Parametric equations 2D planar parabola in an arbitrary plane of $R^3$

Question

I suspect that this is the equation of a parabola (planar 2D curve in an arbitrary plane of $R^3$):

$\vec{x} = \vec{a}∗t^2+\vec{b}∗t+\vec{c}$

I have made parametric plots of this and everything looks fine, but i have no clue of how i can probe that this is the case, the problem seems to be that I don't know how are the parametric equations of a conic section on a arbitrary plane of $R^3$

Thanks and please excuse my English

Answer 1

Sorry, but I think you got a really bad answer – the equations are fine for generating general 3D parabolas in the plane spanned by vectors $a, b$

$b = (0,0,0)$ is a special, degenerate case, in fact the claim that the resulting parametric figure is a line is wrong too since it only gives a ray as $t^2$ is always positive

In fact both purported "fails" could be interpreted as different extreme limiting case parabolas - one infinitely “skinny” the other infinitely “wide”