I'm sorry to bother people with this but I simply do not fully understand about Quadratic forms as I don't think my book explains the more difficult parts of the process well enough which is I assume the same for most people that struggle with things.
$$ 11x_1^2 + 11x_2^2 + 14x_3^2 - 2x_1x_2 - 8x_1x_3 - 8x_2x_3 = 1 $$
Assuming I calculated the eigenvalues correctly I got 6, 12 and 18. The corresponding eigenvectors to be (1,1,1), (-1,1,0) and (1,1,-2) resp.
Now we need to find the closest and farthest points to the origin, now I understand that the points closest to the origin corresponds to the eigenvalue 18 and the farthest to eigenvalue 6, what I don't understand is how to find them, I know I should put like $$\lambda_1 = 0$$ and $$\lambda_2 = 0 $$ and solve for the remaining one to figure it out but how exactly do I do that? A step by step just for this part would be much appreciated.