The cheapest way to go from Norwich to Gatwick Airport is by car.
If you are able to drive all the way to Gatwick you can leave your car in the Airport car parks if you are not going to be away for a long time, checks prices here:
www.gatwickairport.com/parking/
The second best alternative is to take a National Express coach. The duration of the journey from Norwich to London Gatwick is long, from 5 to 6.30 hours and prices are from £23 to £43 for a single journey ticket. You can check and purchase the coach tickets in the following link:
www.nationalexpress.com/