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The context for the question is in the attachments (pg1.png, pg2.png, pg3.png), so there is some reading involved. Although, it is a short and simple read if anything. The inquiry is in (inquiry.png).

My understanding of the situation is that Q(t) abides by the differential equation

Q'(t)Q(t)

So by something akin to Picard's existence theorem on differential equations, we can resolve exact trajectories for Q(t) given initial conditions. The peculiar thing is that

Q(t

can likely be assumed to be true for all

Q(t

should also be true.

Now if I let R = (

R

and the question insists that

R

Although, our choice of the initial condition basis vectors for R is arbitrary, so how can we expect that R necessarily commutes with Q(t)? Unless of course, the question is asking for the matrix expression of Q(t) in the {

(Also, how do I insert latex?)

My understanding of the situation is that Q(t) abides by the differential equation

Q'(t)Q(t)

^{T}+ Q(t)Q'(t)^{T}= 0 .So by something akin to Picard's existence theorem on differential equations, we can resolve exact trajectories for Q(t) given initial conditions. The peculiar thing is that

Q(t

_{0})**z**(t_{0}) =**z**(t_{0})can likely be assumed to be true for all

**z**(t_{0}) ∈ ℝ^{n}, in which caseQ(t

_{0}) = Ishould also be true.

Now if I let R = (

**i**(t_{0}) ,**j**(t_{0}) ,**k**(t_{0}) ), thenR

^{T}R = RR^{T}= I ,and the question insists that

R

^{T}Q(t)R = Q ⇒ Q(t)R = RQ(t).Although, our choice of the initial condition basis vectors for R is arbitrary, so how can we expect that R necessarily commutes with Q(t)? Unless of course, the question is asking for the matrix expression of Q(t) in the {

**i**(t_{0}) ,**j**(t_{0}) ,**k**(t_{0}) } basis. I am just wondering if I am to expect the matrix expression of Q(t) to remain invariant given any orthonormal basis of the same orientation?(Also, how do I insert latex?)

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