Answered step by step
Verified Expert Solution
Link Copied!

Question

1 Approved Answer

a) Find the standard matrix for the linear transformation from R 3 to R 3 that rotates vectors by /4 radians about the xaxis, counterclockwise

a) Find the standard matrix for the linear transformation from R 3 to R 3 that rotates vectors by π/4 radians about the x–axis, counterclockwise looking towards the y–z plane from the point (1, 0, 0). (Note: under this rotation, points on the positive y–axis will rotate to points with positive y– and z–coordinates.)

b) Find the standard matrix for the linear transformation from R 3 to R 3 that rotates vectors by π/4 radians about the y–axis, counterclockwise looking towards the x–z plane from the point (0, 1, 0). (Note: under this rotation, points on the positive x–axis will rotate to vectors with positive x–coordinates and negative z–coordinates.)

c) Use matrix multiplication to find the standard matrix for the linear transformation from R 3 to R 3 that results from applying the rotation in part a) followed by the rotation in part b). (Be careful about the order in which you multiply your matrices!)

d) It’s a fact that composing two rotations, as in part c), always gives another rotation. Find the axis for the rotation you R computed in part c) by using the fact that rotating around an axis leaves vectors on that axis fixed: in other words, vectors pointing along the axis of rotation are simply eigenvectors with eigenvalue λ = 1, so you can find the axis of rotation by computing the λ = 1 eigenspace for the matrix in part c).

e) Now that you know the axis for the rotation R in part c), what about the angle of rotation? To calculate the angle, note that if you rotate about an axis in R 3 , then the plane perpendicular to that axis just axis gets rotated. Find a vector v perpendicular to the axis you computed in d), and calculate the effect of the rotation R on v. Then compute the angle θ between v and R(v) using the formula cos(θ) = v·R(v) |v||R(v)| .

f) What happens if you replace π/4 in parts a) and b) by another angle τ? Will the axis of rotation for the new composite rotation remain the same? Find the axis and angle of rotation for the case τ = π/2.

Step by Step Solution

3.42 Rating (152 Votes )

There are 3 Steps involved in it

Step: 1

a The standard matrix for the linear transformation from R 3 to R 3 that rotates vectors by 4 radians about the xaxis counterclockwise looking towards ... blur-text-image

Get Instant Access to Expert-Tailored Solutions

See step-by-step solutions with expert insights and AI powered tools for academic success

Step: 2

blur-text-image

Step: 3

blur-text-image

Ace Your Homework with AI

Get the answers you need in no time with our AI-driven, step-by-step assistance

Get Started

Recommended Textbook for

Elementary Linear Algebra with Applications

Authors: Howard Anton, Chris Rorres

9th edition

471669598, 978-0471669593

More Books

Students also viewed these Accounting questions

Question

Explain this statement: Goals are dreams with deadlines.

Answered: 1 week ago

Question

Show that, A and B commute if a - d = 7b 5 1

Answered: 1 week ago