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: 
the Gibbs Vector Algebra Cross Product Right Hand Rule is a Convention

Any description of the (nominally) Euclidean 3 dimensional world we perceive runs into Chirality, the distinction between Right and Left handedness, the difference in a Mirror Reflection. 
<https://en.wikipedia.org/wiki/Chirality>

A Mathematical System describing this world has to have a Convention for coordinate order, orientation.  

Justifications for the Right Hand Rule for the Vector Cross Product are a consequence of the choice of Chirality in the the "Standard Basis" and are circular, not deep - you just have to choose.

In fact the other choice has been used:
<http://web.stanford.edu/class/me331b/documents/VectorBasisIndependent.pdf>

> 
> The right-hand rule is a recently accepted universal convention, much like driving
>  on the right-hand side of the road in North America. Until 1965, the Soviet Union used
>  the left-hand rule, logically reasoning that the left-hand rule is more convenient because
>  a right-handed person can simultaneously write while performing cross products.
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As a complement to Bye\_World's illuminating answer, which defines $b \times\ c$ as the vector whose dot product with $a$ equals $\det(a, b, c)$, $\|b\|\|c\|\sin(\theta)$ is the surface of the parallelogram formed by vectors $b$ and $c$. If $a$ is a unit vector orthogonal to the $b$-and-$c$ plane, then $\det(a, b, c)$, which is the (signed) volume of the parallelepiped formed by the 3 vectors, equals the surface of the $b$-and-$c$ parallelogram. In this case, i.e.$\|a\|=1$ and angle from $a$ to $b$ (and to $c$) equal to positive $\pi/2$, one can picture $b \times c$ as $a$ times the surface of the parallelogram.