TITLE: How do you compute the normal vector to a hyperplane in $\mathbb{R}^n$ given $n$ representative points?
QUESTION [8 upvotes]: Given $n$ points (no two identical, no three colinear, no four coplanar, etc.), I'd like to find a formula for the normal vector to the unique hyperplane that intersects each of these points.
In three dimensions, we use a cross product: given $x_1, x_2, x_3$, the normal vector is given by $(x_1 - x_2) \times (x_1 - x_3)$.  How does this generalize?

REPLY [3 votes]: Another way to do this, which I (re)discovered when helping son learn about vectors, is to generalize the representation of the cross-product of two 3-vectors as a 3 by 3 matrix with the first row being the three unit vectors and the other two rows being the components of the two 3-vectors.
In $R^n$, if we have $n-1$ n-vectors, form the n by n matrix with the first row being the n unit vectors, and the next n-1 rows being the components of the vectors. the resulting vector is orthogonal to each of the n-1 vectors.