Find C1 and C₂ for the inner product of R2 given by (u, v) = C₁U₁V₁ + C₂U₂V₂ such that the graph represents a unit circle as shown. (C₁, C₂) = (₂) = ( [
Find C1 and C₂ for the inner product of R2 given by (u, v) = C₁U₁V₁ + C₂U₂V₂ such that the graph represents a unit circle as shown. (C₁, C₂) = (₂) = ( [
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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![**Finding \( c_1 \) and \( c_2 \) for the Inner Product in \( \mathbb{R}^2 \)**
In this exercise, we aim to find the constants \( c_1 \) and \( c_2 \) for the inner product in \( \mathbb{R}^2 \) defined by:
\[ \langle \mathbf{u}, \mathbf{v} \rangle = c_1 u_1 v_1 + c_2 u_2 v_2 \]
such that the graph represents a unit circle as shown in the diagram.
**Equation for Inner Product:**
\[ \langle \mathbf{u}, \mathbf{v} \rangle = c_1 u_1 v_1 + c_2 u_2 v_2 \]
**Condition:**
The graph should represent a unit circle.
**Solution:**
\[
(c_1, c_2) = \left( \boxed{1, 16} \right)
\]
**Graph Description:**
The graph shown is an ellipse centered at the origin with the equation \( ||\mathbf{u}|| = 1 \). The ellipse intersects the x-axis at points \( x = 4 \) and \( x = -4 \), and the y-axis at points \( y = 1 \) and \( y = -1 \).
**Axes Features:**
- The x-axis ranges from \( -4 \) to \( 4 \).
- The y-axis ranges from \( -4 \) to \( 4 \).
The ellipse represents a transformation of the unit circle under a different inner product defined by \( c_1 \) and \( c_2 \).
**Purpose:**
This configuration will help understand how changing the constants \( c_1 \) and \( c_2 \) in the inner product affects the shape of the unit circle in \( \mathbb{R}^2 \).
**Graph Representation:**
```plaintext
y
|
4 | *
3 |
2 |
1 | * -------- \||/ -------- *
0 |-------------------------------------> x
-1 | *
-2 |
-3 |
-4 |
```
The "*" points indicate the intersection points of the ellipse with the axes.
By adjusting \( c_1 \) and \( c_2 \) as](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F031412a4-ce24-4200-b630-467aaa8d4c51%2F2511ef6f-9089-4382-a4b2-c51a5977760f%2F9dd6rit_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding \( c_1 \) and \( c_2 \) for the Inner Product in \( \mathbb{R}^2 \)**
In this exercise, we aim to find the constants \( c_1 \) and \( c_2 \) for the inner product in \( \mathbb{R}^2 \) defined by:
\[ \langle \mathbf{u}, \mathbf{v} \rangle = c_1 u_1 v_1 + c_2 u_2 v_2 \]
such that the graph represents a unit circle as shown in the diagram.
**Equation for Inner Product:**
\[ \langle \mathbf{u}, \mathbf{v} \rangle = c_1 u_1 v_1 + c_2 u_2 v_2 \]
**Condition:**
The graph should represent a unit circle.
**Solution:**
\[
(c_1, c_2) = \left( \boxed{1, 16} \right)
\]
**Graph Description:**
The graph shown is an ellipse centered at the origin with the equation \( ||\mathbf{u}|| = 1 \). The ellipse intersects the x-axis at points \( x = 4 \) and \( x = -4 \), and the y-axis at points \( y = 1 \) and \( y = -1 \).
**Axes Features:**
- The x-axis ranges from \( -4 \) to \( 4 \).
- The y-axis ranges from \( -4 \) to \( 4 \).
The ellipse represents a transformation of the unit circle under a different inner product defined by \( c_1 \) and \( c_2 \).
**Purpose:**
This configuration will help understand how changing the constants \( c_1 \) and \( c_2 \) in the inner product affects the shape of the unit circle in \( \mathbb{R}^2 \).
**Graph Representation:**
```plaintext
y
|
4 | *
3 |
2 |
1 | * -------- \||/ -------- *
0 |-------------------------------------> x
-1 | *
-2 |
-3 |
-4 |
```
The "*" points indicate the intersection points of the ellipse with the axes.
By adjusting \( c_1 \) and \( c_2 \) as
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