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
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
Step 1
Given:
In this problem, the given matrix is
, rotates vectors in counterclockwise by an angle
The direction remains stationary under the rotation.
a) To find the rotation matrix,
b) Find the matrix
c) To verify that the are orthogonal matrices.
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