Models equations of conics are often given in the form of $\mathbb{P}^2$. However say I have 6 points embedded in higher dimension projective spaces, how do I determine if they are in the general position? The background context, if necessary, is I have chosen 6 points from a product of conics, namely $C_1\times C_2\subset\mathbb{P}^8$ under the Segre embedding. I want to determine if they are on the same conic.
Edit: In fact I am only familiar with general positions in $\mathbb{P}^n$. I don't know how to determine general position with $d$-dimensional space embedded in a larger space. Do we have to find first a map between $C_1\times C_2$ and $\mathbb{P}^2$?
First, you need to check if the points are coplanar (on the same plane); this is a rank condition on the collection of their coordinates. Then, if the points are coplanar, the question reduces to the question about six points in the plane; this is a single determinant vanishing condition.