A spaceship flies in $\mathbb{R}^2=\mathbb{C}$ with constant speed in the direction of the x-axis.
Physicist P1 uses the canonical unit vectors $e_1 = (1, 0) = 1$ and $e_2 = (0, 1) = i$ as the coordinate system; in this the location of the spaceship at time $t$ is given by $r (t) = (t, 0) = t$.
Physicist P2 uses a coordinate system rotating around $0$, whose (time-dependent) Axes he holds for the guidelines of the universe, and that are given by $\tilde{e}_1 (t) = (\cos (t), \sin (t)) = \cos (t) + i\sin (t)$, $\tilde{e}_2 (t) = (- \sin (t), \cos (t)) = - \sin (t) + i \cos (t)$
What are the (real) coefficients $a (t), b (t)$ of the representation $r (t) = a (t) \tilde{e}_1 (t) + b (t) \tilde{e}_2 (t)$ in the system of P2 explicit?
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From $r (t) =a (t) \tilde{e}_1 (t) + b (t) \tilde{e}_2 (t)$ we get $(t,0)=a (t) (\cos (t), \sin (t)) + b (t) (- \sin (t), \cos (t)) $.
So we have to solve the system $$t=a(t)\cos t-b(t)\sin t \\ 0 =a(t)\sin t+b(t)\cos t$$ or not? Do we have to differentiate these equations?
Do we use also that the speed is constant?
Hint:
Fix $t$. For a given orthonormal basis $\{\tilde{e}_1 (t),\tilde{e}_2 (t)\}$, find the coordinates for a given vector $r(t)$ with respect to that basis using inner products:
$$ a(t) = r(t) \cdot \tilde{e}_1 (t),\\ b(t) = r(t) \cdot \tilde{e}_2 (t) . $$
Yes, we are solving the system you wrote. But the inverse of an orthogonal matrix is easy!