Conic section's vector equation???

Question

Hello I have curiosity on how will be the vector equation of conic section's on 3d space. For example I have tried with $\vec x = \vec a *t^2+\vec b *t + \vec c$ for parabola but not shure how probe this. Thanks and please excuse my English.

Answer 1

$\newcommand{\Vec}[1]{\mathbf{#1}}\newcommand{\Reals}{\mathbf{R}}\newcommand{\Basis}{\mathbf{e}}$Here is one approach, very similar to yours (with conditions on the vectors $\Vec{a}$ and $\Vec{b}$ to ensure the resulting curve is congruent to a known plane conic):

Suppose $(x(t), y(t)$ is a parametrization of some conic in the Cartesian plane $\Reals^{2}$.

If $\Vec{x}_{0}$ is a point of $\Reals^{n}$, and if $\Basis_{1}$, $\Basis_{2}$ are an orthonormal pair in $\Reals^{n}$, then $$ \Vec{x}(t) = \Vec{x}_{0} + x(t) \Basis_{1} + y(t) \Basis_{2} $$ parametrizes a conic in the plane through $\Vec{x}_{0}$ and spanned by $\Basis_{1}$, $\Basis_{2}$.