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The dot product cannot be sufficient because even making substitutions to minimise the occurrences of $Q$ we get $$\cos(\alpha)=\frac{\overrightarrow{P(t)}\cdot\overrightarrow{Q(t)}}{k|\overrightarrow{P(t)}|^2}$$ there are two solutions in the plane.

I personally would tackle this by complex numbers, but to keep it more geometric we can express the relationship as $$Q(t) = k \begin{pmatrix} \cos(\alpha) & \sin(\alpha) \\ -\sin(\alpha) & \cos(\alpha) \end{pmatrix} P(t)$$