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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61f2bd8c-527a-479e-9deb-09c663979694%2F3e2eeab0-d4fe-4416-9447-1c2c2422525d%2Fdfglu5w_processed.jpeg&w=3840&q=75)
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.
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