Explore BrainMass
Share

Explore BrainMass

    Fundamental Mathematics: Rotation & Reflection

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    We use matrices, eigenvalues and eigenvectors to solve the following:

    Let f be the rotation in the 3-dimensional space about the x-axis through the angle pi/2 and g be the
    rotation about the y-axis through the same angle. Describe the rotation f g. Let h be the reflection in the plane orthogonal
    to the vector 2i - j + 3k. Describe the reflection f h.

    © BrainMass Inc. brainmass.com May 20, 2020, 10:30 pm ad1c9bdddf
    https://brainmass.com/math/linear-algebra/fundamental-mathematics-rotation-reflection-516950

    Attachments

    Solution Preview

    Please see attachment for properly formatted copy.

    Exercise: Let f be the rotation in the 3-dimensional space about the x-axis through the angle pi/2 and g be the
    rotation about the y-axis through the same angle. Describe the rotation f g. Let h be the reflection in the plane orthogonal
    to the vector 2i - j + 3k. Describe the reflection f h.

    It is not specified whether the rotation f is clockwise or counterclockwise, so we assume counterclockwise, that is, it takes
    the vector (0,1,0)^T to the vector (0,0,1)^T (here v^T means transpose). The matrix associated to f is then

    A=...

    since alpha=frac{pi}{2}. Similarly for g, we take the rotation by frac{pi}{2} about the y-axis that takes the
    vector (0,0,1)^T to (1,0,0)^T. It has matrix

    B=...

    Note: if g rotates in the opposite direction them multiply the above matrix by -1 (or ...

    Solution Summary

    Given two rotation in 3-dimensional space, we find the axis of rotation of the composition. Given a reflection and a rotation, we find the description of the composition.

    We use the descriptions of rotations and reflections as matrices. We calculate eigenvalues and eigenvectors and change basis to find the answer.

    $2.19

    ADVERTISEMENT