I've been trying to get the hang of vectors and I read Wolfire's blog about linear algebra and I seem to understand most stuff except when it comes to storing a direction as a vector and using it by multiplying the velocity by it. I know it has to be normalized (have a length of 1), but why does it have to be 1 specifically and how would I use it?
Also how would I represent it in 2D [x,y] coordinates. For example if I wanted a direction such as North-East?
A 2D vector has two values (x and y), and it basically says how far you go from the point of origin in the x- and in the y-direction. For example, a vector of (3,4) goes 3 units in x direction and 4 units in the y direction, resulting in an angled line with a length of 5 (3² + 4² = 9+16 = 25, root of that is 5). So the vector basically gives you two pieces of information:
1) Direction and
2) length
If you now want to calculate where an object moves, you take direction * speed, but you have to normalise the vector to the length of 1 so it doesn't skew your results. If you used the above vector as is, it would multiply your speed by 5 (its length), so your object would move five times as fast as intended.
If you have a normalised vector with a length of 1, the 1 is neutral in multiplications and doesn't cause you any trouble.
I'm sure there are more technical explanations, but that's the gist of it.
up/+y and East is right/+x, the vector (3,4) would point North-East, while the vector (3,-4) would point South-East. Both would have the (positive) length of 5 (squaring deletes the minus).
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Christian's answer already covers most of what you asked for expect your concrete example from the end of your question about representing "North-East":
First of all, it depends on your coordinate system. But if we look onto your world from the top and assume that North/South is the y-axis (+y = North) and West/East is the x-axis (+x = East), you could represent the direction "North-East" as the (un-normalized) vector [1, 1] or the (normalized] vector:
Example visualization in WolframAlpha.
It's just a standard and it makes things easier for everyone.
Why does it have to be 1 specifically?
It doesn't. It's just a standard. If you want so in your game, you can make all your vectors of length 2. I wouldn't call it normalized though because normalization has a mathematical definition of its own.
As long as you have all your vectors in the game standard (all of them of unit length N, doesn't matter what N is, but 1 is pretty standard), you could still do the direction*speed technique.
All that said, I don't know anyone who uses other unit lengths other than 1. I just want you to know that it doesn't have to be so, it's just much easier for everyone.
How would I use it?
A normalized vector would signify direction. That makes it convenient to mash up with speed.
For every frame, instead of doing position = newPosition or position = position + movementVector, you could generalize everything with position = position + direction*speed.
