**Title: Computing Matrix Quantities Using Given Vectors****Objective:**To compute a specified quantity using given vectors and represent it in matrix form.**Given:**Vectors $ \mathbf{u} $ and $ \mathbf{v} $ are provided as follows:\[ \mathbf{u} = \begin{bmatrix} -5 \\ 1 \end{bmatrix} \]\[ \mathbf{v} = \begin{bmatrix} 4 \\ 7 \end{bmatrix} \]**Task:**Compute the quantity:\[ \frac{\mathbf{u \cdot v}}{\mathbf{v \cdot v}} \mathbf{v} \]**Procedure:**1. **Compute $ \mathbf{u \cdot v} $ (Dot Product of $ \mathbf{u} $ and $ \mathbf{v} $):** \[ \mathbf{u \cdot v} = (-5 \cdot 4) + (1 \cdot 7) = -20 + 7 = -13 \]2. **Compute $ \mathbf{v \cdot v} $ (Dot Product of $ \mathbf{v} $ with itself):** \[ \mathbf{v \cdot v} = (4 \cdot 4) + (7 \cdot 7) = 16 + 49 = 65 \]3. **Divide the dot product $ \mathbf{u \cdot v} $ by $ \mathbf{v \cdot v} $:** \[ \frac{\mathbf{u \cdot v}}{\mathbf{v \cdot v}} = \frac{-13}{65} = -\frac{13}{65} = -\frac{1}{5} \]4. **Multiply the result by the vector $ \mathbf{v} $:** \[ \left( -\frac{1}{5} \right) \mathbf{v} = -\frac{1}{5} \begin{bmatrix} 4 \\ 7 \end{bmatrix} = \begin{bmatrix} -\frac{4}{5} \\ -\frac{7}{5} \end{bmatrix} \]**Result:**The computed

This is a problem of vector calculus. We need the concept of dot product of two vectors to solve the…

Answered: -5 4 and v = 7 Compute the quantity…

Math

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-5 4 and v = 7 Compute the quantity using the vectors u = 1 u•v V V•v u•v v = (Simplify your answers.) V•v ロロ

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

**Title: Computing Matrix Quantities Using Given Vectors**

**Objective:**
To compute a specified quantity using given vectors and represent it in matrix form.

**Given:**
Vectors $ \mathbf{u} $ and $ \mathbf{v} $ are provided as follows:
\[ \mathbf{u} = \begin{bmatrix} -5 \\ 1 \end{bmatrix} \]
\[ \mathbf{v} = \begin{bmatrix} 4 \\ 7 \end{bmatrix} \]

**Task:**
Compute the quantity:

\[ \frac{\mathbf{u \cdot v}}{\mathbf{v \cdot v}} \mathbf{v} \]

**Procedure:**
1. **Compute $ \mathbf{u \cdot v} $ (Dot Product of $ \mathbf{u} $ and $ \mathbf{v} $):**
\[
\mathbf{u \cdot v} = (-5 \cdot 4) + (1 \cdot 7) = -20 + 7 = -13
\]

2. **Compute $ \mathbf{v \cdot v} $ (Dot Product of $ \mathbf{v} $ with itself):**
\[
\mathbf{v \cdot v} = (4 \cdot 4) + (7 \cdot 7) = 16 + 49 = 65
\]

3. **Divide the dot product $ \mathbf{u \cdot v} $ by $ \mathbf{v \cdot v} $:**
\[
\frac{\mathbf{u \cdot v}}{\mathbf{v \cdot v}} = \frac{-13}{65} = -\frac{13}{65} = -\frac{1}{5}
\]

4. **Multiply the result by the vector $ \mathbf{v} $:**
\[
\left( -\frac{1}{5} \right) \mathbf{v} = -\frac{1}{5} \begin{bmatrix} 4 \\ 7 \end{bmatrix} = \begin{bmatrix} -\frac{4}{5} \\ -\frac{7}{5} \end{bmatrix}
\]

**Result:**
The computed

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