Why Vectors product the way it is?

[bbbeard]

Hi,

I am wondering why the scalar product of two vectors the way it is, i mean why it wasn't ABtanθ instead of ABcosθ, or XAB, while X is any constant value, why mathematicians define it that way, the same story goes on for the vector product,
why we even had two types of products defined, while we don't have division defined,

and why it was "accidentally!" suitable as the best language to describe nature,
or i got the hole thing wrong and i have to think about it as it was defined as a new mathematics to best describe nature, in this case all of my questions will be answered as it had to be that way because we describe nature already and nature behave that way

my original thought that vector algebra is a topic of pure mathematics and it was developed not in mind describing nature, and after that physicists use it to describe nature

i need your opinion...

Well, here's my 2 cents. Maybe this is too technical, but the sooner you learn this, the better.

From the standpoint of physics, what makes something a vector is how it behaves under some set of coordinate transformations. In ordinary 3d space (as opposed to 4d Minkowski space, for example), the coordinate transformations we're usually interested in are rotations, which you are invited to think of in terms of rotation matrices. A vector is a "rank 1 tensor", i.e. a one-index object, say $V_i$ where i=1,2,3 represent x,y,z. A rotation matrix is a two-index object, say $R_{ij}$. A rotation matrix acts on a vector, producing another vector

$V'_i=\sum\limits_{j=1}^3 R_{ij}V_j$

where the sum is over j=1,2,3. We have a convention, called the Einstein summation convention, which says if we write a product like $R_{ij}V_j$ with a repeated index j, we are supposed to assume a summation over the repeated index, which allows us to write the expression above

$V'_i=R_{ij}V_j$

(That Einstein guy saved the world a lot of ink.) Anyway, we say any three quantities that transform under a rotation $R_{ij}$ the way that the coordinates do is a "vector". Things like forces and momenta are vectors, but not every set of three quantities is a vector. For example, if we think about something like

X=(1/x,1/y,1/z),

this X is not a vector. That is, $1/x_i$ is not a vector -- it doesn't obey the vector transformation law $V'_i=R_{ij}V_j$.

We can also have two-index objects that obey a vector-like, or "tensor" transformation law. But we have to "rotate" both indices, that is, in order for $T_{ij}$ to be a tensor, it has to obey a law like

$T'_{ij}=R_{ik}R_{jm}T_{km}$.

And we can continue making higher-order tensors, but each index gets its own copy of the rotation matrix. An important rank three tensor is the fully-antisymmetric tensor, the so-called "epsilon" tensor (also called the Levi-Civita symbol), $\epsilon_{ijk}$. You can read more at the link, but the basic idea is that $\epsilon_{ijk}$ switches signs when two indices are swapped, so that $\epsilon_{123}=-\epsilon_{213}$. By convention $\epsilon_{123}=+1$. Necessarily $\epsilon_{ijk}=0$ if any two of the indices are equal, e.g. $\epsilon_{112}=0$.

Now, here's the point: the outer product of two tensors is a tensor. So if we have two vectors $V_i$ and $W_j$, the object

$T_{ij}=V_i W_j$.

is a tensor. If we sum any two indices of a tensor, the result is a lower-rank tensor, e.g. if $T_{ij}$ is a rank two tensor, then $T_{ii}$ is a tensor of rank zero, also called a scalar (this is actually the trace of T if we think of T as a matrix). A scalar has no indices left over, so it doesn't change under rotations. In particular, if $T_{ij}=V_i W_j$ as above, then

$T_{ii}=V_i W_i=V \bullet W$

is the inner product (also called the scalar product or dot product) of V and W. Now, the stuff about the magnitudes and the cosine of the angle is interesting, but the real utility of the inner product is that it takes two vectors and gives us a scalar.

Likewise, it turns out that we can take the epsilon tensor and two vectors V and W and make something that is a vector product:

$U_{i}=\epsilon_{ijk}V_j W_k=V \times W$.

Again, the stuff about the magnitudes and the sine of the angle is interesting, but the real interesting thing is that we have created a vector from two vectors. Actually, the cross-product is what we call a "pseudo-vector" because it changes sign under spatial inversion, but that's an aspect we don't have to explore here.

Now, hopefully, from this point you can see that there are lots of ways to make products from tensors. We could, say, make a rank-four tensor from a single vector using

$T_{ijkm}=V_i V_j V_k V_m$.

but there's no guarantee that this will be useful! There are rules for combining epsilons with different indices, and there's this thing called the Kronecker delta which bears a striking resemblance to the identity matrix, and it turns out that the del operator is also a rank one tensor, but that discussion is for another day...

BBB

 

[D'Alembert]

Mathematicians alway try to present mathematics as independent from experience and the nature. Einstein once raised the same question like you how mathematics can explain nature so well when it has nothing to do with it.
The clear reason for it is, that all axioms of mathematics are basing on experience. Vector algebra bases on arithmetics and arithmetics are derived from the nature.

 

[IWantToLearn]

"This geometric representation of $$\bf{A} \times \bf{B}$$ is of such common occurrence that it might well be taken as the definition of the product."

He then goes on to state that "the vector product appears in mechanics in connection with couples." and later, "The product makes its appearance again in considering the velocities of the individual particles of a body which is rotating with an angular velocity given in magnitude and direction by A.

I would like to point out that in the nineteenth century, there were three other related algebraic systems being developed that became overshadowed by vector analysis.

Geometric Algebra is well worth your attention. It unifies a wide array of mathematics into one unified system including complex numbers, vector analysis, differential forms, and Pauli spinor algebra, just to mention a few. Geometric Algebra may be used in all branches of physics with some occasionally startling results.

when it comes to physics, the more tools that you have to work with the better, and Geometric Algebra, while not usually taught in universities, is a powerful tool.

Many Thanks AEM, you had enlightenment me
you had provided us with useful information, and you had opened new windows of study for me, i liked the historical analysis of the development of the subject, and i liked the idea that mathematician invent their mathematics almost inspired from nature
now i am searching for some online lectures on geometric algebra
Thanks Again

It is a projection to test for how much "one thing" is "like" "another thing."

i liked that too much

(mathematicians call a lot of things vectors that you wouldn't normally think of as vectors).

Thanks nucl34rgg,

From the standpoint of physics, what makes something a vector is how it behaves under some set of coordinate transformations. In ordinary 3d space (as opposed to 4d Minkowski space, for example), the coordinate transformations we're usually interested in are rotations, which you are invited to think of in terms of rotation matrices.

X=(1/x,1/y,1/z),

this X is not a vector. That is, $1/x_i$ is not a vector -- it doesn't obey the vector transformation law $V'_i=R_{ij}V_j$.
Now, hopefully, from this point you can see that there are lots of ways to make products from tensors. We could, say, make a rank-four tensor from a single vector using

$T_{ijkm}=V_i V_j V_k V_m$.

but there's no guarantee that this will be useful!

Thanks bbbeard, as you said it is too much technical, but you enlightenment me too much, specially by introducing the vector transformation law

Mathematicians alway try to present mathematics as independent from experience and the nature. Einstein once raised the same question like you how mathematics can explain nature so well when it has nothing to do with it.
The clear reason for it is, that all axioms of mathematics are basing on experience. Vector algebra bases on arithmetics and arithmetics are derived from the nature.

Thanks D'Alembert
i agree with you 100%