Someone asked a very interesting question on the role of vectors in Physics. He was curious to know if dot product of vectors is natural but vector product is just syncretism, — that is make shift or unnatural manipulation.
Every vector can be resolved into two components. The cosine and sine components (any two vectors would constitute a plane) while cos part can represent the projection defined through dot PDT we can’t leave out the sine part. It plays its role through the vector or cross PDT.
The vector direction is no more along same direction as original vectors because of orthogonality. To preserve symmetry of both orthogonal components (or equal footing of both vectors, vector a and b eg) we need the 3rd dimension. Hence such a definition of cross PDT.
Eg the emf generated in a changing “mag field” (Faraday’s law) depends on change in mag field if area is held const. It also depends on change in “area vector” if mag field is held const. So there are two vectors involved and their transverse values matter (and not their longitudinal values). To preserve equivalent role of both area and mag field vectors the resultant vector must be in a 3rd orthogonal direction.
Also think of this; a scalar is not necessarily directionless. (think electric current or even temperature or heat gradient etc) They just do not have the full fledged capacity of vectors. Its like flower bud vs fully blossomed flower.
So scalars can’t be added like vectors. We tend to make a mistake here. We say scalars don’t have a direction. That’s totally erroneous. They do have direction and it matters. Which direction you want to stick to if the current flows along certain direction only
Lets make it still more clear.
If there are two directions in which there are electric currents, we say they are both equal, the direction won’t matter. That’s where we make the mistake. We should say they are equivalent and not equal. Equality is ideal, its mathematical. But equivalence is physical. Its the effects of both currents in a certain sense that make them equivalent, but their strict equality does not follow.
Talking about equivalence: what matrices are in ideal or mathematical situations, tensors are in physical situations. Just like vectors were row or column matrices in their ideal formulation.
So when forming dot product either of the vectors can lend its direction and the projection of the other vector is multiplied with magnitude of the reference one.
Although scalars then become directionless because they are just magnitude of two vectors multiplied together, they still have an innate sense of direction, based on the reference vector.
They no more remain as valiant as before when the original vectors were considered. But A.B can be found either along A or B, giving equivalent but not strictly equal result.
But projection is just the cosine component of one of the vector along the other. This discards the other (i.e. sine) component because in the physical nature of things it wouldn’t matter.
If you apply a force perpendicular to some object all it will do is change the original direction of motion, transverse to the direction of force. But it will not change all possible inertia, that is the speed of the motion. As a result it does no work, since the displacement is zero, given the speed did not change. (Force did not produce additional displacement) Magnetic forces are notorious for that. They are lazy. They do no work. They only take you round and round telling you stories, like the HRD ministry. (Frictional forces are the opposite in a sense, they spoil your work, like religious groups.)
So when two vectors are as important as they can collude to act along or opposite to each other, eg displacement and force vector, all it matters, is to know or employ their longitudinal components.
Such components of forces correspond to change of inertia of speed and not direction. In general inertia is just velocity vector, change velocity the inertia changes, as does the momentum, hence the force, that’s the essence of Newtons first and second law. That it changes is Newtons first law and how much it changes second law.
What would happen if a charge which is moving is placed in a magnetic field?
It would experience a magnetic force. Such forces arise from the effect of two vectors. One; the velocity vector of the charge. Two; the magnetic field vector. And if the velocity is along the field direction, it is seen that the inertia does not change atall. Thus while we expected some amount of work we don’t get any work done, because now there is no force.
The only other possibility that remains is when the charge is moving in a transverse direction to the field. All general cases are superposition of both independent or orthogonal cases, the longitudinal or along the line and the transverse or perpendicular to the line, cases. So we need be concerned only, about what happens for the transverse case.
In this case the force depends on two vectors. 1. a. The magnitude of field and b. its direction. 2. a. The magnitude of velocity and b. velocity or motion direction.
The resulting force does not discriminate between the field vector or the velocity vector. And we already saw that there is no force when these two vectors are along a line or opposite to it. We are concerned only about knowing what happens when the velocity vector and field vector are perpendicular or transverse to each other. Equivalence of two vectors would lead to a symmetry. Their magnitude as well as their directions matter equally well.
The total effect (like in the case of dot product and work done) is a product of magnitude of two vectors, one with the perpendicular component of the other with respect to the first one as a reference. Thus this satisfies the symmetric situation in terms of magnitude. (i.e. it does not matter which vector you take as the reference, there is only one angle between them, both vectors as reference will give equivalent results)
The only other symmetry that is required is that of direction. The only possible direction which is not biased wrt one of the vectors is “orthogonal to both vectors direction”.
Right Handed or left handed is merely convention left as an anthropic liberty which must be used consistently. Cross product is much more involved than dot product but rightly so. Its not an artificially inseminated idea, just to satisfy our quest of finding any kind of glory in doing so.