Suppose we have a fixed system of coordinates $O→(x,y,z)$, that from now on will be called world frame, and a body $P$ representing an aircraft whose coordinates in this system are $(P_x, P_y, P_z)$.
In the body $P$ is recognized a relative system of coordinates $O'→(P,R,Y)$. The Roll axis points from the origin towars the nose of $P$, the Pitch axis points toward the right of $P$, the Yaw axis points toward the bottom. At the beginning the body was orientated with the nose of the body in the same direction of the $y$ versor, but now is oriented otherwise.
My goal is to describe the position of a second body $Q=(Q_x, Q_y, Q_z)$ into the coordinates system of the body $P$ using the conventions most used with aircrafts. If $C$ is a change of coordinates, the $x$ axis must be sent into the $P$ axis, the $y$ axis must be sent into the $R$ axis, the $z$ axis must be sent into the $Y$ axis.
First question: why three angles are necessary to describe the orientation of $P$?
Suppose for a moment that $P$ is in the origin of the world frame. In order to describe the orientation of $P$, one needs only two axis: the third can be deducted thanks to the right-hand rule.
Second question: what is the correct affine transformation that transforms the coordinates $Q=(Q_x, Q_y, Q_z)$ of the world frame into the coordinates $Q'=(Q_P, Q_R, Q_Y)$ into the system relative to $P$?
Using vectorial and matricial representation, $Q' = A·Q + c$ with $A$ a rotation matrix and $c$ a vector. From the known orientation of $P$ we obtain three rotation matrices: in what order should those matrices be multiplicated to obtain $A$? The correct value of $c$ is $-P$ or $-A·P$?