Q2E Find the matrix of a rotation th... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q2E (page 71)

Find the matrix of a rotation through an angle of $60^{o}$ in the counterclockwise direction.

Short Answer

Expert verified

Matrix for T will be $\frac{1}{2} [\begin{matrix} 1 & 鈭� \sqrt{3} \\ \sqrt{3} & 1 \end{matrix}]$ .

Step by step solution

Transformation

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

Let $尾$ be any point of $R^{2}$ .

Then we can write the transformation.

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

Value of angle

We have given the angle is of $60^{鈭�}$ ,so put the value of angle in above equation we will get.

$\begin{matrix} {[T]}_{尾} = [\begin{matrix} \cos 60^{鈭�} & 鈭� \sin 60^{鈭�} \\ \sin 60^{鈭�} & \cos 60^{鈭�} \end{matrix}] \\ = [\begin{matrix} \frac{1}{2} & 鈭� \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{matrix}] \\ = \frac{1}{2} [\begin{matrix} 1 & 鈭� \sqrt{3} \\ \sqrt{3} & 1 \end{matrix}] \end{matrix}$

Hence, the matrix for T will be $\frac{1}{2} [\begin{matrix} 1 & 鈭� \sqrt{3} \\ \sqrt{3} & 1 \end{matrix}]$ .

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the matrix of a rotation through an angle of $60^{o}$ in the counterclockwise direction.

Short Answer

Step by step solution

Transformation

Value of angle

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Geometry

Logic and Functions

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

91影视

Find the matrix of a rotation through an angle of 60oin the counterclockwise direction.

Short Answer

Step by step solution

Transformation

Value of angle

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Geometry

Logic and Functions

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find the matrix of a rotation through an angle of $60^{o}$ in the counterclockwise direction.