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Question 3

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A brief Word note by Phil dated 2.26.17, labeled Question 3 in a set of hard questions on a frames document. It asks how to get the components of a point r in all the Euler angle frames, referring to Section G.5. It uses the product Rz(φ)Rx(θ), picks off a column, and gets the ζ' component as x sinθ sinφ - y sinθ cosφ + z cosθ, noting the matrix rows give all three at once. The equations are partly garbled in extraction.

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Question 3 PhL 2.26.17 Question 3: If you have some point r in space, how to you compute its components in all the Euler Angle frames of reference. We are looking at Section G.5. You want to compute for example r ' = rζ' You know that ' = Rz(φ)Rx(θ) = Rz(φ)Rx(θ) = picks off the right column of matrix The matrix Rz(φ)Rx(θ) is the transpose of what is shown below in Maple, so = = sinθsinφ - sinθcosφ + cosθ Alternatively use (b) = Rx(-θ) Rz(-φ) // using (4) Then ' = sinθsinφ - sinθcosφ + cosθ So now consider rζ' = r ' = x sinθsinφ - y sinθcosφ + z cosθ and this is the component sought after. For a vector V, (V)ζ' = V ' = sinθsinφ(V)x - sinθcosφ (V)y + cosθ(V)y Now the rows of the above matrix are the so you get all three at once.