I'm looking for examples of vector multiplication where the resultant vector is at a right angle to the two initial vectors. This convention seems to have arisen in mathematical analysis of certain phenomena where multiplying two vectors gives a resultant that runs at a right angle to them and commonly referred to as a "Cross Product" such as A X B = C

Faraday's law is a prime example where the motion vector (M) is multiplied by the magnetic vector (B) to give the resultant vector of voltage (V) or current (I) which is at a right angle to the plane of M and B.

Another example might be where the principle stress (such as tension T) in a bar creates another stress (compression C) that runs at a right angle to the tension. This vector relationship is often known as a "tensor".

However, there are many cases where multiplying vectors creates a resultant vector that runs parallel to the initial vectors. Force X Displacement = Work is one example.

 

The Lorentz force: https://en.wikipedia.org/wiki/Lorentz_force

 

Hi,

The direction is in the direction of the vector normal to a and b in axb.


What is very interesting to me is that when we push on something in common experience if we push it in one direction it goes in that direction, and if it is moving in a straight line at speeds less than the speed of light and we quickly push on the left 'side' of that object it deflects in the SAME direction as we pushed (to the right).
The Lorentz force however shows that when we have an electron moving in a straight line and we apply a magnetic field on the left, it deflects DOWN (or up depending on the orientation of the mag field). This is very different from ordinary common experience with regular objects. It's almost like a wedge action or a spin action rather than a direct pushing action.

 

The gyroscope is another example of vector multiplication; this one involving angular momentum.

\(\bold L\;=\;\bold r\;\times\;\bold p\)

Where r is the radial position vector and p is the momentum vector. L will always be perpendicular to r and p.
In 3 dimensions:

1. A scalar is a tensor of rank 0, and has one component.
2. A vector is a tensor of rank 1, and has three components.
3. A tensor of rank 2 has nine components

 

The Lorentz force however shows that when we have an electron moving in a straight line and we apply a magnetic field on the left, it deflects DOWN (or up depending on the orientation of the mag field). This is very different from ordinary common experience with regular objects. It's almost like a wedge action or a spin action rather than a direct pushing action.

One way to look at the magnetic component (the ‘Coriolis-like’ point of view) of the Lorentz force is to see it as a relativistic effect of the electric force. The electric and magnetic fields are related by coordinate transformations like forces on merry-go-rounds, tops and wheels.

https://arxiv.org/pdf/1109.3624.pdf

This pretty argument, however, does not work if the test charge moves transversally to the wire. Also, it does not reveal the underlying reason for the “strange directional character” of the magnetic force, perpendicular to the velocity, hence doing no work, similarly to a Coriolis force – of kinematic (rather than dynamic) origin.
...
But it is true that in general there is no need to mention Thomas rotations. Still, from a conceptual point of view, it is nice to understand wherefrom comes the peculiar Coriolis-like nature of magnetic forces (and, also, that electromagnetic fields transform as second-rank tensors simply because they parametrize generators of Lorentz transformations).

http://farside.ph.utexas.edu/teaching/302l/lectures/node72.html

However, this angle is always

for the force exerted by a magnetic field on a charged particle, since the magnetic force is always perpendicular to the particle's instantaneous direction of motion. It follows that a magnetic field is unable to do work on a charged particle. In other words, a charged particle can never gain or lose energy due to interaction with a magnetic field. On the other hand, a charged particle can certainly gain or lose energy due to interaction with an electric field. Thus, magnetic fields are often used in particle accelerators to guide charged particle motion (e.g., in a circle), but the actual acceleration is always performed by electric fields.

A uniform magnetic field cannot change the speed (kinetic energy) of a charged particle. If it could then we would be able to build real 'free-energy' machines using magnets just like the bogus ones on youtube. The magnetic force is always perpendicular to the particle's velocity(particle's speed together with its direction), therefore it cannot change the particle's speed, only its direction.

http://farside.ph.utexas.edu/teaching/302l/lectures/node73.html

 

One way to look at the magnetic component (the ‘Coriolis-like’ point of view) of the Lorentz force is to see it as a relativistic effect of the electric force. The electric and magnetic fields are related by coordinate transformations like forces on merry-go-rounds, tops and wheels.

https://arxiv.org/pdf/1109.3624.pdf


http://farside.ph.utexas.edu/teaching/302l/lectures/node72.html


A uniform magnetic field cannot change the speed (kinetic energy) of a charged particle. If it could then we would be able to build real 'free-energy' machines using magnets just like the bogus ones on youtube. The magnetic force is always perpendicular to the particle's velocity(particle's speed together with its direction), therefore it cannot change the particle's speed, only its direction.

http://farside.ph.utexas.edu/teaching/302l/lectures/node73.html

Hi,

I was thinking about relativistic effects also, in simple terms, that if we push on a particle that is traveling at the speed of light it can not change in that same push direction unless maybe energy is lost somehow because that would imply that it suddenly could go faster than the speed of light. Given that as a limit, it must move in a direction that does not violate that limit, and there's only one direction left, and that is in the direction orthogonal to both of the other directions.
I'll try to read some of your links.

 

Given that as a limit, it must move in a direction that does not violate that limit,

Or you just add to it's mass. But, if it was traveling at the speed of light, it was massless to begin with.

 

Force X Displacement = Work is one example.

It's been a long time and I may be way off, but I don't believe Work is a vector.

 

It's been a long time and I may be way off, but I don't believe Work is a vector.

You are correct -- it is not a vector. It has magnitude, but no direction. Dot products are scalar quantities, or if you prefer tensors of rank 0.

 

Hi,

I was thinking about relativistic effects also, in simple terms, that if we push on a particle that is traveling at the speed of light it can not change in that same push direction unless maybe energy is lost somehow because that would imply that it suddenly could go faster than the speed of light. Given that as a limit, it must move in a direction that does not violate that limit, and there's only one direction left, and that is in the direction orthogonal to both of the other directions.
I'll try to read some of your links.

What's amazing is that length contraction, time dilation and relativistic effects happen at the slow speeds of electrons in wires due to the speed limit of c. The electrostatic forces needed to separate charges in a good conductor are enormous so while the individual effect on a electron is small the numbers of conduction electrons moving is astronomical resulting in the easily seen effect of magnetism.

https://en.wikipedia.org/wiki/Relativistic_electromagnetism#Principle

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

By the way, it's remarkable that we can measure magnetic forces at all, since the average drift velocity in a household wire is only a snail's pace: v/c is typically only 10e-13, so the Lorentz factor differs from 1 only by about one part in 10e26. We can still measure this effect because the total charge of all the conduction electrons in a meter-long wire is tens of thousands of coulombs; two such charges separated by only a few millimeters would exert enormous electrostatic forces on each other.