I'm trying to find the Eigenvectors for the equation of the second degree (for example Elliptic cone). The estimated values $V_1$, $V_2$ and $V_3$ must satisfy the right-hand rule. How can we verify if the three vectors satisfy the right-hand rule or not? And what does that mean?
Suppose you have the following matrix:
$$A = \begin{pmatrix} 2.6844 & -0.1247 & -1.2741 \\ -0.1247 & 4.2198 & -1.2351 \\ -1.2741 & -1.2351 & 1.0000 \end{pmatrix} $$
So, the Eigenvectors for the above matrix are:
$$ \text{Eigen_Vectors} = \begin{pmatrix} 0.4232 & 0.8888 & 0.1759 \\ 0.2659 & -0.3074 & 0.9137 \\ 0.8661 & -0.3399 & -0.3664 \end{pmatrix}$$
Where is each column represent a vector. The question is: I want to know if each vector satisfy the right-hand rule or not?
Three vectors have the same chirality (i.e. handedness) as the coordinate axes if their determinant is positive. If your coordinate system is assumed to be right-handed, then so will be the vectors. So compute the determinant of your matrix of eigenvectors. If it is positive, you are fine. If it is not, then you can either change the signs of one vector, or of all three, or swap two vectors. Chirality is always a property of three vectors, so asking whether a single vector is right-handed doesn't make a lot of sense.