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: 
There's a whole algebra ("Geometric Algebra", or Clifford algebra) behind the concept of multiplying vectors. Look there to find out where the handedness comes from, why $ \cos \theta $ for $a \cdot b$ but $\sin \theta$ for $a \times b$, and many other gems. [Here](http://faculty.luther.edu/~macdonal/GA&GC.pdf)'s a great reference.

Essentially, the cross product is a vector that describes a plane. A plane is described by two vectors (in any number of dimensions), but in $\mathbb{R}^3$ we have the handy situation that a plane can also be described by a vector. So, in 3-d only, vector $c = a \times b$ describes the plane described by $a$ and $b$. Since a plane has two sides, there's a two-fold arbitrariness ("orientation") that the choice of sign handles.

The $\sin$ and $\cos$ bit is slightly more complicated, but not much. Check out the reference.
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.