Suppose $\vec{a}(t)$ is a time dependent vector and $\vec{b}$ is a constant vector. I want to show $$\frac{d}{dt}\left[\vec{a} \cdot \left(\frac{d \vec{a}}{dt} \times \vec{b}\right)\right] = \vec{a} \cdot \left[\frac{d^2\vec{a}}{dt^2} \times \vec {b}\right].$$
I'm not sure how to go about doing this. The proof need not be rigorous; I'm just confused on which properties to elicit where. Thanks!
Apply the derivative
$$\frac{d}{dt} \left[\vec{a} \cdot \left( \frac{d\vec{a}}{dt} \times \vec{b}\right) \right] = \underbrace{\frac{d\vec{a}}{dt} \cdot \left( \frac{d\vec{a}}{dt} \times \vec{b}\right)}_{0} + \vec{a} \cdot \left( \frac{d^2\vec{a}}{dt^2} \times \vec{b}\right)$$
since $\vec{a}\,' \times \vec{b}$ is orthogonal to $\vec{a}\,'.$