# 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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