Understanding symmetry in electric field calculations

[Vaentus]

Homework Statement

Consider the specific case of a point above the axis of a circular ring of charge, how do the calculations follow to cancel the radial components? I understand the concept of the symmetry but don't understand how to express it in the expression without just removing the term.

Homework Equations

$$E=\frac{1}{4\pi\epsilon_0}\int{\frac{1}{r^2}\hat{r}dq}$$

The Attempt at a Solution

Integration of the integral is very straight forward due to the constant radius of the circle and constant height of the point as well as that dl can be simplified to RdΦ such that dΦ is from 0 to 2π but I still have a term in the radial direction that doesn't disappear.

 

[Simon Bridge]

That question has already been answered in your other thread:
Electric field above a circular loop
... note: "the radial direction" is not just one direction - it is all directions perpendicular to the axial (z) direction. Some of those directions are opposite each other.