Sorcerer is perfectly fine for an Eldritch Knight/Arcane Archer build.
The lack of armor is nowhere near as debilitating as you'd think at first glance, especially if you work around it a bit.
You could go with a build like ... Divine Hunter (Paladin) 2/Sorcerer 8, Arcane Archer 10.
With SLAs, you could grab EK 10 in there sooner (so Paladin 2/ Sorc 1/ EK10 /AA4 /finish with 3 levels of sorc) This puts you at 18 BAB, 16th level sorc casting (kinda bleh, but still servicable). You can mix the AA levels in somewhere with EK to get imbue arrow sooner, but it's not really that needed.
The second option will work best, but I would suggest taking the early entry into eldritch knight until you meet the arcane archer requirements and then going back into eldritch knight after level four in the AA prestige to preserve caster levels.
You ought to take magical knack as well.
With this set up you could do some pretty cool stuff, like confusion arrows and other aoe's like cloud spells. Pretty fun sounding to me :)