# Composition of intrinsic rotations yaw, pitch, roll.

I am using the æronautical meanings of yaw, pitch and roll (intrinsic rotations, not extrinsic). Say I know the yaw ($=\theta_z$ radians about the intrinsic $z$ axis of an object), pitch ($=\theta_x$ radians about the intrinsic $x$ axis), and roll ($=\theta_y$ radians about the intrinsic $y$ axis) of an object. Is there a simple way to write the matrix for the composition of these three rotations? Is the composition even commutative? If it isn't, then why do æronautical engineers use this system? Should I be converting into quaternions?

Say I have a telescope on a airplane, and I want to convert the azimuth & elevation of a star w.r.t. the earth into the azimuth & elevation of that star w.r.t. the airplane (so that I can point the plane's telescope at it). I could think of this data for the star as a vector in the same three variables as the airplane, with one variable being irrelevant (say roll, if it is done last). If I had a matrix which represented the overall rotation of the plane away from some original position $(\theta_z=0,\theta_x=0,\theta_y=0)$, then I could apply this matrix's inverse to the star's vector, and get the new vector for the star w.r.t. the airplane.

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 These rotations do not commute. Converting to quaternions does not change this: they don't commute either. – Marc van Leeuwen Apr 14 '12 at 17:09 You can get the matrix for the composition by multiplying the matrices for the individual rotations in the appropriate order. – Jim Belk Apr 14 '12 at 17:33 see Euler angles and Conversion formulae between formalisms. Good luck! – Raymond Manzoni Apr 14 '12 at 18:06 Thank you for your responses. I guess I am really wondering why the position of the plane is expressed as (Yaw, Pitch, Roll), without any specific order, when the order of these is important. I guess I may be misunderstanding how these data are presented. It would be so much easier if they just gave the orientation of the plane in terms of a single matrix (but, if I decide on an order, then I suppose I have such a matrix, don't I?). – Posty Apr 15 '12 at 14:45