A common Euler angle sequence used in aircraft dynamics to describe the relative

orientation of the aircraft body frame with respect to some reference frame is the

asymmetric $3-2-1$ sequence $($so called "yaw, pitch, roll"$).$ The standard Euler angle

sequence that we will use for orbital mechanics to describe the relative orientation

of an orbit with respect to some reference frame is the $3-1-3$ sequence. Assume that

you have been given a set of $3-2-1$ Euler angles with ${}_1=45,{}_2=30,$ and ${}_3=10.$

Determine the following:

The corresponding direction cosine matrix for the relative frame orientation.

The equivalent set of $3-1-3$ Euler angles: ${}_1^{\text{'}},{}_2^{\text{'}},$ and ${}_3^{\text{'}}.$