Why does Fleming's right hand rule work? Fleming's right hand rule shows that if you have a force in a certain direction and a magnetic field to in another direction, current will flow in a direction perpendicular to both. If you had a motion upwards, and a field to the left, current would flow towards you. but why not away from you? If it flows away from you it would still be perpendicular to both motion and magnetic field, so what makes it flow towards you instead of away from you?
I am looking for a physical explanation more than a mathematical one.

 A: The short explanation is that it works by convention. In other words, there are quantities that are unambiguous and everyone agrees on, like forces and velocities1. The magnetic field is, believe it or not, not one of those. The force produced by a magnetic field is calculated using the right hand rule, which seems arbitrary. The important part, though, is that we calculate the magnetic field using the same arbitrary right hand rule. Interestingly, when used in pairs like this, the arbitrary nature of the right hand rule cancels out.
In algebra, we say that the magnetic field of a wire is proportional a current by
$$\vec{B}=\frac{\mu_0 I}{2\pi r} \left(\hat{I}\times\hat{r}\right),$$
where $\hat{I}$ points in the direction of current flow and $\hat{r}$ is perpendicular to the wire and points from the wire to the point we're measuring the magnetic field at. Rather than work out what direction that is, let's drop this into the Lorentz force law
\begin{align}
\vec{F}&=q\vec{v}\times\vec{B}\\
&=q\frac{\mu_0 I}{2\pi r} \vec{v}\times\left(\hat{I}\times\hat{r}\right) \\
&=q\frac{\mu_0 I}{2\pi r}\left[\left(\vec{v}\cdot\hat{r}\right)\hat{I}-\left(\vec{v}\cdot\hat{I}\right)\hat{r}\right].
\end{align}
All of the cross products, and accompanying arbitrary right hand rules, have vanished! 

1. Note that there is actually an arbitrariness in the definition of velocities and forces. We draw velocities as pointing from where the object was and to where it will be, but we could have chosen the opposite convention and not much would have worked out differently, just some minus signs being shuffled around. Similarly, we could have chosen the opposite convention for forces.
A: The basic thing in electrodynamics is lorentz force. It has a cross product of v×B so use the cross product rules to find direction of force everywhere it will be more helpful
A: The basic ingredients in determining the nature of the force acting on a moving particle are (restricting to classical physics) the collection of variables $(q, \mathbf{v}, \mathbf{B})$, with $q$ being the charge assigned to the particle, $\mathbf{v}$ its initial velocity vector and $\mathbf{B}$ the magnetic field the particle lives in. 
It is experimentally true that the force happens to be orthogonal to the plane where the pair $(\mathbf{v}, \mathbf{B})$ lies, therefore it must be either in the direction of $\mathbf{v}\times\mathbf{B}$ or the opposite. It is in addition experimentally true that, given the definitions that we chose for electric charges (which are completely arbitrary and determined mostly by measurement processes) the force happens to be proportional to $q(\mathbf{v}\times\mathbf{B})$: had we chosen different conventions for the electric charges, we would have re-written the above as $-q(\mathbf{v}\times\mathbf{B})$ (and equivalently many other laws of physics where we do have a minues sign would have been re-written accordingly).
A: The magnetic field $\textbf{B}$ is not a vector, instead it is an anti-symmetric tensor. In 3D such creature has three independent components and can be assigned an almost vector nature. Because the "almost" we call it an axial vector. When an anti-symmetric tensor is multiplied (here to calculate the Lorenz force) with a true vector such as velocity you get the so-called vector product of an axial vector (representing the anti-symmetric tensor) and a true vector that is sometimes called polar vector to distinguish it from an axial vector. The three components of the anti-symmetric tensor represents a planar rotation either to the right or to the left and that is unambiguous but when you represent it as an axial vector you need the some convention to set the direction of the rotation that is in the plane perpendicular to the axis. That is the right (or left) hand rule.
