Q30P For the matrices G聽and K in聽(7... [FREE SOLUTION]

Chapter 3: Q30P (page 131)

For the matrices Gand K in $(7.21)$ , find the matrices $R = GK$ and $S = KG$ . Note that $R 鈮� S$ . (In3dimensions, rotations about two different axes do not in general commute.) Find what geometric transformations are produced byRand, S.

Short Answer

Expert verified

The geometric transformations are that the rotation axis of R its z axis and the rotation angle is $90^{鈭�}$ and that of S is x axis and the rotation angle is $90^{鈭�}$

Step by step solution

Matrix Transformations

Matrix transformation can be of two types: rotation and reflection only for square matrices. When the determinant value of the matrix is1then it is termed rotation and if the value is -1 then, it is a reflection.

Given Parameters

$G = (\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}) and K = (\begin{matrix} 0 & 0 & 1 \\ - 1 & 0 & 0 \\ 0 & - 1 & 0 \end{matrix})$

Calculating R and S

$R = GK = (\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}) (\begin{matrix} 0 & 0 & 1 \\ - 1 & 0 & 0 \\ 0 & - 1 & 0 \end{matrix}) = (\begin{matrix} 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) S = KG = (\begin{matrix} 0 & 0 & 1 \\ - 1 & 0 & 0 \\ 0 & - 1 & 0 \end{matrix}) (\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}) = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix})$

Therefore, $R 鈮� S$ and G ,K do not commute.

Determinants of R and S need to be calculated.

$d e t R = 1$ and $d e t S = 1$ hence both matrices are rotations.

The rotation axis of R its z axis and the rotation angle is $90^{鈭�}$ and that of S is x axis and the rotation angle is $90^{鈭�}$ .

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 Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

For the matrices Gand K in $(7.21)$ , find the matrices $R = GK$ and $S = KG$ . Note that $R 鈮� S$ . (In3dimensions, rotations about two different axes do not in general commute.) Find what geometric transformations are produced byRand, S.

Short Answer

Step by step solution

Matrix Transformations

Given Parameters

Calculating R and S

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Geometrical and Physical Optics

Dynamics

Fluids

Particle Model of Matter

Famous Physicists

Study anywhere. Anytime. Across all devices.

91影视

For the matrices Gand K in(7.21), find the matricesR=GK andS=KG. Note thatR鈮�S. (In3dimensions, rotations about two different axes do not in general commute.) Find what geometric transformations are produced byRand, S.

Short Answer

Step by step solution

Matrix Transformations

Given Parameters

Calculating R and S

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Geometrical and Physical Optics

Dynamics

Fluids

Particle Model of Matter

Famous Physicists

Study anywhere. Anytime. Across all devices.

For the matrices Gand K in $(7.21)$ , find the matrices $R = GK$ and $S = KG$ . Note that $R 鈮� S$ . (In3dimensions, rotations about two different axes do not in general commute.) Find what geometric transformations are produced byRand, S.