What is the difference between a conic and a quadric? I'm guessing that this depends on your ambient space? I think that conics are just special quadrics and are a codimension 1 object and a quadric is the space that gets cut out by a quadric polynomial in any ambient space? Is that correct? For example, would
$$\left\{[u:v:w]\in \mathbb{C}\mathbb{P}^2 \; \vert \; u^2 +3v^2 -vw = 0\right\}$$
be a conic (and quadric) hypersurface, but
$$\left\{[u:v:w:z]\in \mathbb{C}\mathbb{P}^3 \; \vert \; u^2 +3v^2 -vw = 0\right\}$$
is a quadric hypersurface? Or is it too much to talk about projective space? Or maybe I have to be in Real space? Thanks for the help.