There are several ways to detect a colision in AS3, and you have to choose the one that fits better with your needs.
ActionScript3 provides two native methods for testing colisions:
- obj1.hitTestObject( obj2 );
This function will return true if the bounds of the objects are colliding. Be careful, this will cause fake colisions if you are using shapes like circles, because the function will test the area of the shapes, not the pixels.
- obj1.hitTestPoint( obj2.x, obj2.y, true );
This function will return true if a coordinate colides with the shape of an object. Be careful you can only test one point, you may want to determine several points of colision in a shape, then test all of them to a better result.
Of course these are only natives functions, if you want to save you some time, you can import and use a colision kit. They can be very useful, since they have a lot of features and examples. A good one can be found here.
About your other question, tonypa have a great tutorial about vectors, that can be very useful when developing phisic games. The tutorials are written in AS3, and provides a lot of graphics representations. These can be found here.
If you do not want to write the full code, you can learn how to use boxD2, that provides a full environment whith phisics. But maybe you will want to learn some theory before using a phisic library.
Good luck in your studies.