Does F = k·u × v produce circular or spiral motion?

[Anastasis Pk]

Good afternoon!
Exercise: Prove that if a body moves under the action of a force F = k⋅u × v, where u is an arbitrary unit vector and v the velocity, the motion is circular with angular velocity ω = k ⋅ u or, in a more general case, a spiral parallel to u.

Source: (Alonso & Finn: Fundamental university physics)

I attempted to prove that F is a central force, so that the angular momentum is constant. Is that wrong?
Thank you in advance!

 

[vela]

Would showing F is a central force buy you anything? The force of the Sun on a planet is a central force. but the orbits aren't circular.

 

[haruspex]

I attempted to prove that F is a central force

That won't do it. F = k.v would satisfy that. Get an equation for the acceleration.

 

[Anastasis Pk]

Thank you for your responses! In order to prove that it's motion is circular wouldn't it be sufficient to prove that ω is constant?
Could you "boost" me a bit?

 

[haruspex]

Thank you for your responses! In order to prove that it's motion is circular wouldn't it be sufficient to prove that ω is constant?

Not really. The question is asking you to demonstrate that the equation of motion corresponds to a particle moving at constant speed in a helix. (It says spiral, but it means helix.)
It might help if you could write down the general equation for such a motion, so that you could see what the answer has to look like. Hint: pick a suitable set of orthogonal unit vectors.
Anyway, the first (easy) step, as I posted originally, is to obtain an equation for the acceleration of the particle.