I have always had this doubt in my mind that when things like force due to moving electric and magnetic field were being discovered, how did physicists know that the Force acting on a charged particle moving in a Magnetic Field (or a moving magnetic field in an electric field) will be a Cross Product?
Since, from what I know these terms had to be previously defined.
To summarize, I can understand intuitively what is a dot product and why it is needed. Sometimes we need a component of a vector acting along the direction of another vector (that is being dotted with) to get the value of a resultant vector.
For eg, while considering Work, we know that it is always supposed to be in the Direction of Displacement, and so the Force needs to be in the direction of displacement, so when we have a force acting in a direction not equal to the direction of the displacement, we just have to take the component of force acting in the direction of displacement. and that's why we have the formula
W=FScos(theta) force and displacement always have to be in the same direction
In order to project one vector on to another and then just multiply with that component, which is very similar to one projecting a vector onto x y or z-axis in a right-handed cartesian coordinate system.
But there is no intuitive way I can imagine why two coplanar vectors can ever result in a third dimension. And how do we know whether 3 more such quantities in the physical world are actually related in such a manner?
like I said I didn't understand why force on a moving charged particle in a magnetic field would be in some other dimension.
I hope I am clear with my explanation. I appreciate every answer.