1 4 Compute -u, where u = u•u 3 1 u•u (Type an integer or simplified fraction for each matrix element.)
1 4 Compute -u, where u = u•u 3 1 u•u (Type an integer or simplified fraction for each matrix element.)
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
![### Problem Description
Compute \(\frac{1}{\mathbf{u} \cdot \mathbf{u}}\), where \(\mathbf{u} = \begin{bmatrix} -4 \\ 3 \end{bmatrix}\).
### Mathematical Expression
\[
\frac{1}{\mathbf{u} \cdot \mathbf{u}}
\]
### Solution Process
To solve this, follow these steps:
1. **Compute the dot product** \(\mathbf{u} \cdot \mathbf{u}\).
For \(\mathbf{u} = \begin{bmatrix} -4 \\ 3 \end{bmatrix}\), the dot product \(\mathbf{u} \cdot \mathbf{u}\) is calculated as follows:
\[
\mathbf{u} \cdot \mathbf{u} = (-4)^2 + (3)^2 = 16 + 9 = 25
\]
2. **Compute the reciprocal** of this value:
\[
\frac{1}{\mathbf{u} \cdot \mathbf{u}} = \frac{1}{25}
\]
\[
\frac{1}{\mathbf{u} \cdot \mathbf{u}} = \boxed{\frac{1}{25}}
\]
### Final Answer
\[
\frac{1}{\mathbf{u} \cdot \mathbf{u}} = \frac{1}{25}
\]
(Type an integer or simplified fraction for each matrix element.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f9df24d-bd71-4c3f-9ec7-f4cd302c8c7d%2F2857964b-d73f-4c0d-938d-078799a731ff%2Fknrbk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Description
Compute \(\frac{1}{\mathbf{u} \cdot \mathbf{u}}\), where \(\mathbf{u} = \begin{bmatrix} -4 \\ 3 \end{bmatrix}\).
### Mathematical Expression
\[
\frac{1}{\mathbf{u} \cdot \mathbf{u}}
\]
### Solution Process
To solve this, follow these steps:
1. **Compute the dot product** \(\mathbf{u} \cdot \mathbf{u}\).
For \(\mathbf{u} = \begin{bmatrix} -4 \\ 3 \end{bmatrix}\), the dot product \(\mathbf{u} \cdot \mathbf{u}\) is calculated as follows:
\[
\mathbf{u} \cdot \mathbf{u} = (-4)^2 + (3)^2 = 16 + 9 = 25
\]
2. **Compute the reciprocal** of this value:
\[
\frac{1}{\mathbf{u} \cdot \mathbf{u}} = \frac{1}{25}
\]
\[
\frac{1}{\mathbf{u} \cdot \mathbf{u}} = \boxed{\frac{1}{25}}
\]
### Final Answer
\[
\frac{1}{\mathbf{u} \cdot \mathbf{u}} = \frac{1}{25}
\]
(Type an integer or simplified fraction for each matrix element.)
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