v•u -2 Compute u· u, v•u, and using the vectors u = and v = 4 7 n.n u•u= - 20 (Simplify your answer.) (Simplify your answer.) V•u= v•u (Type an integer or a simplified fraction.) u•u
v•u -2 Compute u· u, v•u, and using the vectors u = and v = 4 7 n.n u•u= - 20 (Simplify your answer.) (Simplify your answer.) V•u= v•u (Type an integer or a simplified fraction.) u•u
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: Dot Products
## Problem Statement
Compute \( \mathbf{u} \cdot \mathbf{u} \), \( \mathbf{v} \cdot \mathbf{u} \), and \( \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \) using the vectors \( \mathbf{u} = \begin{bmatrix} -2 \\ 4 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} 5 \\ 7 \end{bmatrix} \).
### Given:
\[ \mathbf{u} = \begin{bmatrix} -2 \\ 4 \end{bmatrix} \]
\[ \mathbf{v} = \begin{bmatrix} 5 \\ 7 \end{bmatrix} \]
### Computations:
1. **Compute \( \mathbf{u} \cdot \mathbf{u} \):**
\[ \mathbf{u} \cdot \mathbf{u} = -20 \]
Simplify your answer.
2. **Compute \( \mathbf{v} \cdot \mathbf{u} \):**
\[ \mathbf{v} \cdot \mathbf{u} = \_\_\_\_ \]
Simplify your answer.
3. **Compute \( \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \):**
\[ \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} = \_\_\_\_ \]
Type an integer or a simplified fraction.
### Answer Steps:
* Calculate the dot product \( \mathbf{u} \cdot \mathbf{u} \):
\[
\mathbf{u} \cdot \mathbf{u} = (-2)(-2) + (4)(4) = 4 + 16 = 20
\]
* Calculate the dot product \( \mathbf{v} \cdot \mathbf{u} \):
\[
\mathbf{v} \cdot \mathbf{u} = (5)(-2) + (7)(4) = -10 + 28 = 18
\]
* Calculate](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47d7a0f8-4e66-40ed-ae66-faa7d8f664d3%2F7d5ccd34-3bea-4e77-8ae5-546c2c979848%2Fufqr8gb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Vector Operations: Dot Products
## Problem Statement
Compute \( \mathbf{u} \cdot \mathbf{u} \), \( \mathbf{v} \cdot \mathbf{u} \), and \( \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \) using the vectors \( \mathbf{u} = \begin{bmatrix} -2 \\ 4 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} 5 \\ 7 \end{bmatrix} \).
### Given:
\[ \mathbf{u} = \begin{bmatrix} -2 \\ 4 \end{bmatrix} \]
\[ \mathbf{v} = \begin{bmatrix} 5 \\ 7 \end{bmatrix} \]
### Computations:
1. **Compute \( \mathbf{u} \cdot \mathbf{u} \):**
\[ \mathbf{u} \cdot \mathbf{u} = -20 \]
Simplify your answer.
2. **Compute \( \mathbf{v} \cdot \mathbf{u} \):**
\[ \mathbf{v} \cdot \mathbf{u} = \_\_\_\_ \]
Simplify your answer.
3. **Compute \( \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \):**
\[ \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} = \_\_\_\_ \]
Type an integer or a simplified fraction.
### Answer Steps:
* Calculate the dot product \( \mathbf{u} \cdot \mathbf{u} \):
\[
\mathbf{u} \cdot \mathbf{u} = (-2)(-2) + (4)(4) = 4 + 16 = 20
\]
* Calculate the dot product \( \mathbf{v} \cdot \mathbf{u} \):
\[
\mathbf{v} \cdot \mathbf{u} = (5)(-2) + (7)(4) = -10 + 28 = 18
\]
* Calculate
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