Rotating Rotation Axis onto Coordinate Axis

(1) Rotate about x-axis into xz plane

same as rotation of yz projection onto z
u' = (0, b, c)
u₂ = (0, 0, d)
cos [alpha] = (u' ^. u₂) / ( | u' |^. | u₂ | ) = cd/dd = c/d

where d = sqrt(b² + c²)

sin [alpha] = b/d

u'' = (a, 0, d)