# Vector Operations: Dot Products## Problem StatementCompute $ \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

A vector is a quantity that has both magnitude and direction. It can be also defined as the…

Answered: v•u -2 Compute u· u, v•u, and using the…

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

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Question

# 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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