Please do part A,B,C,D and please show step by step and explain 1.10 Example To represent a rotation map to: R² → R² that turns all vectors inthe plane counterclockwisethrough an angleū=+x/5we start by fixing the standard bases E₂ for both the domain and codomainbasis, Now find the image under the map of each vector in the domain's basis.(*)Represent these images with respect to the codomain's basis. Because this basisis E2, vectors represent themselves. Adjoin the representations to get the matrixrepresenting the map.3COSsin(1)-() () (sincosRepɛ2,8₂ (te) =tπ/6 (ū)cos (sin 0- sin Ꮎcos 0The advantage of this scheme is that we get a formula for the image of anyvector at all just by knowing in (*) how to represent the image of the two basisvectors. For instance, here we rotate a vector by 0 = π/6.333.5983.598: (√³/2² √3/2) (-2) = (-²0232) - (²0.232E282More generally, we have a formula for rotation by 0 = π/6.(√3/2)x-(1/2)y(:) == (√3/2² -√2/2) (*) - (1/2) + (√3/21M)= ✓ 1.34 Example 1.10 shows how to represent rotation of all vectors in the plane throughan angle about the origin, with respect to the standard bases.(a) Rotation of all vectors in three-space through an angle 0 about the x-axis is atransformation of R³. Represent it with respect to the standard bases. Arrangethe rotation so that to someone whose feet are at the origin and whose head isat (1,0,0), the movement appears clockwise.(b) Repeat the prior item, only rotate about the y-axis instead. (Put the person'shead at ₂.)(c) Repeat, about the z-axis.(d) Extend the prior item to R¹. (Hint: we can restate 'rotate about the z-axis'as 'rotate parallel to the xy-plane'.)

Answered: 1.34 Example 1.10 shows how to…

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

Related questions

Question

Please do part A,B,C,D and please show step by step and explain

1.10 Example To represent a rotation map to: R² → R² that turns all vectors in
the plane counterclockwise
through an angle
ū
=
+x/5
we start by fixing the standard bases E₂ for both the domain and codomain
basis, Now find the image under the map of each vector in the domain's basis.
(*)
Represent these images with respect to the codomain's basis. Because this basis
is E2, vectors represent themselves. Adjoin the representations to get the matrix
representing the map.
3
COS
sin
(1)-() () (
sin
cos
Repɛ2,8₂ (te) =
tπ/6 (ū)
cos (
sin 0
- sin Ꮎ
cos 0
The advantage of this scheme is that we get a formula for the image of any
vector at all just by knowing in (*) how to represent the image of the two basis
vectors. For instance, here we rotate a vector by 0 = π/6.
3
3
3.598
3.598
: (√³/2² √3/2) (-2) = (-²0232) - (²0.232
E2
82
More generally, we have a formula for rotation by 0 = π/6.
(√3/2)x-(1/2)y
(:) == (√3/2² -√2/2) (*) - (1/2) + (√3/21M)
=

✓ 1.34 Example 1.10 shows how to represent rotation of all vectors in the plane through
an angle about the origin, with respect to the standard bases.
(a) Rotation of all vectors in three-space through an angle 0 about the x-axis is a
transformation of R³. Represent it with respect to the standard bases. Arrange
the rotation so that to someone whose feet are at the origin and whose head is
at (1,0,0), the movement appears clockwise.
(b) Repeat the prior item, only rotate about the y-axis instead. (Put the person's
head at ₂.)
(c) Repeat, about the z-axis.
(d) Extend the prior item to R¹. (Hint: we can restate 'rotate about the z-axis'
as 'rotate parallel to the xy-plane'.)

Expert Solution

Introduction

As per the question, we were asked to derive rotation matrices for rotating vectors in three-dimensional and four-dimensional space.

Specifically, we were asked to find rotation matrices for rotating vectors about the x-axis, y-axis, and z-axis, and represent them with respect to the standard bases.

In addition, we were asked to arrange the rotations so that they appear clockwise when viewed from a certain perspective.

Finally, we extended the question to four-dimensional space and found the corresponding rotation matrix for rotating vectors parallel to the xy-plane.