L, M and N are the second fundamental coefficients, and E, F and G are the first fundamental coefficients.
Since $\kappa~d\vec r+d\vec N$ is perpendicular to both $\vec r_1$ and $\vec r_2$ I can say that it is along normal to the tangent plane at that point since $\vec r_1$ and $\vec r_2$ are the basis for the tangent plane.
As is given in the book that it lies in the tangent plane it has to be a zero vector since it can't be both normal and tangential to the surface.
I have a problem with the statement that it lies in the tangent plane.
$d\vec r=du~\vec r_1+dv~\vec r_2$, so $d\vec r$ lies in the tangent plane. Now, the vector $\kappa~d\vec r+d\vec N$ will also lie in the tangent plane if $d\vec N$ lies in the tangent plane. How can I prove that $d\vec N$ lies in the tangent plane?
$\vec N\cdot \vec N = 1$, so $2d\vec N\cdot\vec N = 0$.