/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q5E The matrix[鈭�0.8鈭�0.60.6鈭�0.8... [FREE SOLUTION] | 91影视

91影视

Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology
Features
Features

Discover all of these amazing features with a free account.

Flashcards

91影视 AI

Notes

Study Plans

Study Sets
What鈥檚 new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
91影视
Discover

All the hacks around your studies and career - in one place.

Find a degree

Magazine
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

The largest student job board with the most exciting opportunities.

Verified student deals from top brands.

Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q5E (page 71)

The matrix
$[\begin{matrix} 鈭� 0 .8 & 鈭� 0 .6 \\ 0 .6 & 鈭� 0 .8 \end{matrix}]$
Represents a rotation. Find the angle of rotation (in radians).

Short Answer

Expert verified

The angle of rotation is 2.5 radians approx.

Step by step solution

01

Transformation

Given T is a linear transformation from $R^{2} 鈫� R^{2}$ .

Then we can write the transformation.

$\begin{matrix} T (r \cos 伪, r \sin 伪) = (r \cos (伪 + 胃), r \sin (伪 + 胃)) \\ = (r \cos 伪 \cos 胃鈭� r \sin 伪 \sin 胃, r \sin 胃 \cos 伪 + r \cos 胃 \sin 伪) \\ = [\begin{matrix} \cos 胃 & 鈭� \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}] [\begin{array}{l} r \cos 伪 \\ r \sin 伪 \end{array}] \\ {[T]}_{尾} = [\begin{matrix} \cos 胃 & 鈭� \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}] \end{matrix}$

02

Value of angle

Now, equate the obtained transformation with the given matrix.

$[\begin{matrix} \cos 胃 & 鈭� \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}] = [\begin{matrix} 鈭� 0.8 & 鈭� 0.6 \\ 0.6 & 鈭� 0.8 \end{matrix}]$

Now on to solving the above equations.

$\begin{matrix} \cos 胃 = 鈭� 0.8 \\ 胃 = 2.498 radians \end{matrix}$

Hence, the angle of rotation is 2.5 radians approx.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Is the product of two lower triangular matrices a lower triangular matrix as well? Explain your answer.

If matrices A and B commute, then the formula A²B = BA² must hold.

Question:Consider the matrices of the form $A = (\begin{matrix} a & b \\ b & 鈭� a \end{matrix})$ , where aand bare arbitrary constants. For which values of aand b is A^-1= A?

Consider the transformation Tfrom $R^{2}$ to role="math" localid="1659714471562" $R^{2}$ that rotates any vector $\overset{鈫�}{x}$ through a given angle胃in the counterclockwise direction. Compare this with Exercise 33. You are told that Tis linear. Find the matrix of Tin terms of胃.

Consider a linear transformation Tfrom $R^{2}$ to $R^{2}$ . Suppose that $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ are two arbitrary vectors in $R^{2}$ and that $\overset{鈫�}{x}$ is a third vector whose endpoint is on the line segment connecting the endpoints of $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ . Is the endpoint of the vector $T (\overset{鈫�}{x)}$ necessarily on the line segment connecting the endpoints of $T (\overset{鈫�}{v)}$ and $T (\overset{鈫�}{w)}$ ? Justify your answer.

See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.