Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
100%
Correct and detailed answer will be Upvoted else downvoted. Thank you
![### Problem Description
**Task:** Cancel the cross product term from the quadratic form
\[ 2x^2 + 10xy + 2y^2 \]
### Detailed Explanation
#### Step 1: Identify the Cross Product Term
In the given quadratic equation:
\[ 2x^2 + 10xy + 2y^2 \]
The term \( 10xy \) is the cross product term.
#### Step 2: Transform the Equation
To cancel the cross product term, we can use a linear transformation. A common approach is to rotate the coordinate system by an angle \( \theta \) to eliminate the \( xy \) term. The transformed coordinates can be found using:
\[ x = x' \cos \theta - y' \sin \theta \]
\[ y = x' \sin \theta + y' \cos \theta \]
#### Step 3: Calculate the Angle
We need to choose an angle \( \theta \) such that:
\[ \tan(2\theta) = \frac{2 \cdot \text{coeff}(xy)}{\text{coeff}(x^2) - \text{coeff}(y^2)} = \frac{2 \cdot 10}{2 - 2} \]
However, the denominator becomes zero in this case, indicating that we should rotate by \( \theta = \frac{\pi}{4} \) or 45 degrees.
#### Step 4: Apply the Transformation
When we apply the rotation \( \theta = 45^\circ \):
\[ x = x' \cos 45^\circ - y' \sin 45^\circ \]
\[ y = x' \sin 45^\circ + y' \cos 45^\circ \]
Since \( \cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2} \), we get:
\[ x = \frac{\sqrt{2}}{2} x' - \frac{\sqrt{2}}{2} y' \]
\[ y = \frac{\sqrt{2}}{2} x' + \frac{\sqrt{2}}{2} y' \]
By substituting these into the original quadratic form and simplifying, we would eliminate the cross product term.
### Conclusion
After applying a coordinate system rotation, the cross product term is effectively cancelled, simplifying the quadratic form to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b914819-4152-4353-8ebc-b14bc0f05ba7%2F7f9f5ab3-7685-4b65-ab34-0630308de7de%2F16m1o8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Description
**Task:** Cancel the cross product term from the quadratic form
\[ 2x^2 + 10xy + 2y^2 \]
### Detailed Explanation
#### Step 1: Identify the Cross Product Term
In the given quadratic equation:
\[ 2x^2 + 10xy + 2y^2 \]
The term \( 10xy \) is the cross product term.
#### Step 2: Transform the Equation
To cancel the cross product term, we can use a linear transformation. A common approach is to rotate the coordinate system by an angle \( \theta \) to eliminate the \( xy \) term. The transformed coordinates can be found using:
\[ x = x' \cos \theta - y' \sin \theta \]
\[ y = x' \sin \theta + y' \cos \theta \]
#### Step 3: Calculate the Angle
We need to choose an angle \( \theta \) such that:
\[ \tan(2\theta) = \frac{2 \cdot \text{coeff}(xy)}{\text{coeff}(x^2) - \text{coeff}(y^2)} = \frac{2 \cdot 10}{2 - 2} \]
However, the denominator becomes zero in this case, indicating that we should rotate by \( \theta = \frac{\pi}{4} \) or 45 degrees.
#### Step 4: Apply the Transformation
When we apply the rotation \( \theta = 45^\circ \):
\[ x = x' \cos 45^\circ - y' \sin 45^\circ \]
\[ y = x' \sin 45^\circ + y' \cos 45^\circ \]
Since \( \cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2} \), we get:
\[ x = \frac{\sqrt{2}}{2} x' - \frac{\sqrt{2}}{2} y' \]
\[ y = \frac{\sqrt{2}}{2} x' + \frac{\sqrt{2}}{2} y' \]
By substituting these into the original quadratic form and simplifying, we would eliminate the cross product term.
### Conclusion
After applying a coordinate system rotation, the cross product term is effectively cancelled, simplifying the quadratic form to
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education