In computing the electric field at some point P in the x-y plane of a Cartesian coordinate system due to a linear charge distribution in the x-y plane each contribution to the electric field at point P from each infinitesimal element of the charge distribution is a vector.

Hence, to fine electric field   one must do a vector sums of an infinite numbers of infinitesimal vectors How does one deal with such an enormous vector addition problem?

Ans:

What one does is to find the x-component of the electric field at point P because of an infinitesimal element of the linear charge distribution and sum up all such x-components to get the total x-component. A second alike procedure yields the total y-component of the electric field at point P. The two components of is able to be added like vectors to yield the total electric field.