Given 3 orthogonal vectors how to calculate the ellipsoid induced by them? Given 3 orthogonal vectors, how can I define their inscribing ellipsoid?
$$ax^2 + by^2 + cz^2= 1$$
or 
$$(\mathbf{x-v})^\mathrm{T}\! A\, (\mathbf{x-v}) = 1$$
Meaning, given 3 principal orthogonal directions how to craft the ellipsoid that inscribes them in the two form above.

The equation of a triaxial ellipsoid of principal semi-axis $\alpha,\beta,\gamma$, referred to the symmetry axis, is:
$$
\frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}+\frac{z^2}{\gamma^2}=1
$$
so, in your first equation:
$$
a=\frac{1}{\alpha^2}\qquad b=\frac{1}{\beta^2} \qquad c=\frac{1}{\gamma^2}
$$
In the second equation the eigenvalues of the matrix $A$ are the inverses of the square of the semi-axis ( the analogous of $a,b,c$ in your first equation) , and the eigenspaces of the matrix are the axis of symmetry.
