Question: Solution Of Electromagnetic Problem

I'm going through what I appears to be a rite of passage for anyone who wants to call themselves physicists right now.  The infamous John David Jackson "Classical Electrodynamics".

 

I'm going through a solution to a problem that I found but have trouble understanding the last step of the solution for 9.8.  It's the integral involving the value (n.p)(nxp*), and how the author handles that integration and this vector quantity. The integral is over all solid angles.

The best guess I can come up with is some sort of vector identity that equates

(n.p)(nxp*) with (n.n)(pxp*) where the cross product is a constant vector outside of the integration.

Please help,

thanks

v/r,

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