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?
> 
> 
> 

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: 
The cross product doesn't actually follow any such rule. It is defined arithmetically as a determinant:

$$\mathbf{u\times v} = \begin{vmatrix}
 \mathbf{i}&\mathbf{j}&\mathbf{k}\\
 u\_1&u\_2&u\_3\\
 v\_1&v\_2&v\_3\\
\end{vmatrix}$$

There is no hand rule there. The right hand rule comes from the conventions about how we visualize the result in 3-D vector space.

Firstly, the resulting scalar value is assigned as the length of a vector which is orthogonal to $\mathbf{u}$ and $\mathbf{v}$. That does not give us a right hand rule yet.

If we identify the first vector component with the $x$ axis, the second with the $y$ axis and the third with the $z$ axis, such that the positive ray of the $x$ axis shoots left, $y$ shoots upward, and $z$ points out of the page toward us, then we end up with the right hand rule.

Why? Because the cross product of a unit vector lying in the positive $x$ axis direction, with a unit vector similarly lying in positive $y$ produces a unit vector lying in positive $z$.

Based on how we laid out the axes, this gives us the right hand rule: if we extend our thumb in the direction of $z$, then the fingers curl from $x$ to $y$.

Your question is really about why we lay out the axes this way, and that is merely a convention, like driving on the right side of the road (in the majority of the world's countries, not all, obviously).

The important thing about conventions is only a) to have them and b) have them be consistent. The choice between two equivalent conventions, like whether to choose one system or its exact mirror image, is unimportant; there is no external reason to go one way or the other.
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.