Equations of a Plane/Hyperplane

[AngelofMusic]

Homework Statement

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?

Homework Equations

Equation of a Plane: ax + by + cz + d = 0 (1)
Equation of a Hyperplane: $$a_1 x_1 + ... + a_n x_n + d=0$$ (2)

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!

 

[tiny-tim]

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?

Hi AngelofMusic!

Hint: find the normal (and don't use coordinates, use whole vectors) …

try the 3D case first: for three vectors a b and c, what can you say about the normal?

 

[njama]

The plane is the determinant:

$$\begin{vmatrix}<br /> x-x_1 & y-y_1 & z-z_1\\ <br /> x_2-x_1 & y_2-y_1 & z_2-z_1 \\ <br /> x_3-x_1 & y_3-x_1 & z_3-z_1 <br /> \end{vmatrix}=0$$

Where $M_1(x_1,y_1), M_2(x_2,y_2), M_3(x_3,y_3)$.

 

[AngelofMusic]

Hi AngelofMusic!

Hint: find the normal (and don't use coordinates, use whole vectors) …

try the 3D case first: for three vectors a b and c, what can you say about the normal?

One such normal would be n = (a-b)x(b-c), right? So for the n-dimensional case, would I just repeatedly take cross products of the vectors? The wikipedia page on surface normals has a neat solution where $$n=(AA^T + bb^T)^{-1} b$$, but they don't show how that is derived.

Thanks a lot for the help so far! I can deal with 3-D cases relatively well, but my mind just isn't wrapping around the n-dimensional algebra very well at the moment.