(a) Find the eigenvalues and eigenvectors of the rotation matrix
R_(hat(n))(\theta )=(1)/(3)([1,-2,-2],[2,-1,2],[-2,-2,1])
(b) What is the axis of rotation hat(n) and the angle of rotation \theta ?
(c) What is the generator of this rotation J_(hat(n)) ? Recall that the generator is defined as
R_(hat(n))(\theta )=e^(\theta J_(n))
and that it can be constructed as
J_(hat(n))=n_(x)J_(x) n_(y)J_(y) n_(z)J_(z)
where J_(x),J_(y), and J_(z) are the generators of rotation along x,y, and z axis respectively. They are:
J_(x)=([0,0,0],[0,0,-1],[0,1,0]),J_(y)=([0,0,1],[0,0,0],[-1,0,0]), and ,J_(z)=([0,-1,0],[1,0,0],[0,0,0])
(d) Write down the plane whose normal is hat(n) (the origin is a point on this plane) and then write down the matrix that describes reflections with respect to this plane i.e., the plane you got is your reflection plane ("mirror"). Hint: The matrix you get should change hat(n) to -hat(n).