Animation of rotating axes

I am reading an introductory textbook on theoretical mechanics. In the
chapter on moving coordinate systems the author sets a problem to solve:

An xyz coordinate system is rotating with respect to an XYZ coordinate
system having the same origin and assumed to be fixed in space (i.e. it is
an inertial system).

The angular velocity of the xyz system relative to the XYZ system is given
by w = 2ti - t^2j + (2t+4)k where t is the time.

The position vector of a particle at time t as observed in the xyz system is
given by r = (t^2+1)i - 6tj + 4t^3k.

Find

a) the apparent velocity and
b) the true velocity at time t=1.

Answer: (not relevant to this posting, included for completeness)

a) Apparent velocity at time t is dr/dt = 2ti -6j + 12t^2k.
At time t=1, this is 2i - 6j + 12k

b) True velocity at time t is dr/dt + w x r = 34i - 2j +2k at t=1


I am trying to use animation in MuPad to demonstrate visually the full
motion of the particle, by showing its motion relative to the xyz coordinate
system and the motion of the xyz coordinate system relative to the XYZ
system.

Using MuPad, the motion of the particle relative to xyz is simply...

x_axis := plot::Arrow3d([1, 0, 0], Color = RGB::Red):
y_axis := plot::Arrow3d([0, 1, 0], Color = RGB::Red):
z_axis := plot::Arrow3d([0, 0, 1], Color = RGB::Red):
r := plot::Arrow3d([t^2+1, 6*t, 4*t^3], t = 0..1, Color = RGB::Blue):
plot(x_axis, y_axis, z_axis, r);


Next, I need to use plot::Rotate3D to rotate this whole system with angular
velocity w = 2ti - t^2j + (2t+4)k

plot::Rotate3d is described as:

plot::Rotate3d(angle, [cx, cy, cz], [dx, dy, dz], object) rotates a 3D
object by the given angle around the rotation axis defined by the point [cx,
cy, cz] and the direction [dx, dy, dz].

The angular velocity w in the problem is specified relative to the xyz
system. It appears to me that, to use Rotate3d to rotate the system, I need
W which is w relative to the XYZ system.

In the XYZ system, let the unit vectors along the axes be I, J, K and
W = W1I + W2J + W3K. Since w and W are the same vector w = W, so that

W1I + W2J + W3K = 2ti - t^2j + (2t+4)k

Successively take the dot product of both sides with I, J and K to obtain

W1 = 2ti.I - t^2j.I + (2t+4)k.I

W2 = 2ti.J - t^2j.J + (2t+4)k.J

W3 = 2ti.K - t^2j.K + (2t+4)k.K

i.I is the cosine of the angle between i and I and similarly for the other
dot products.

This gives me the components of W, i.e. W1, W2 and W3, but not in a form I
can use with Rotate3D.
Every idea I have tried has not born any fruit.

If anyone has some advice on how to perform this animation it would be much
appreciated.

Cheers,
Brad


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