Just based on general 3D knowledge, I would assume:

- Normalize normalises a vector, e.g. returns a vector with the same direction but magnitude (length) of 1. http://en.wikipedia.org/wiki/Normalized_vector
- Dot gets the dot product of two vectors. http://en.wikipedia.org/wiki/Dot_product
- Cross gets the cross product of two vectors, and you're pretty much on the money with your answer. http://en.wikipedia.org/wiki/Cross_product

TransformCoord... not sure. I'm not a D3D guy.

Normalize - divides each component of the vector by the vector's total length (the length of a vector v = (x, y, z) is computed by sqrt(x*x + y*y + z*z), also can be written as |v|). The length of the new vector is now equal to 1
.
TransformCoord - transforms the vector using the transformation defined by the matrix argument. The w component is assumed to be equal to 1. Read about matrix maths if you aren't sure what this means

Dot and Cross are two different ways of multiplying vectors together. Dot returns a single number, whereas Cross returns a vector

Dot - calculates the dot product of 2 vectors, which is the younger sibling of the cross product, speaking mathematically. If v1 = (a, b, c) and v2 = (d, e, f), then Dot(v1, v2) returns (a*d + b*e + c*f), the sum of the multiples of components from each vector. This is also equal to |v1||v2|cos(t), where t is the angle between the two vectors in 3D space, and also notice that Dot(v1, v1) = |v1||v1|cos(0) = |v1|[sup]2[/sup] - the dot product of the vector with itself is equal to the length squared, if you look at the two formulas this is easy to see. This is useful for many many things

Cross - calculates the cross product of 2 vectors. The returned vector is, as you said, perpendicular to the two vectors. If I set v3 = Cross(v1, v2), then the length of v3 is equal to: |v3| = |v1||v2|sin(t), where t is the angle between v1 and v2. Now if v1 and v2 are parallel then notice that t = 0, therefore the returned vector is (0, 0, 0).

Now the really interesting thing about Cross is that the answer is different if you pass in the two vectors in different orders, In other words, Cross(v1, v2) is not equal to Cross(v2, v1). (To be precise, Cross(v1, v2) = -Cross(v2, v1)). If you think about it, if v1 and v2 are (non-parallel) vectors, there are 2 possible vectors v3 and v4 perpendicular to v1 and v2 - they are the negative version of each other (v3 = -v4). For example, if v1 = (1, 0, 0) and v2 = (0, 1, 0) then you can easily see that v3 = (0, 0, 1) is perpendicular to both v1 and v2, but also v4 = (0, 0, -1) is as well! the Cross product of two vectors 'decides' which direction to return based on the order, so if you pass in the two arguments the other way around you will get the opposite direction.

Ok that was quite a lecture, if you still don't understand something then speak now or forever hold your peace

Is the length the distance from the origin? So if I had like (0, 10, 0) it's length is 10? Or (0, 10, 5) is 15?

A vector is often used to indicate a direction rather than a point, but the distance from the origin idea is close enough for magnitude. Think of it as the distance (as the crow flies) from the origin to the point. For example, (0, 1, 0) has length 1, (0, 1, 1) has length square root of 2 (roughly 1.4142).

[quote name='jefferytitan' timestamp='1338512556' post='4945144']
Normalize normalises a vector, e.g. returns a vector with the same direction but magnitude (length) of 1.

Is the length the distance from the origin? So if I had like (0, 10, 0) it's length is 10? Or (0, 10, 5) is 15?
[/quote]

Yes, if you picture yourself the vector as a point in 3D space, then magnitude of the vector is distance from the origin.
But your examples are wrong!

Magnitude of vector (0, 10, 0) really is 10
But magnitude of vector (0, 10, 5) isn't 15 but 11.18034...
15 is the distance from origin, but not the shortest (straight) distance, you simply went 10 units on the y axis and then 5 units parallel with the z axis, which made the resulting 15. But if you went straight from the origin to the point (0, 10, 5), you would get there in 11.18034 units.

How did I get this weird number?
Using thePythagorean theorem in 3D space:
magnitude[sup]2[/sup] = x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup]
Where x, y, z are the elements of the vector. So in your example:
magnitude[sup]2[/sup] = 0[sup]2[/sup] + 10[sup]2[/sup] + 5[sup]2[/sup]
magnitude[sup]2[/sup] = 125
magnitude = 11.18034...