Step 1: Recall the rule connecting line and field. At every point on an electric line of force, the direction of the electric field is along the tangent to the line at that point. This fixes exactly one direction per point.
Step 2: Test the crossing idea. Suppose two lines actually met at some point \(P\). You could then draw two tangents at \(P\), one for each line, which means the field at \(P\) would be pointing in two directions at once. A vector quantity like \(\vec{E}\) cannot simultaneously have two directions at the same location.
Step 3: Interpret the situation. Therefore mutually intersecting field lines signify a self-contradiction, two values of field direction at a single point, so such a configuration is not physically permitted; electric field lines are always drawn so that they do not cross.
\[\boxed{\text{Intersection would mean two directions of }\vec{E}\text{ at one point, which is impossible.}}\]