The spin is a unique property belonging to quantum particles;

the effect of spin in atoms can be observed by deflecting flying atoms using a magnetic field. Spin can be represented by a vector, z, in three dimensions, where zx, zy, zz , are the spin components in the x, y, and z directions. When radio waves or magnetic fields interact with the spin of nuclei, the interaction can be represented by a rotation by angle θ. For this example we will approximate the spin’s interaction as the rotation matrix A = 





1 0 0 0 cos(θ) − sin(θ)
0 sin(θ) cos(θ)






.
Suppose the initial spin vector is (0,0,1), and that the vector produced by this rotation is (
0 √
2 , 1 √
2 , 1 √
2 ). Calculate the angle through which the vector has been rotated and thus find the value of θ.