# Why Vectors product the way it is?

IWantToLearn
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...

## Answers and Replies

sambristol
Mathematics is the physics of all logically possible universes and inter alia provides the language , – you pick the mathematical tool best suited to your problem and use the leverage of the math to make your calculation simpler. It is quite possible to use ordinary algebra (say Cartesian coordinates x,y,z) and solve the same problem but the algebra tends to be daunting.

Vectors and vector algebra just are, very useful, short hand to make the math easy so you can see the physics clearly separated from the math.

You may find some solace in the fact the vector product only exists for 3 dimensional vectors – in less than three dimensions there is no 'right angle' for the product and in more than 3 dimensions the nearest equivalent is an entity called an antisymmetric tensor.

Hope this helps

Regards

Sam

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Functor97
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...

These operations are defined. They express useful ideas, and by making a definition we may make the process of cognition rigorous.

IWantToLearn
You are absolutely right in saying that we use mathematics in the best way that simplify the description of the physical world, but if vector algebra is a branch in mathematics that developed independently from physics and may historically before discussing the concept of motion and the physical quantities involved in it. then how it is very very suitable to describe nature...

The second approach is that we assume that vector algebra designed specifically for describe the aspects of physical world...

i don't know

Functor97
You are absolutely right in saying that we use mathematics in the best way that simplify the description of the physical world, but if vector algebra is a branch in mathematics that developed independently from physics and may historically before discussing the concept of motion and the physical quantities involved in it. then how it is very very suitable to describe nature...

The second approach is that we assume that vector algebra designed specifically for describe the aspects of physical world...

i don't know

Before the 20th century mathematics and physics developed very much hand in hand. Gauss, Euler, Newton worked on physical problems just as often as pure mathematical ones.

jetwaterluffy
Vector algebra was invented to deal with length. It's not hard to see what works for length works for velocity, as velocity is based on length.

chrisbaird
The dot product of two vectors is defined as the length of the projection of one vector onto the other vector. If you draw a little triangle with vector A as the hypotenuse and its projection on vector B as the adjacent side, you will see that the dot product has to be A B cos θ and not A B tan θ. The dot product can also be defined as the sum of the product of all corresponding components, and this again leads to A B cos θ.

Homework Helper
You are absolutely right in saying that we use mathematics in the best way that simplify the description of the physical world, but if vector algebra is a branch in mathematics that developed independently from physics and may historically before discussing the concept of motion and the physical quantities involved in it. then how it is very very suitable to describe nature...

The second approach is that we assume that vector algebra designed specifically for describe the aspects of physical world...

i don't know

You've raised a good point. Things can happen one of two ways:

1)People saw physical phenomena then came up with maths to describe those phenomena. But of course the maths is an abstract concept that is not limited to describing physical problems.

2)People sometimes create maths that is totally abstract and they have no real-life application in mind for their maths. Either these theories stay abstract, or someday someone happens to find a physical phenomena that is described by that maths.

So it can happen either way, but I think usually people make maths with a physical application in mind. For the specific case of vector algebra: Do you mean linear algebra, or vector calculus? In either case, I think the people contributing to the maths would have known that there was some real-life application for the maths they were creating.

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IWantToLearn
"chrisbaird", yes i know that it was defined to be the scalar product of one of the projection two vectors into the other multiplied by the length of the other,
but this is not my question, my question is why it was defined originally like this

"sambristol", u are absolutely right in saying that doing the math using vector algebra will be very much easier than uses only analytically geometry
and i liked and quoted your statement "Mathematics is the physics of all logically possible universes"
but i still don't agree with you saying "physics clearly separated from the math", it will be more logical for me to believe that all the subject of vector algebra is designed to ease the description of the mathematical world.

"Functor97", i do agree with u
"BruceW", i do agree with u

Homework Helper
We are most interested in vector products that are unchanged by rotation (SO(3)). The only possibilities (up to multiplication by a constant) are the dot and cross products. It is quite obvious that such products would be useful, dot and cross products of a system of vectors tell us everything about them that is unchanged by rotation (SO(3)).

IWantToLearn
i did a research over the internet about the history of Vector analysis, and i had found useful information that support the idea of (vector analysis being invented to be a tool to ease the description of physical world)

here is a quote from a research paper,

Three Early Sources of the Concept of a Vector and of Vector Analysis
When and how did vector analysis arise and develop? Vector analysis arose only in the period after 1831, but three earlier developments deserve attention as leading up to it. These three developments are (1) the discovery and geometrical representation of complex numbers, (2) Leibniz’s search for geometry of position, and (3) the idea of a parallelogram of forces or velocities.

In Year 1545
Jerome Cardan publishes his Ars Magna, containing what is usually taken to be the first publication of the idea of a complex number. In that work, Cardan raises the question: “If someone says to you, divide 10 into two pails, one of which multiplied into the other shall produce 30 or 40, it is evident that this case or question is impossible.” Cardan then makes the surprising comment “Nevertheless, we shall solve it in this fashion” and proceeds to find the roots 5 + √(-15) and 5 -√(-15). When these are added together, the result is 10. Then he stated: “Putting aside the mental tortures involved, multiply 5 + √(-15) by 5 -√(-15) making 25 - (-15) which is +15. Hence this product is 40”. As we shall see, it took more than two centuries for complex numbers to be accepted as legitimate mathematical entities. During those two centuries, many authors protested the use of these strange creations.

In Year 1679
In a letter to Christian Huygens, Gottfried Wilhelm Leibniz proposes the idea (but does not publish it) that it would be desirable to create an area of mathematics that “will express situation directly as algebra expresses magnitude directly” Leibniz works out an elementary system of this nature, which was similar in goal, although not in execution, to vector analysis.

In Year 1687
Isaac Newton publishes his Principia Mathematic, in which he lays out his version of an idea that was attaining currency at that period, the idea of a parallelogram of forces. His statement is “A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately.” Newton did not have the idea of a vector. He was, however. Getting close to the idea, which was becoming common in that period, those forces, because they have both magnitude and direction, can be combined, or added, so as to produce a new force.

""

Science Advisor
my question is why it was defined originally like this
Because this definition turns out to be useful.

In physics for example you often want to combine two vector quantities in a way that makes the result maximal when the vectors are parallel, then the dot product is useful. If you want the maximal result if the vectors are perpendicular, then you use the cross product.

IWantToLearn
Because this definition turns out to be useful.

In physics for example you often want to combine two vector quantities in a way that makes the result maximal when the vectors are parallel, then the dot product is useful. If you want the maximal result if the vectors are perpendicular, then you use the cross product.

Interesting point, also support the idea that vector analysis, is invented to ease the description of nature

AlexChandler
We want the dot product to give a measure of how much the vectors go in the same direction. It is just the product of the lengths of the projections of the vectors along a common direction (one that coincides with the direction of one of the vectors). If you wish you can invent the product where you use tan theta and call it something else.. but don't be disappointed if nobody uses it ;)

IWantToLearn
"lurflurf" i didn't understand this statement, "We are most interested in vector products that are unchanged by rotation (SO(3))", rotation of what?, suppose we have to vectors, if u mean rotation of both vectors together with same rate, then the angle between them will never change, and also the magnitude will never change, then both the value of both the scalar product and vector product will never change, but why in the beginning we are interested only in "vector products that are unchanged by rotation (SO(3))",
and what is (SO(3))?

IWantToLearn
We want the dot product to give a measure of how much the vectors go in the same direction. It is just the product of the lengths of the projections of the vectors along a common direction (one that coincides with the direction of one of the vectors). If you wish you can invent the product where you use tan theta and call it something else.. but don't be disappointed if nobody uses it ;)

:) sure i will not do that,
and your statement "We want the dot product to give a measure of how much the vectors go in the same direction." support my idea of that vector analysis being designed to describe nature

AlexChandler
"lurflurf" i didn't understand this statement, "We are most interested in vector products that are unchanged by rotation (SO(3))", rotation of what?, suppose we have to vectors, if u mean rotation of both vectors together with same rate, then the angle between them will never change, and also the magnitude will never change, then both the value of both the scalar product and vector product will never change, but why in the beginning we are interested only in "vector products that are unchanged by rotation (SO(3))",
and what is (SO(3))?

SO(3) is just the group of all possible 3 dimensional rotations. The idea is that when you have two different coordinate systems, the components of a specific vector are different in the two systems but it is still the same vector represented by different basis vectors. So when we say 'unchanged by rotations', we are not rotating the vector, but rotating the coordinate systems that we are using to measure the vectors.

jetwaterluffy
"chrisbaird", yes i know that it was defined to be the scalar product of one of the projection two vectors into the other multiplied by the length of the other,
but this is not my question, my question is why it was defined originally like this

"sambristol", u are absolutely right in saying that doing the math using vector algebra will be very much easier than uses only analytically geometry
and i liked and quoted your statement "Mathematics is the physics of all logically possible universes"
but i still don't agree with you saying "physics clearly separated from the math", it will be more logical for me to believe that all the subject of vector algebra is designed to ease the description of the mathematical world.

"Functor97", i do agree with u
"BruceW", i do agree with u

Why did you comment on everyone's post except mine?

IWantToLearn
Why did you comment on everyone's post except mine?

i am really sorry, i didn't mean to not reply, i just forget it,
and your comment also prove that vector analysis is invented to describe nature, since both lengths and velocities are aspects of the physical world.

sorry again

IWantToLearn
I think I can state the following
• Vector product should be invented so that is has a unique value regardless of coordinate system that is used to measure them, and regardless of the rotation or transition of the coordinate system that is used to measure them, unless it will be meaningless to talk about vector product which will have different values in different coordinate systems.
• Vectors analysis and vector notion is designed to ease the description of the physical world, and it had to be a derived from the behavior of the physical quantities in the physical world.
• Any type of multiplication for vectors must include the magnitude of the vectors.
• Any type of multiplication for vectors must include the angle (specifically the trigonometric ratios of this angle, since these ratios is dimensionless) between them because it is the component that represents the concept of direction.
• In the physical world there are situation where two vector quantities multiplied together to result in a scalar quantity, for example multiplying force (Vector Quantity) with displacement (Vector Quantity) to result in work (scalar quantity), while amazingly in other situations the same two quantities multiplied together to result in a vector quantity which is torque, so we had to have two types of vector multiplication, Scalar (dot) product and Vector (cross) product.
• In the case of the scalar product of force with displacement, we found that work is maximum when both force and displacement are on the same direction θ=0, so the trigonometric function that should be used is cosθ.
• In the case of the vector product of force with displacement, we found that torque is maximum when both force and displacement are perpendicular to each other θ=90, so the trigonometric function that should be used is sinθ.

IWantToLearn
SO(3) is just the group of all possible 3 dimensional rotations. The idea is that when you have two different coordinate systems, the components of a specific vector are different in the two systems but it is still the same vector represented by different basis vectors. So when we say 'unchanged by rotations', we are not rotating the vector, but rotating the coordinate systems that we are using to measure the vectors.

if i understand u right,
then there are infinite number of possible rotations for a 3D Cartesian coordinate system, and i can say that all of these rotation will not effect the value of the vector product(dot or cross), since the product depend on magnitudes of the vectors and the angle between them, and even if the cross product is defined as a.b=ab tanθ, it will not be changed.....!!!

so "lurflurf" statement, "We are most interested in vector products that are unchanged by rotation (SO(3))", also applied for the form a.b=ab tanθ
!!!

Homework Helper
ab tanθ is unchanged by rotation, it could be expressed with dot and cross product, but it would not be a good product in that it does not distribute. We want
(u,av+bw)=a(u,v)+b(u,w)

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IWantToLearn
ab tanθ is unchanged by rotation, it could be expressed with dot and cross product, but it would not be a good product in that it does not distribute. We want
(u,av+bw)=a(u,v)+b(u,w)

"(u,av+bw)=a(u,v)+b(u,w)" what is that?, and what is u and v stand for

IWantToLearn
two thing left why vector division is not defined, isn't multiplying scalar value to a vector is best to replace the idea of vector division?

and why cross product is not defined in more than 3D space

Functor97
two thing left why vector division is not defined, isn't multiplying scalar value to a vector is best to replace the idea of vector division?

and why cross product is not defined in more than 3D space

Because if the cross product were define in spaces of greater than 3D it would not be the cross product. The cross product returns the vector perpendicular to two given vectors. How would you define this in 4d, 5d?

IWantToLearn
Because if the cross product were define in spaces of greater than 3D it would not be the cross product. The cross product returns the vector perpendicular to two given vectors. How would you define this in 4d, 5d?

very convincing logic, thank you

but what about division?

Homework Helper
The inverse of a matrix is sort of like 1/matrix, since a matrix times its inverse is equal to the unity matrix.

But I don't think there is a direct way to 'divide by a vector', or there is no such useful operation.

chogg
You can divide by a vector, but the result isn't another vector. In general a vector divided by a vector gives the sum of a scalar plus a bivector (either of which can be zero).

Dividing by vectors is actually useful. For two vectors a and b, the quotient (a/b) is an operator which performs rotations in the direction that takes a into b, by twice the angle between them. [*]

This comes from Geometric Algebra. A good reference is this book:
http://www.geometricalgebra.net/ [Broken]

[*] Well, up to a scaling factor, anyway. The point is that the quotient of vectors is a rotation operator.

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Homework Helper
"(u,av+bw)=a(u,v)+b(u,w)" what is that?, and what is u and v stand for

u,v, and w are some vectors and a and b are some scalars. (u,v) is some product of u and v.

two thing left why vector division is not defined, isn't multiplying scalar value to a vector is best to replace the idea of vector division?

and why cross product is not defined in more than 3D space

There are generalizations of cross product to higher dimensional space. The trouble is that 3D is extra special in that several products of interest are the same. Cross product is at once a map
V2->V
V2->Vn-2
Vn-1->V
where
Vn is a collection of n vectors
In general (n not 3) they would all be different.

The trouble with defining vector division is the generality of it.
av/v=a is sensible
(av+bu+cw)/v=a is not

The sensible division of vectors is a reciprocal basis. It acts not on individual vectors, but on sets of them.
Vn->Vn
It is equivalent to matrix inversion.

IWantToLearn
Thank you all for help, it was a great help, it settles down the debate of the foundation of vector concept and the very fundamental processes like vector dot product and cross product, and it opened a new window for me to read more in vector analysis, and more theoretical staff, that I wasn’t aware about, I will do my best to understand it, and may come back again later to discuss more sophisticated issues.
Best Regards

Amazement
In regards to the products, most sources I read describing the basics of three dimensional vectors merely state what the definition of a cross product and dot product are without a geometric/trignometric proof.

I tried to prove the equation for the dot product (A.B=ABcosθ through the construction of a parallelogram with sides A and B with diagonal D.

Now to calculate D i arrive with an equation describing the diagonal as the addition of two cosines of two similiar triangles formed by transposing A to the tip of B (i.e. half the parallelogram.)

D=Acosθ+Bcosγ​

Where θ is the angle of the right hand triangle with hypotenuse A and likewise for γ.

I personally cannot see from this point how you obtain the dot product equation, perhaps I am attempting it in completely the wrong manner. Thx in advance.

Oh and p.s. Im assuming its much more obvious in matrix form and I have read the matrix proof on PF but when I suggested matrices in class the professor was outraged that I made such a heretical suggestion.

Homework Helper
chogg said:
You can divide by a vector, but the result isn't another vector. In general a vector divided by a vector gives the sum of a scalar plus a bivector (either of which can be zero).

Dividing by vectors is actually useful. For two vectors a and b, the quotient (a/b) is an operator which performs rotations in the direction that takes a into b, by twice the angle between them.[*]

This comes from Geometric Algebra. A good reference is this book:
http://www.geometricalgebra.net/ [Broken]

[*] Well, up to a scaling factor, anyway. The point is that the quotient of vectors is a rotation operator.
Wow, I never knew this. Its above my level of mathematical knowledge, I guess :)

Amazement said:
I tried to prove the equation for the dot product
For a parallelogram defined by vectors $\vec{A}$ and $\vec{B}$, the squared length of the diagonal created is equal to $(\vec{A} + \vec{B}) \cdot (\vec{A} + \vec{B}) = D^2$ (simple vector rules). So this will mean that $A^2 + B^2 + 2\vec{A} \cdot \vec{B} = D^2$

We can also work out the squared length of the diagonal geometrically (i.e. by just drawing the picture). You can do this by imagining another triangle fixed on the end of the parallelogram, then working out the lengths by using the angle. When you work it out, you get: $A^2 +B^2 + 2ABcos(\theta) = D^2$ And we can compare this with our other equation to get the result $\vec{A} \cdot \vec{B} = ABcos(\theta)$

Of course, this isn't a rigorous mathematical proof, but it is nice to see that it all works when you try it out yourself.

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AlexChandler
In regards to the products, most sources I read describing the basics of three dimensional vectors merely state what the definition of a cross product and dot product are without a geometric/trignometric proof.

In math you do not need to prove a definition. We simply notice that there is this combination of symbols that we often see, so we decide to give it a name: "dot product" or "cross product". From these definitions you can prove that the cross product is perpendicular to the original vectors for instance, or that a dot product is zero only for perpendicular vectors.

AEM
I have come to this thread rather late, however I'd like to make a few comments. Vector analysis was worked out in the early 1880s by the American mathematical physicist, J. Willard Gibbs. In the period 1881-1884, he circulated a pamphlet he had written to interested people and in 1901 Yale University press published a book based upon lectures Gibbs gave at Yale University. As well as Gibbs, some aspects of what we now call vector analysis was also developed by Oliver Heaviside an others in England who were working over Maxwell's electrodynamics.

Looking at Gibbs' book "Vector Analysis", one can get an idea of what his thinking might have been in the early days when he was working things out. On page 62 it is pointed out that the vector product

$$\bf{A} \times \bf{B} = A B Sin{\theta}$$

represents the area of the parallelogram whose sides are A and B. After pointing this out he says,

"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. These were William Rowan Hamilton's Quaternions, Hermann Grassmann's algebra (now called Grassmann algebra), and William Kingdon Clifford's reworking of Grassmann's algebra into Geometric Algebra. All of these had their own version of the vector product (which I'll call the wedge product). It also was skew symmetric. The difference they had with Gibbs' vector product is that the order of multiplication serves to define an orientation of the plane determined by the two vectors, but not a third vector perpendicular to the plane. This means that the wedge product can be defined in a space of any dimension.

The improvement that Clifford made on Grassmann's algebra was among other things was to combine the scalar product and wedge product into one product that has two parts --one symmetric (the dot product) and one antisymmetric (the wedge part). This is analogous to combining real numbers and imaginary numbers into a single complex number.

In the 1960's, David Hestenes, now at Arizona State University rediscovered Clifford's Geometric Algebra and has written many articles and several books on Geometric Algebra. Many of these are available on the Internet at http://geocalc.clas.asu.edu/

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. It currently is widely used in computer graphics, as indicated by the link given by Chogg.

Two more links where you can find additional information are

http://faculty.luther.edu/~macdonal/

and

http://www.mrao.cam.ac.uk/~clifford/

The reason that I've made this post so long is that I firmly believe that 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.

nucl34rgg
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...

Conceptually, the dot product represents the multiplication of one vector by another projected onto it. What I mean by "projected' is qualitatively "how much along the other vector is this vector." This is the meaning of the inner product. It is a projection to test for how much "one thing" is "like" "another thing." Physically, with two vectors pointing in different directions, if you take the component of one of the vectors along the other and multiply them, you will have your dot product. Geometrically, this is taking the magnitude of one of the vectors and multiplying it by the cosine of the angle between them, since this value will give you the length of the component.

One cool thing you will see is that inner products aren't limited to what we normally call vectors (mathematicians call a lot of things vectors that you wouldn't normally think of as vectors). For example, the fourier transform value at a particular frequency is an example of a function that gives u what is basically the inner product of your original function in the time domain with sine and cosine waves of the given frequency. In a sense, it shows you how much the frequency of the original signal is "like" that of the test frequency. (The peaks in the Fourier spectra are the frequency components of the source signal that made the most contribution.)