Cross and Dot product of vectors. 1

Someone asked a very interesting question on 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.

April 16. Reply

Reminiscing a dreadful day in my life. Today 16th April, a decade ago, 2007.

It was early morning. Snow flakes were making the surrounding really beautiful. Especially because I was sitting inside of a cozy air conditioned student center, looking out of the glass wall. The hazelnut flavored milk-less coffee used to be one of my favorite, especially when I would be sitting alone. I was reading a paper on “phenomenology of quantum mechanical phases related to my thesis research”. These were the days I had started writing or at-least musing about writing blogs. So I was even perhaps in the mood of writing a poem or something.

The beautiful silence was broken by a strange noise of a crowd. I looked around. A man with a long gun, as long perhaps as himself, of-course a bit smaller, wielding it to the ground was hailing at the folks around “stay indoor”. More…

Interference by wave-front and amplitude splitting. Reply

Optics Series Lecture, Lecture – XIV, XV, XVI.

“Color of thin films, Newton’s rings, Lloyd’s mirror and Phase changes during reflection” These lecture were delivered on 16th February, 21st February and on 17th March. The lecture sessions were of 1 and 1/2 hours. The lectures were delivered to both Physics honors as well as Physics elective students on different days.

We have previously discussed what is interference and what is wave-front splitting and amplitude splitting interference. We have also discussed in much details two wave-front splitting interference viz. Young’s double slit interference (Lecture – IX) and Fresnel’s bi-prism (Lecture – XI). Today we will discuss one more wave-front splitting interference namely Lloyd’s mirror interference before moving onto the amplitude splitting interference of the Newton’s Rings. Also we will discuss two interesting and related concepts; i. Phase change on reflection and ii. Color of thin films. More…

A magic formula in MS Word. Reply

If you have come across a daunting problem in MS word where a new blank page appears without your volition and you lose your mind cos you just can’t delete it, relax, the following will restore your peace if it already was available to you before MS misbehaved. We have covered your back. Sorry I copied the following magical solution from a comment section from the web after suffering silently the fool that “software helplessness” is. Yes, the following works like a charming mother in law. Thanks MILs. More…

Waves. Reply

Optics Series Lecture, Lecture – XII and – XIII.

“Traveling waves, Differential wave equations, Particle and wave velocities.” These lectures were delivered on 17th and 20th February 2017, in two lecture sessions of 1 and 1/2 hours each. The web version has been named “Waves.” and the lectures were delivered to Physics honors students.

In one of our earlier optics session lecture I had hinted at having waves defined by their pulse shape called as wave profile — or alternatively wave shape or wave form, and transcribing them into forms that represent actual wave motion. The later are then called as traveling or progressive waves. The former, the so called wave shape or wave profile are then time-snapshots of the full fledged time varying waves that we just called traveling waves. Remember that stationary or standing waves are not wave profiles or any snapshots of a single traveling wave, they are rather the superposition of an advanced and a retarded wave — that is one traveling wave moving forward and another exactly shaped traveling wave moving in the reverse direction. We studied advanced and retarded waves, here. We have also already dealt with traveling waves in much detail, eg, here and here. This lecture will justify what we have been espousing all along. More…

Fresnel’s Bi-prism: measurement of wavelength of light. 1

Optics Series Lecture, Lecture – XI.

“Fresnel’s Bi-prism: measurement of wavelength of light by it.” This lecture was delivered on 16th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics elective students and later to honors students. This does not strictly pertain to 1 and 1/2 hours of regular lecturing session that we have mostly been employing. Thats because it was created with another part which can be optionally appended to other related subject matter. In the web-version thats what we will do. Our guiding principle is more in line with the honors course, where the subject matter is quite extensive and deep which brings more flexibility and choices into the lecture compositions.

Today we will discuss another interesting interference set-up, now that we have discussed the Young’s double slit experiment, in lecture – IX. A few words about the general mechanism behind interference. There are two kinds of interference basically that we will be discussing in our lectures. We discussed the Young’s DS interference pattern based on our understandings of intensity or irradiance patterns that we studied here: lecture – VII. Interference is sustained and visible if the corresponding sources of light are coherent among themselves, that is, if the sources have phase differences that are not arbitrarily or abruptly changing, as a consequence we can safely assume the phase differences are constant and therefore predictable. Incoherent light makes this impossible. Incoherent light is that light source whose production itself is arbitrary and abrupt and unpredictable, hence nothing can be definitively said on its phase, as a result the coherence is only short lived. If two light sources are so generated that their respective coherence time (or coherence length) are well within each others span, they are said to be coherent light. More…