How to prove a particle moves in a circle with vectors?

Question

What would be needed to show that a particle moves in a circle, with vectors? If you were to take the dot product between the position vector and the normal vector, how can you show that the angle between the vectors is constant?

The information I have is that the velocity vector is equal to w cross x, where w is angular velocity and x is position vector, the magnitude of the position vector is constant, and that the normal vector is constant.

Answer 1

HINT: Try with a coordinate change. Is there a coordinate system more "affine" to the motion you are trying to show?

Also, since you know the velocity vector is given by a cross product, by the definition of cross product you should be able to tell the angle between position and velocity.

Answer 2

If $$\underline {v}=\underline {w}\times \underline {x}$$ Then $$\underline {v}\cdot\underline{x}=0$$ $$\implies \frac {d}{dt}(\underline{x}\cdot\underline{x})=0$$ $$\implies |\underline{x}|^2=constant$$ $$\implies |\underline{x}|=constant$$