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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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![### Linear Algebra Vector Operations
In this section, we explore some fundamental operations involving vectors. Let’s consider the following vectors for our examples:
\[ \mathbf{u} = \begin{bmatrix}
-1 \\
2
\end{bmatrix} , \quad \mathbf{v} = \begin{bmatrix}
2 \\
3
\end{bmatrix} , \quad \mathbf{w} = \begin{bmatrix}
3 \\
-1 \\
-5
\end{bmatrix} , \quad \mathbf{x} = \begin{bmatrix}
6 \\
-2 \\
3
\end{bmatrix} \]
We will examine and compute the following operations:
1. **Dot Product Operations**:
- \( \mathbf{u} \cdot \mathbf{u} \)
- \( \mathbf{v} \cdot \mathbf{u} \)
- \( \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \)
2. **Dot Product with Different Vectors**:
- \( \mathbf{w} \cdot \mathbf{w} \)
- \( \mathbf{x} \cdot \mathbf{w} \)
- \( \frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \)
3. **Scalar Multiplications**:
- \( \frac{1}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w} \)
4. **Projection of \( \mathbf{u} \) onto Itself**:
- \( \frac{1}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} \)
5. **Projection of \( \mathbf{u} \) onto \( \mathbf{v} \)**:
- \( \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v} \)
### Detailed Explanations
- **Dot Product**: The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a} \cd](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324f16c4-c4e5-4b06-b074-f261b39e021a%2F81a08481-90ad-4d79-9c08-d390cdc86f1f%2Fvsm26z_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Vector Operations
In this section, we explore some fundamental operations involving vectors. Let’s consider the following vectors for our examples:
\[ \mathbf{u} = \begin{bmatrix}
-1 \\
2
\end{bmatrix} , \quad \mathbf{v} = \begin{bmatrix}
2 \\
3
\end{bmatrix} , \quad \mathbf{w} = \begin{bmatrix}
3 \\
-1 \\
-5
\end{bmatrix} , \quad \mathbf{x} = \begin{bmatrix}
6 \\
-2 \\
3
\end{bmatrix} \]
We will examine and compute the following operations:
1. **Dot Product Operations**:
- \( \mathbf{u} \cdot \mathbf{u} \)
- \( \mathbf{v} \cdot \mathbf{u} \)
- \( \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \)
2. **Dot Product with Different Vectors**:
- \( \mathbf{w} \cdot \mathbf{w} \)
- \( \mathbf{x} \cdot \mathbf{w} \)
- \( \frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \)
3. **Scalar Multiplications**:
- \( \frac{1}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w} \)
4. **Projection of \( \mathbf{u} \) onto Itself**:
- \( \frac{1}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} \)
5. **Projection of \( \mathbf{u} \) onto \( \mathbf{v} \)**:
- \( \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v} \)
### Detailed Explanations
- **Dot Product**: The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a} \cd
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