Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
8. We have seen that the matrix R(0) = ( cos sin 0 around the y-axis rotates vectors in R2 counterclockwise by an angle 0. In three dimensions, every rotation has an axis, i.e. a direction that remains stationary under the rotation. sin (a) Find the 3 x 3 rotation matrix R₂ (a) that rotates vectors v = counterclockwise around the x-axis cos angle a counterclockwise around the z-axis (8) 0 Hint: R₂(a) leaves the z-component of v alone and rotates the x and y components by angle a. (b) Similarly, find the matrix R, (B) that rotates v by an angle ß counterclockwise 0 (8). 1 and the matrix R. (y) that rotates v by an angle y 0 1 (6). 0 0 (:) by an (c) Verify that R₂(a), Ry(B), and R. (y) are orthogonal matrices. (d) It turns out that any three-dimensional rotation, around any axis, can be written as a composition R = R₂(a)Ry(B) Rz(y) for some angles a, ß, and y. (We take this fact for granted.) a, ß, and y are called the yaw, pitch, and roll angles. Explain why the resulting 3 x 3 matrix R is orthogonal. Hint: You do not have to explicitly calculate the product of these three matrices.

8. We have seen that the matrix R(0) = ( cos sin 0 around the y-axis rotates vectors in R2 counterclockwise by an angle 0. In three dimensions, every rotation has an axis, i.e. a direction that remains stationary under the rotation. sin (a) Find the 3 x 3 rotation matrix R₂ (a) that rotates vectors v = counterclockwise around the x-axis cos angle a counterclockwise around the z-axis (8) 0 Hint: R₂(a) leaves the z-component of v alone and rotates the x and y components by angle a. (b) Similarly, find the matrix R, (B) that rotates v by an angle ß counterclockwise 0 (8). 1 and the matrix R. (y) that rotates v by an angle y 0 1 (6). 0 0 (:) by an (c) Verify that R₂(a), Ry(B), and R. (y) are orthogonal matrices. (d) It turns out that any three-dimensional rotation, around any axis, can be written as a composition R = R₂(a)Ry(B) Rz(y) for some angles a, ß, and y. (We take this fact for granted.) a, ß, and y are called the yaw, pitch, and roll angles. Explain why the resulting 3 x 3 matrix R is orthogonal. Hint: You do not have to explicitly calculate the product of these three matrices.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
8. We have seen that the matrix R(0) = ( cos rotates vectors in R2 counterclockwise by an angle 0. In three dimensions, every rotation has an axis, i.e. a direction that remains stationary under the rotation. around the y-axis sin 0 (a) Find the 3 x 3 rotation matrix R₂ (a) that rotates vectors v = - sin cos angle a counterclockwise around the z-axis counterclockwise around the x-axis (i) 0 Hint: R₂ (a) leaves the z-component of v alone and rotates the x and y components by angle a. (b) Similarly, find the matrix Ry (6) that rotates v by an angle ß counterclockwise (8).- 2 (:) and the matrix Rz(y) that rotates v by an angle y 1 0 0 by an (c) Verify that R₂ (a), R, (B), and R. (y) are orthogonal matrices. (d) It turns out that any three-dimensional rotation, around any axis, can be written as a composition R = R₂(a) R, (B) Rz(y) for some angles a, 8, and y. (We take this fact for granted.) a, ß, and y are called the yaw, pitch, and roll angles. Explain why the resulting 3 x 3 matrix R is orthogonal. Hint: You do not have to explicitly calculate the product of these three matrices.
Transcribed Image Text:8. We have seen that the matrix R(0) = ( cos rotates vectors in R2 counterclockwise by an angle 0. In three dimensions, every rotation has an axis, i.e. a direction that remains stationary under the rotation. around the y-axis sin 0 (a) Find the 3 x 3 rotation matrix R₂ (a) that rotates vectors v = - sin cos angle a counterclockwise around the z-axis counterclockwise around the x-axis (i) 0 Hint: R₂ (a) leaves the z-component of v alone and rotates the x and y components by angle a. (b) Similarly, find the matrix Ry (6) that rotates v by an angle ß counterclockwise (8).- 2 (:) and the matrix Rz(y) that rotates v by an angle y 1 0 0 by an (c) Verify that R₂ (a), R, (B), and R. (y) are orthogonal matrices. (d) It turns out that any three-dimensional rotation, around any axis, can be written as a composition R = R₂(a) R, (B) Rz(y) for some angles a, 8, and y. (We take this fact for granted.) a, ß, and y are called the yaw, pitch, and roll angles. Explain why the resulting 3 x 3 matrix R is orthogonal. Hint: You do not have to explicitly calculate the product of these three matrices.
Expert Solution
steps

Step by step

Solved in 7 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS