College And Vectors

Ok, I did well in highschool physics. This is pretty much because the teacher slacked off. I did well in AP physics, too, mostly because the teacher taught the same thing he did in regular physics. Now I'm in college, and vectors, which I always thought were simple - you have an X component and a Y component, right? - have just gotten a lot more complicated.
Namely: Scalars, Scalar Products and Cross (Vector) Products.
I do not understand at all. I get that we have a new component (Z), and I get (I think) how scalars work (they are a way to express that a magnitude is actually a vector, i->x, j->y, k->z, they don't do anything else), and I understand the right hand rule (or a simplified version of it), but I don't understand how the process of multiplying vectors works. If someone could link to examples of dot/cross vector multiplication, or explain it, I would be very grateful.

http://en.wikipedia.org/wiki/Dot_product

http://en.wikipedia.org/wiki/Cross_product

What do you mean you have a new component? (I did Ib, so I don't know) did you not have 3 dimensional vectors in AP Physics or Maths? If you didn't, doesn't really matter, but just remember you can have an infinite amount of components (that will come in handy later).

Scalars scale things (sorry if that seems flippant, but that is really all they do).

If you did some linear algebra courses in the past which dealt with matrices, I would advise you find some book/website (using google) that explains vectors in terms of matrices because it can be easier for some to grasp vectors that way. If you didn't, don't because it will probably just add confusion.

Multiplying vectors is just like repeatedly adding vectors. Try to do some drawing of vectors of the form A + B (A,B vectors) and then try A + 2B, etc. It might seem patronising, but it might help (and what else would you expect from self-important tossers).

A dot (scalar) product can be thought of as a projection (you should have done that in physics), wikipedia talks about that.

A cross product is a bit different, it is not like dot product  because the answer is a vector (not a number). It gives you the vector which is perpendicular to BOTH vectors that were crossed (again, wikipedia does this nicely). If we use A x B as an example: and A and B lay on the x & y axes respectively, then A x B is on the z axis.

Your'e just performing linear algebra on simultaneous equations most of the time tbh.