In two dimensions one can derive a simple formula for the angle of the eigenvector of a matrix if it is symmetric as in the case of the covariance matrix.
Since the equation is quadratic there are two solutions and one can substitute these into the equation for V to find the eigenvalues.
In three dimensions the initial equation in the derivation is that of a plane whose parametric equation is the composition of two independent unit vectors and corresponding parameters. One can use the eigenvectors corresponding to the two largest eigenvalues. The deviations from the plane are in the direction normal to it which corresponds to the smallest eigenvalue. For a line the parametric equation is the same as in two dimensions but there are deviations in the two directions normal to that of the line.