Problem $1.(10$ points$)$ To prepare data to train an $I$ to look for cosmic topology it was necessary

to estimate the rotation to a specific coordinate system $($defined by the orientation of the topology

axes$).$ The main part of the algorithm to determine this rotation starts with two unit vectors hat$(v{)}_1$

and hat$(v{)}_2$ given and defines a coordinate system based on these vectors as follows.

Let $c-=$hat$(v{)}_1$hat$(v{)}_2$ and rotate this vector to the $z-$axis using two rotations by angles $$ and $.$

Given the rotation from the previous part, rotate the two original vectors. Calling these

vectors hat$(v{)}_1^{\text{'}}$ and hat$(v{)}_2^{\text{'}}$ we define $v^{\text{'}}-=$hat$(v{)}_1^{\text{'}}+$hat$(v{)}_2^{\text{'}}.$ Rotate $v^{\text{'}}$ to the $x-$axis. This requires one rotation

by angle $.$

These three angles define the Euler angles of the rotation required to prepare the data.

$($i$)$ "Obviously" the vectors hat$(v{)}_1^{\text{'}}$ and hat$(v{)}_2^{\text{'}}$ lie in the $xy-$plane. Why is this obvious? $[$Note: This is a

consistency check used in practice to test the implementation of the algorithm.$]$

$($ii$)$ Determine the three Euler angles in the ZYZ convention. This is casiest to do for an active

rotation $($which is sufficient$).$ In fact, it is better to write the angles in terms of the sine$,$

cosine, or tangent of the angle. The expressions can be written in terms of the components

of $c$ and $v^{\text{'}}.($For those of us familiar with numerically working with inverse trig functions,

we know when we can use sine or cosine and when we need to use the tangent.$)$