A twist on finding angle between planes

Calculus Level 4

Given a vector function \( \vec{r} (t) = \left \langle 2t, t^2, \dfrac{t^3}{3} \right \rangle \), if the smaller angle between the osculating planes at \( t=1 \) and \( t=2 \) can be represented in the form \( \arccos \left( \dfrac{\alpha}{\beta} \right) \), where \( \alpha, \beta \in \mathbb{R} \) and are coprime positive integers, find \( \alpha + \beta \).

Clarification

  • The osculating plane is the plane given by vectors \( \vec{N} \) and \( \vec{T} \), where \( \vec{T} = \dfrac{\vec{r}'(t)}{\left|\vec{r}'(t)\right|} \) and \( \vec{N} = \dfrac{\vec{T}'(t)}{\left|\vec{T}'(t)\right|} \).
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