Is the the cross product of two Bezier curves a Bezier curve? For some geometric problem that I am trying to solve, I am interested 
in characterizing the cross product between a Bezier curve and its second order derivative.
So, let's $\mathbf{c}(t)$ be a Bezier curve, and $\ddot{\mathbf{c}}(t)$ it's derivative.
Is $\mathbf{w}(t) = \mathbf{c}(t)  \times \ddot{\mathbf{c}}(t)$ a Bezier curve?
If it is the case, is it also the case in general ?
Thanks a lot for your time and answers. Sincerely,
Steve

Every parametric curve where each coordinate is a polynomial in the parameter can be expressed as a Bézier curve.
Thus also the cross product of Bézier curves. The order of the resulting curve will in general (except in degenerate cases) be the sum of the orders of the two curves you multiply -- so you can't, for example, multiply two cubic curves and expect to get a cubic curve back.
