Why can't two electrical field lines intersect each other? I know the textbook answer that the tangent to field line give the direction of electric field and intersection of two field lines would mean two directions of electric field which is not possible, but why not!?. Electric field at a point tells the direction of 'force' at that point. Well force! When force on a object is applied in more than 1 directions we take the vector sum of the forces. Similarly if forces(electrical) with different directions were acting on a charge kept in an electric field we could take the vector sum of the forces and we would get the directon of resultant force(or resultant field) 
P. S. : I know, i  sound totally stupid but this thought is not letting me live in peace. I totally agree that I could be horribly wrong. Even if i am please be generous enough to explain in detail. 
Thank you
 A: $\newcommand{\vect}[1]{{\bf #1}}$
Formally a field is a function of the coordinates, in this case $\vect{E} = \vect{E}(\vect{x})$, the key here is the word function, remember that for any value of the argument $\vect{x}$ there can only be one single value as a result of applying $\vect{E}(\vect{x})$. Therefore, at any position $\vect{x}$, the field must have only one value, or said differently, no two lines can intersect each other. What can happen is that they can be added, resulting in a new value, but that is equivalent to say $\vect{E}(\vect{x}) = \vect{E}_1(\vect{x}) + \vect{E}_2(\vect{x})$  and $\vect{E}(\vect{x})$ still is a well defined field.
More intuitively speaking, there are all sorts of troubles if you allow this to happen, for instance if a particle is moving according to $\vect{E}_1(\vect{x})$ what would happen at a location where $\vect{E}_2(\vect{x})$ is defined? Does it keep moving along according to $\vect{E}_1(\vect{x})$? $\vect{E}_2(\vect{x})$? You already know the answer: $\vect{E}_1(\vect{x}) + \vect{E}_2(\vect{x})$
A: I see the frustration - but I think it's resolved easily.
Electric field lines do not exist in reality. They are simply tools created by humans to help humans understand how electric fields work. If you have two electric field lines cross each other, you could better represent the physical scenario by having the antiparallel components cancel and drawing one non-intersecting line instead.
A second explanation would be to use the definition of electric field as $-
   \nabla\Phi(\vec{r})$. Then I think it's easy to see there'd be no intersecting lines if $\Phi$ is continuous/analytic.
