Problem 1:

Part (a)

Since the second rotation is with respect to the moving frame {B}, we post-multiply, rather than premultiply successive rotations.

aRb = RZ[θ] . RX[ϕ] ; aRb // TraditionalForm

( cos(θ) -cos(ϕ) sin(θ) sin(θ) sin(ϕ) ɯ ... s(θ) sin(ϕ) 0 sin(ϕ) cos(ϕ)

Part (b)

Since the rotation is now with respect to the fixed frame {A}, we pre-multiply successive rotations.

aRb = RX[ϕ] . RZ[θ] ; aRb // TraditionalForm

( cos(θ) -sin(θ) 0  ... (ϕ) -sin(ϕ) sin(θ) sin(ϕ) cos(θ) sin(ϕ) cos(ϕ)

Created by Mathematica  (September 29, 2003)