Hey I'm doing a course in mechanics and these keep cropping up!
So for this question I'm working in 3d, and so far have
$$m \mathbf{k} \cdot (\mathbf{q} \times \ddot{\mathbf{q}} )=0$$ so I need to integrate this with respect to $t$ to get: $$ m \mathbf{k} \cdot (\mathbf{q} \times \dot{\mathbf{q}} ) =\text{a constant}$$
I know why this is constant but have no idea how you integrate what's in the brackets.