2} \).
If you're computing (c) or (d), be sure to carefully combine the vectors as specified before calculating the norms correctly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad35cd49-88f1-4759-9e4c-232c0c792d3e%2Fd6b3cdc6-2e0b-4de8-ba54-f76a79a64166%2F5lkhsxv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Vector Operations
Let the vectors be defined as follows:
\[ \mathbf{u_1} = \begin{bmatrix} 5 \\ 0 \\ -1 \\ 3 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} 2 \\ 8 \\ 4 \\ 0 \end{bmatrix}, \quad \mathbf{u_3} = \begin{bmatrix} -4 \\ 0 \\ 3 \\ -3 \end{bmatrix} \]
Compute each of the following. (A graphing calculator is recommended.)
#### (a) Compute \( \mathbf{u_2} \cdot \mathbf{u_3} \)
- **Solution:** \( 32 \) ✔️
#### (b) Compute \( \| \mathbf{u_1} \| \)
- **Solution:** \( \sqrt{39} \) ✔️
#### (c) Compute \( \| 3 \mathbf{u_1} + 2 \mathbf{u_3} \| \)
- **Solution Attempt:** \( \sqrt{955} \) ❌
#### (d) Compute \( \| 5 \mathbf{u_1} - 4 \mathbf{u_2} - \mathbf{u_3} \| \)
- **Solution Attempt:** \( \sqrt{1479} \) ❌
**Explanation of Calculations:**
- **Dot Product:** The dot product is calculated as the sum of the products of corresponding entries of the two sequences of numbers.
- **Magnitude (Norm):** The magnitude of vector \( \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix} \) is calculated by \( \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2} \).
If you're computing (c) or (d), be sure to carefully combine the vectors as specified before calculating the norms correctly.
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