1. The problem statement, all variables and given/known data Given 3 points in 3-dimensional space, Find the equation of a plane containing those 3 points. How would you generalize this to n points in n-dimensional space? 2. Relevant equations Equation of a Plane: ax + by + cz + d = 0 (1) Equation of a Hyperplane: [tex]a_1 x_1 + ... + a_n x_n + d=0[/tex] (2) 3. The attempt at a solution For the 3-D case, I simply substituted the 3 points into the equation. For example, p1 = (x1, y1, z1), p2 = (x2, y2, z2), p3=(x3, y3, z3). ax1 + by1 + cz1 = -d ax2 + by2 + cz2 = -d ax3 + by3 + cz3 = -d ax1 + by1 + cz1 = ax2 + by2 + cz2 => a(x1-x2) + b(y1-y2) + c(z1-z2) = 0 Similarly, a(x1-x3) + b(y1-y3) + c(z1-z3) = 0 and a(x2-x3) + b(y2-y3) + c(z2-z3) = 0 Which is 3 equations with 3 unknowns that can be solved. Where I'm stumped is how I would generalize this to the n-dimensional case. I have a feeling that maybe I'm doing this the 'dumb' way and there's a far more elegant solution (perhaps involving matrices?) for solving the 3-D case that will extend more easily to the n-dimensional case. Any help would be appreciated!