We have the following quadratic form:
$$q := 6 x^2_1 + 3 x^2_2 + 3 x^2_3 - 4 x_1 x_2 + 4 x_1 x_3 - 2 x_2 x_3$$
whose eigenvalues are $\lambda_1=\lambda_2=2$ and $\lambda_3=8$
I am not really sure how could we get the procedure to find the orthogonal basis for this form... For now, I know two things: 1. The Orthogonal Basis of the quadratic form was assumed randomly 2. Somehow we used some sort of "Witchcraft" from the eigenvectors.
Which one is true
Not quite sure exactly what it is you’re asking here. There are a couple of ways that you can find an orthogonal eigenbasis.
One way is to go through the normal process of finding a basis for each eigenspace. The eigenspace of $2$ will be two-dimensional, so orthogonalize the basis that you found for it using the Gram-Schmidt process.
Alternatively, recall that the eigenspaces are pairwise orthogonal, so since the ambient space is three-dimensional, the eigenspace of $2$ must be the orthogonal complement of the eigenspace of $8$. Find an eigenvector for the latter, generate a vector that’s orthogonal to it (I hope you can do at least that), and then take the cross product of these two vectors to complete the basis.