Modelling Motion (Angular Speed etc.) (1 Viewer)

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I've done the first bit (showing [itex]x[/itex] and [itex]\dot{x}[/itex] are always perpendicular) but how do I show that the motion lies in a plane? Also, how do I do the final part?
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That should say I've already shown [itex]x[/itex] and [itex]\dot{x}[/itex] are always perpendicular to [itex]b[/itex] . I did this by deriving a 2nd order linear ODE, finding the general solution and then applying the initial conditions.

I'm just stuck on how to proceed with the next parts of the question. Can anyone help?

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