homework2.nb

Problem 6:

We will solve this problem by trying to visualize the axes of rotations that can be achieved for certain values of φ. The complete set of rotation axes can be viewed as points on a unit sphere, and for a given value of φ, we can generate rotation matrices for random values of θ1, θ2 and θ3. These rotation matrices can then be converted to an equivalent angle-axis representation and plotted in 3d. Below, the function visualize[...] does just that, giving a 3d view of a random set of rotation axes for a given value of φ, and three separate planar views in the (kx, ky), (kx, kz) and the (ky, kz) planes, in red, green and blue, respectively.

Definitions

$kv[rm_, θ_, ϕv_] := <br /> Module[{vars, rn, kdir}, <br />&nb ... ]] - rn[[1, 2]]} ; <br /> kdir/Sqrt[kdir . kdir]] ;$

$RowBox[{RowBox[{visualize[rm_, ϕv_, n_:1000], :=, <br />, , RowBo ... , ,, 1.1}], }}]}], }}]}], ,, , AspectRatio->1}], ]}]}], ;, , {g1, g2, g3, g4}}]}], ]}]}], ;}]$