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.
[Physics] Is it possible to express Fleming’s Left Hand Rule and Right Hand Rule in terms of vectors
electric-currentelectric-fieldselectricityelectromagnetism
Related Solutions
The formula for the force of a particle due to its magnetic field is $F = q \vec v \times \vec B$. The cross product has the property that its result is always perpendicular to both arguments.
Its direction is simply a result of how the cross product function is defined and the sign of electric charge (an electron is defined as negative).
It is important to note that the order of the arguments matters as the cross product function is not commutative; generally $ \vec A \times \vec B \neq \vec B \times \vec A$. For vectors the direction will be reversed; for matrices both sides can be wildly different.
Note that a current carrying wire produces a circular magnetic field that's why it doesn't matter how you hold your hand ie how you rotate your hand around your arm as long as your thumb shows the direction of the current.
Edit after comments:
See the illustration I've added below. Now use your hand in the way that you've learned and convince yourself that what I've drawn below is correct.
Best Answer
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:
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.