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 cross product originally came from the *quaternions*, which extend the complex numbers with two other 'imaginary units' $j$ and $k$, that have noncommutative multiplication (i.e. you can have $uv \neq vu$), but satisfy the relations

$$ i^2 = j^2 = k^2 = ijk = -1 $$

AFAIK, this is the exact form that Hamilton originally conceived them. Presumably the choice that $ijk = -1$ is simply due to the convenience in writing this formula compactly, although it could have just as easily been an artifact of how he arrived at them.

Vector algebra comes from separating the quaternions into scalars (the real multiples of $1$) and vectors (the real linear combinations of $i$, $j$, and $k$). The cross product is literally just the vector component of the ordinary product of two vector quaternions. (the scalar component is the negative of the dot product)

The association of $i$, $j$, and $k$ to the unit vectors along the $x$, $y$, and $z$ axes is just lexicographic convenience; you're just associating them in alphabetic order.
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>

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> 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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