Vectors and vector notation.

It is a sad fact that many so-called "theoretical" physicists are terribly confused about vectors. They do not know the difference between a vector and a scalar, and some imagine a vector is simply a list in the form (x,y,z,t) which they confuse with an event.

The components of a vector are themselves vectors, something that is fairly intuitive to the high school student familiar with the above diagram.  The diagram can represent many things; examples are the forces acting on a point or the velocity of a point. The numbers represent the "length" of a vector, they are not the vector itself; of course neither force or velocity can have length. Force and velocity vector diagrams merely use length as a convenient aid to visual understanding; a point (x,y) or (x,y,z) is in fact a vector and has fixed distance from the origin of a cartesian coordinate system. Make no mistake, position is a vector. These numbers are called scalars, but they have no direction. They are the magnitude of the basis vectors only. The basis vectors (in our three dimensional universe) are (1,1,1), and they have to be independent of each other. We use a cartesian coordinate system to represent the space, and rotations and offset of this coordinate system is arbitrary, we merely select a chosen direction and call it the X-direction. This is the same image with rotation (the background is the sky).

The requirements of a vector are as follows:

1. Commutativity: x+y = y+x.

We see that with this diagram.

We take the green first and then the red and arrive at the same point as if we had taken the red first and then the green. Note the use of colour here, I've used it to stress what is a vector and what is a scalar.

The non-mathematician will say "Well, that's trivially obvious, of course x+y = y+x", but beware, that it isn't always true.

The left Androcube (borrowed from Rubik, of course) has the green face rotated first, the right Androcube begins with the purple face. Hence

x+y ≠ y+x

Notice the top of the left cube has two blue faces, the top of the right cube has two purple faces. The final results are different, care must be taken to rotate in the correct order.

"Ah", you say, "what about a polar coordinate such as (1, p/4), isn't that a vector?"

Well, yes, it is! So what's gone wrong? 

The polar coordinate (1,0p) can be rotated around the z-axis to (-1,1p), but that only works in the x-y plane. There is nothing to show how deeply into the screen or out of it the vector is. We need another rotation for the x-z and y-z planes.  The rotation of the Androcube is the proof. 

 To denote the axis about which the rotation take place, the notation (1,0p, ^x), (1,0p,^y), (1,0p,^z) could be used, as could (1,0p)_x, (1,0p)_y (1,0p)_z  but it is not conventional to do so and would only lead to further confusion. There are enough notational difficulties as it is without adding more to them. Astronomers use the terms "Right Ascension" (R.A.) with units of time (hours, minutes, and seconds) and "Declination" (units in degrees) with units of distance in parsecs (parallax seconds), although recently the light-year has gained ground. Astronomy may be the oldest science, it is also the most stupid.  

Notice that I mixed scalars with vectors but used bold type. That is conventional, I make no apology and agree with some way of distinguishing the two. What I have done above is misled you when I say x+y ≠ y+x, what I should have said is x_*y ≠ yox, where "_*" and "o" are different rotation operators.

Much confusion arises out of the symbol "+", it doesn't always mean simple addition of numbers. In Boolean logic it means OR and in vector algebra it means the combination of independent vectors. It is better to write xoy ≠ yox  where "o" is a general operator but convention has already dictated otherwise, we have inherited "+" and have to live with what is, after all, merely a notation difficulty which leads to a severe notion difficulty. Mathematical notation is shorthand, it must be read very carefully. When Sir Isaac Newton said in his "Principia"

he wrote a paragraph that can be reduced to F = dp/dt, so long as we understand the terms and realise when he used the phase "are directly contrary to each other; or obliquely joined" he was talking about vectors.

The four axioms of a group (borrowed from Wackypedia) state

The Androcube and Rubik's cube are groups and meet these requirements, as do vectors, but it must be stressed that even though a * b   ≠  b* a  for the cube example given, there is some b such that  a * b = b * a = e. The rotations can be reversed to return to the identity. Not only that, but one can continue rotating through many configurations to form a path from the identity back to the identity, much as a plane can fly from London in any direction on a great circle and return to London after circumnavigating the world. Or it could merely taxi out to the runway and return to the gate. The spaces in which aircraft fly and fish swim are closed. One obvious limitation of the Androcube and Rubik's cube which doesn't apply to fish and aircraft is that the cube cannot rotate in more than one plane simultaneously. That could lead to a discussion of metrics, but I'll leave that for another time.  What I want to stress here is that the operator (_*, o, +) representing the three operations of rotation of the cube is itself a vector and not a single operator.

 Notation can be very misleading and I cannot produce "+" in bold type, it looks like "+". 

Increasing font size shows there is a just barely noticeable difference.

I can produce + in bold type with a larger font, it looks like +. 

The scalar length of the vector (3+4) is 5, because 5 = (3^² + 4^²)^-1 (by the Euclidean metric).

The addition of scalars (3+4) is 7.

The addition of vectors (3+4) is (3x,4y). Vector addition is not scalar addition.

It is conventional notation that (3,4) means (3x + 4y) and that x and y are unit vectors.

2. Associativity: (x+y) + z = x + (y+z)

We see that here:

I have used colour to show the vector (x+y) and the vector (y+z) are the combination of vectors.

3. Additive identity: For all x, 0+x = x+0 = x

4. Existence of additive inverse: For any x there exists a -x such that x + (-x) = 0

It is here that the "theoretical" physicist first runs into trouble.

Time does not "run" backwards!

A force can be counterbalanced by an opposing force, a velocity or a distance by an opposing velocity or distance, but there is no way to return to when you were as you can to where you were.

'Really, this is what is meant by the Fourth Dimension, though some people who talk about the Fourth Dimension do not know they mean it. It is only another way of looking at Time. There is no difference between Time and any of the three dimensions of Space except that our consciousness  moves along with it.' -- Herbert George Wells - "The Time Machine" - 1895.

When H.G. Wells wrote that he was writing a work of fiction for our amusement, not a physics paper. Mathematics is a wonderful tool to describe the workings of Nature, but as with all tools it must be used correctly. One does not use a hammer to turn a screw or a vector to reverse time. No matter how you write -t, it isn't going to happen; whatever else time may be, it is not a vector. It cannot be slowed or accelerated, no scalar can modify it. It cannot be hyperbolically rotated as if were a length, one simply cannot arbitrarily assign the symbol t in place of z or y, call it a vector and willy-nilly shuffle symbols like a teenager taking his first math examination, it is meaningless as calling an apple an orange or a loaf of bread. Any so-called "physicist" who attempts to do so is off his rocker, he has no knowledge of mathematics. 

Around 1886 Albert Einstein began his school career in Munich. As well as his violin lessons, which he had from age six to age thirteen, he also had religious education at home where he was taught Judaism. Two years later he entered the Luitpold Gymnasium and after this his religious education was given at school. He studied mathematics, in particular the calculus, beginning around 1891.

In 1894 Einstein's family moved to Milan but Einstein remained in Munich. In 1895 Einstein failed an examination that would have allowed him to study for a diploma as an electrical engineer at the Eidgenössische Technische Hochschule in Zurich.

Ref: http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Einstein.html

It cannot be proven but it is a curious fact that Einstein's weird fascination with time coincides with H.G. Wells "Time Machine", his teenage years, his weak schoolboy mathematics and his clerkship with the Swiss patent office where he must have observed applications for  Swiss cuckoo clock patents. His theories have more holes in them than the Swiss cheese he was eating.  The fermentation process caused gas to expand within his brain, causing large bubbles which become holes. Far from being a genius, the guy was cuckoo and incompetent in mathematics, there can be small wonder he was not invited to join the Manhattan Project. Caveat emptor.

5. Associativity of scalar multiplication:  p(qx) = (pq)x

Fairly simple, 3 * (10x) = 30x

6. Distributivity of scalar sums:  (p+q)x = px + qx

This is simply 2x + 3x = 6x, tacking the tail of one vector onto the nose of another. It is very different from 2x+3y, the plus sign (+) in vector addition takes on a different meaning to the usual one. Vectors are mutually independent.

7. Distributivity of vector sums: p(x+y) = px + py

Once again the theoretical physicist runs into serious trouble, for it cannot make much sense to say that 1 mile + 1 hour  = 2 miles + 2 hours. 2(x+t) = 2x+2t. We can double speed but we cannot double time and if we did we'd still have the same speed.

Time is not a vector or a component of a vector.

Time is a scalar in vector algebra.

8. Scalar multiplication identity:  1x = x

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A raving lunatic:

 http://math.ucr.edu/home/baez/boosts.html

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A mere idiot and my comments below:

"Timo A. Nieminen" <timo@physics.uq.edu.au> wrote in message news:Pine.WNT.4.64.0703060620030.1044@serene.st...
> There is sometimes, even oftentimes, a disconnect between the teaching of
> physics and the teaching of mathematics. An example can be found in
> vectors.
>
> In introductory courses in physics, students are often told that a vector
> is a magnitude combined with a direction. This is an essentially geometric
> definition, and aims to take advantage of both everyday understanding of
> geometry and the purpose of using vectors in introductory physics.
>
> In the maths course, the student is given a much more formal definition,
> with the definition of a vector space, and so on. It's the same
> mathematical entity as the physicist's vector, but students are
> hard-pressed to see that.


Physics students are hard-pressed, math students take it in their stride.
Heck, physics students don't even know how far it is from A to A.

"Distance travelled by photon from A to A is not A-A. End of story." - Jan Beilawski.

> For physics students, I think that the first version is more worthwhile,

It doesn't matter what Nieminen Ph.D. hallucinates, time is not a vector, he's an idiot.

> since geometric insight goes a long way in physics (although it later
> tends to become wrecked on the rocks of infinite-dimensional vector
> spaces, Hilbert spaces, etc; out of which can emerge a new geometric
> understanding, eg of vector spaces where the basis vectors are a complete
> set of solutions to a classic PDE of mathematical physics).
>
> But that's not the main point. The main point is that the word "scalar" is
> terribly misused, by both physicists and mathematicians (or at least,
> physics and maths teachers). We have two distinct usages:
>
> Scalar = non-vector. A number without a direction attached. Surely, we
> could say "real number" or "complex number" instead.

We do not write (x+y) as a vector, but we still write (x+iy)
because it is CLEAR that real and imaginary are independent.
Much of the problem is notation. (x+iy) is a vector with scalars x and y.
So is (x, iy), same thing.
The confusion arises with (x,y) being a vector with x and y indicating
direction and (ax+by) would then mean a and b were scalars. The
essential difference is x and y are INDEPENDENT.
The resultant is sqrt(x^2+y^2) but that doesn't mean we can say
(x,y,sqrt(x^2+y^2)) is a vector even though sqrt(x^2+y^2) is
a vector. It is not independent. However, we can say (x,sqrt(x^2+y^2))
and (y,sqrt(x^2+y^2)) are independent vectors, although that would
be clumsy.

(x,y,z, ct) is not independent, c = x/t and then we have (x,y,z,x) which is nonsense.
http://www.fourmilab.ch/etexts/einstein/specrel/www/figures/img40.gif
is total garbage, c = 2AB/(t'A-tA) = 0/0,
"but the ray moves relatively to the initial point of k with the velocity c-v."
Einstein was an incompetent moron.


>
> Then there's the geometric definition of a scalar - a quantity that's
> invariant under (a restricted class of) coordinate transformations. Rank-0
> tensor. Not the same thing at all.
>
> The conflict between these definitions is clear if we consider the
> components of a vector. Scalars by the first definition, not scalars by
> the second.

The components of a vector are themselves vectors. Every mathematician knows this.
Even wackypedia says so and that is hardly a credible source, anyone can write it.
http://en.wikipedia.org/wiki/Vector_(spatial)

> Ah well, at least something to slap advanced students about the face
> with.

I've been doing that for years, student. It's time you woke up from
the dream time in the outback. Theoretical physicists are failed math
students, Einstein never amounted to anything.
http://www.androcles01.pwp.blueyonder.co.uk/SRvNM.htm