How do you integrate Cross Products?

Question

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.

Answer 1

just notice that $$\frac{d}{dt}(\mathbf{q}\times\frac{d}{dt}\mathbf{q})=(\frac{d}{dt}\mathbf{q})\times(\frac{d}{dt}\mathbf{q})+ \mathbf{q}\times\frac{d^2}{dt^2}\mathbf{q}$$ and that the first term is zero, as $\mathbf{u}\times\mathbf{u}=0$.