Is it possible to express Fleming's Left Hand Rule and Right Hand Rule in terms of vectors? I recently studied Fleming's Left Hand Rule and Fleming's Right Hand Rule for electromagnetism (For locating direction of Force, Magnetic Field and Current). Using my hands to find the direction is very tedious. Is it possible to use these rules by using vectors? If yes, how? I searched in Google for this but could not get a satisfactory answer. Please help. 
 A: You could just calculate the vector components. These hand rules are used whenever vector quantities are related by a cross product. Let's take as an example the force on a moving charge due to a magnetic field:
$$\vec{F}=q \vec{v} \times \vec{B}$$
Let's write this out in terms of its components (assuming we're working in 3D):
$$\left( \begin{array}{} F_x \\ F_y \\ F_z \end{array} \right) = q \left( \begin{array}{} v_x \\ v_y \\ v_z \end{array} \right) \times \left( \begin{array}{} B_x \\ B_y \\ B_z \end{array} \right) = q \left( \begin{array}{} v_y B_z - v_z B_y \\ v_z B_x - v_x B_z \\ v_x B_y - v_y B_x \end{array} \right)$$
In principle, this gives you the direction of the force... but I doubt it will be faster than using a hand rule. I generally only use one hand rule, namely the right hand rule for cross products:

Given two vectors $\vec{A}$ and $\vec{B}$, point the fingers of your
  right hand in the direction of $\vec{A}$, then curl them towards
  $\vec{B}$. Your thumb then points in the direction of the outer
  product $\vec{A}\times\vec{B}$.

If you know the equation for the cross product quantity you're trying to calculate, you can always use this rule and don't have to think about whether you need to use a left-hand or right-hand rule.
A: If you have trouble remembering which is which (or which is the witch), the way I remember it is: If the electrons go CLOCKWISE around the coil, the north pole faces towards you, AND, a conductor or electron beam in the field is pushed to the side where the electrons are going in the same direction. If it is pushed in that direction through a magnetic field, the current generated in it goes the opposite way to oppose the magnetic field (that's why generators get harder to turn when there is a load on them). Now, if you are going to use conventional current, you have to throw your brain into reverse (because they stuffed up about 300 years ago & didn't realize it until too late to re-write all the books & papers), and substitute conventional current (which goes the opposite way), for real current. I prefer to work in electron current most of the time.
