MATH152 April 2016
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Question A 29
The matrix below represents rotation in 3D about a line through the origin.
Find a vector in the direction of the line of rotation.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Consider the eigenvector. The direction vector of the rotation is just the vector unchanged after rotation, i.e.,
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If a vector lies in the line of rotation, after multiplying the rotational matrix on the vector, the vector stays the same.
That is to say if we denote this vector by , then we have
Equivalently we are looking for eigenvector corresponding to eigenvalue 1. We need to solve characteristic equation
By row reduction we have
Thus and , i.e., So the eigenvector is
which is the vector direction of axis.