Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 29/Solution 1

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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

That is

By row reduction we have

Thus and , i.e.,
So the eigenvector is
which is the vector direction of axis.