Question: > 
> $\mathbf{a}\times \mathbf{b}$ follows the right hand rule? Why not left hand rule? Why is it $a b \sin (x)$ times the perpendicular vector? Why is $\sin (x)$ used with the vectors but $\cos(x)$ is a scalar product?
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So why is cross product defined in the way that it is?
I am mainly interested in the right hand rule defintion too as it is out of reach?

Answer: 
I have answers for your question. If it was defined relative to the left-hand rule, you'd currently be asking why not the right-hand rule. It's how we deal with ambiguity: by establishing conventions. Depending on how you choose the definition of cross product, the result could be one of two different vectors. This led to the establishment of a convention. If there were no conventions with cross products, things would get very confusing because some people would be coming up with vectors in the opposite direction from other people, and their results would be different, which would lead to confusion. It's also likely that some theorems wouldn't work out because they're based on one definition and not the other, etc; one would keep having to convert between definitions all the time; and it would get very annoying. 

The answer to your second questions? Upon establishing the definition of cross product, it just happened that the geometry worked out with sine and not cosine. It's just the behaviors of geometry in light of the constraints placed on it as a result having decided on a definition. 

The same answer goes for the scalar product. Once the definition was made, that's how the geometry worked out. 

However, it could be, though that the definitions were made so that they'd coincide with sin and cosine the way they did to unite a variety of approaches to the same set of problems, some people using trig, others using algebra to attack the same set of problems until some brilliant person united all the approaches with one definition. Such a thing often happens in mathematics, as anyone who's familiar with the concept of research might tell you. 

Hope that helps.
The real reason is because it is convention to use a right-handed coordinate system. If you switched to a left-handed coordinate system for some reason (I'm not sure why), then you would need to do your cross-products using a left-hand rule to get the same answers. That's really as simple as it gets. It's a result of our conventions in defining our coordinates the way we do.