# Why Vectors product the way it is?

## Main Question or Discussion Point

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

Related Classical Physics News on Phys.org
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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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

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

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 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.

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.

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.

BruceW
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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"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

lurflurf
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)).

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.

""

A.T.
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.

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

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 ;)

"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))?

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

"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.

"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?

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

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.

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
!!!

lurflurf
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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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

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

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?