The incompressible three-dimensional ideal flows develop very thin pancake-like regions of increasing vorticity, which evolve with the scaling $\omega_{\max}\sim\ell^{-2/3}$ between the vorticity maximum and the pancake thickness. In this work we describe this process from the point of view of the vortex lines representation (VLR). Based on two numerical simulations in anisotropic grids with $1536^3$ total number of nodes, we examine the structure of characteristic matrices for the VLR (the Jacobi matrix and the Hessian matrix of the Jacobian), and link these matrices with the emergence of the scaling law $\omega_{\max}\sim\ell^{-2/3}$.