Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**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](https://content.bartleby.com/qna-images/question/2f9df24d-bd71-4c3f-9ec7-f4cd302c8c7d/634d6faa-82b2-45d4-a547-9d45adc83248/xxzz3ep_processed.jpeg)
Transcribed Image Text:**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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