Geometric algebra and simple geometric operations I'm not an expert on the subject, but If it is worth I'd like to start on getting some grasp on the subject.
Assuming geometric algebra framework, Is there somewhere a list of formulas where for example the intersection between a line and a plane, line and sphere, if the point belong to a plane etc?
I want to compare those formulas against the one would classically use to see whether or not possibly there's a benefit.

For a thorough introduction to Geometric Algebras have a look at:
Geometric Algebra for Physicists by Chris Doran and Anthony Lasenby
For a thorough look at the conformal model:
Geometric Algebra for Computer Scientists by Dorst,Fontijne and Mann.
This book contains a lot of very useful practical information on implementing and using geometric algebra in practice.
For a handy reference for the conformal model look at:
A Covariant Approach to Geometry using Geometric Algebra by Anthony Lasenby, Joan Lasenby and Rich Wareham (this is a very nice one, I work a lot with geometric primitive intersection and have this open most of the time. Full disclaimer: Joan Lasenby is my PhD supervisor so I am probably biased towards liking her papers as we use the same notation and think about GA in a similar way)
A very good collection of useful material comes from Guide to Geometric Algebra in Practice edited again by Leo Dorst and Joan Lasenby. This one contains a lot of good papers from one year of the AGACSE conference.
The big reference text for a GA is Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics by David Hestenes and Garret Sobczyk, this contains a huge amount of stuff but is fairly heavy going if you are just starting the subject.
If you are interested in playing around with an interactive environment for GA then ganja.js has a lot of examples and good visualisation capabilities (Disclaimer: Steven De Keninck, the author of this library, is a friend of mine)
