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Fundamental Mathematics: Rotation & Reflection

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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.

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.

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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 ...

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