Dot product in Euclidean Space

[Trying2Learn]

TL;DR Summary

What postulates of Euclid enables the geometric dot product

Hello

As you know, the geometric definition of the dot product of two vectors is the product of their norms, and the cosine of the angle between them.
(The algebraic one makes it the sum of the product of the components in Cartesian coordinates.)

I have often read that this holds for Euclidean Space.

I know that there are 5 postulates of Euclidean Space.

However, I am unable to connect the geometric definition of the dot product as deriving from those five postulates.

Can someone explain why those five postulates lend themselves to an understanding of the geometric definition of the dot product

 

[fresh_42]

Can you reference or list the five postulates you are thinking of?

The dot product is the generalization of the law of cosines which is a generalization of Pythagoras.

 

[Trying2Learn]

Can you reference or list the five postulates you are thinking of?

The dot product is the generalization of the law of cosines which is a generalization of Pythagoras.

Here, these five...



1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment , a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

If a space obeys those rules, it is called Euclidean.

However, I cannot connect those five rules to the definition of the dot product (even as a generalization of Pythagoras) (and while at it: the cross product, too)

 

[fresh_42]

You confuse geometric geometry (Euclid) with analytic geometry (Euclidean spaces).

In Euclidean geometry we only have lengths as relation to other lengths or to a given prototype as the distance between two intersections. A norm is nowhere defined, and neither is the cosine - only angles. So in order to break down Cartesian systems to Euclid's postulates, we first have to define the tools. I'm sure this can be done, as every norm is a multiple of a given unit length, and a cosine only the relation between two triangle sides. But this is probably not easily done in a rigorous way. If the terminology is translated, then we can go the way I described above: Pythagoras $\to$ Law of Cosines $\to$ Higher Dimensions.

 

[Trying2Learn]

You confuse geometric geometry (Euclid) with analytic geometry (Euclidean spaces).

In Euclidean geometry we only have lengths as relation to other lengths or to a given prototype as the distance between two intersections. A norm is nowhere defined, and neither is the cosine - only angles. So in order to break down Cartesian systems to Euclid's postulates, we first have to define the tools. I'm sure this can be done, as every norm is a multiple of a given unit length, and a cosine only the relation between two triangle sides. But this is probably not easily done in a rigorous way. If the terminology is translated, then we can go the way I described above: Pythagoras $\to$ Law of Cosines $\to$ Higher Dimensions.

OK then: it is NOT the dot product I am after. It is something else.

In many intro texts on statics or dynamics, they often talk about parallel translating vectors, or moving an inertial frame to another point in space. Both of those hinge on space being Euclidean.

THAT is what I am after, then: how do the five postulates of Euclidean space, enable vector addition (in the traditional sense of parallel addition, not in the modern algebraic definition of a vector (or maybe))?

 

[fresh_42]

You mean the intercept theorems? They should be an essential part in every book about classical geometry. But why do you want to break it down to Euclid's postulates? If you study physics you will soon meet geometries without parallel axiom anyway.

 

[Trying2Learn]

You mean the intercept theorems? They should be an essential part in every book about classical geometry. But why do you want to break it down to Euclid's postulates? If you study physics you will soon meet geometries without parallel axiom anyway.

No, not the intercept theorems. The question is more simple than than (and perhaps my problem is that I do not know what I am asking).

That said... Whenever I open a classical text in dynamics, I eventually encounter the phrase "Assume space is Euclidean..." and a moment later, the author is shifting origins of frames and taking dot and cross products or translating the reference frame.

The segue is almost immediate, in almost all books. And this is leading me to ask: "why, in traditional (classical) dynamics of Newton (Newton's and Euler equations for, say robotics or single rigid bodies), do they make a point of asserting that this work holds in Euclidean space.

When I look up the definition of Euclidean Space, I get those five postulates: if the space obeys those postulates, it is Euclidean. But I am unable to make the transition from those postulates, to the fundamental issues the books raise immediately after: dot product, cross product, pointwise principle, translating the frame to another joint, etc.)

The books could just as well be saying: "Assume the space is sugar-free, and apply the pointwise principle. I would still know what to do, but I would ignore the reason for saying "sugar-free." Do you see what I mean? Why do they make a point of specifying "euclidean."

 

[fresh_42]

Euclidean in this context has little to do with Euclid's postulates. O.k., not really, they are closely related, but nobody thinks about Euclid if he uses the term Euclidean space. It more or less just means: flat, no curvature, and a reasonable metric. The distinction is between a curved surface (manifold, non Euclidean) and its flat approximation (tangent space, Euclidean). If you read Euclidean space, then you most often can think of it as $\mathbb{R}^n$. The room you are currently in is likely a better suited concept of what Euclidean space means, than classical geometry is.

 

[Trying2Learn]

Euclidean in this context has little to do with Euclid's postulates. O.k., not really, they are closely related, but nobody thinks about Euclid if he uses the term Euclidean space. It more or less just means: flat, no curvature, and a reasonable metric. The distinction is between a curved surface (manifold, non Euclidean) and its flat approximation (tangent space, Euclidean). If you read Euclidean space, then you most often can think of it as $\mathbb{R}^n$. The room you are currently in is likely a better suited concept of what Euclidean space means, than classical geometry is.

So would I be wrong in asserting that when they assert: "Assume space is Euclidean... blah blah blah... we can now parallel translate the frame to the revolute joint..." that the textbooks are being slightly pedantic?

I THINK I get what you mean when you say: "O.k., not really, they are closely related, but nobody thinks about Euclid if he uses the term Euclidean space. "

You seem to be saying, "yes, if we wish to be a nitpicker (please excuse what is ONLY an 'ostensible' devaluation of your input) then, we can draw a bridge between the five postulates and the ability to translate a frame. Regardless, however, when they say 'because of Euclidean space,' they are being a bit overwrought."

 

[Trying2Learn]

Euclidean in this context has little to do with Euclid's postulates. O.k., not really, they are closely related, but nobody thinks about Euclid if he uses the term Euclidean space. It more or less just means: flat, no curvature, and a reasonable metric. The distinction is between a curved surface (manifold, non Euclidean) and its flat approximation (tangent space, Euclidean). If you read Euclidean space, then you most often can think of it as $\mathbb{R}^n$. The room you are currently in is likely a better suited concept of what Euclidean space means, than classical geometry is.

Because it would seem to me that "a reasonable metric" (necessary when using the word 'Euclidean') is ESSENTIAL to the definition of the dot and cross product.

And then, it would only mean I would want to draw a line from the five postulates to the dot product, but you seem to be saying: "fugetaboutit." (sorry, that word, works). I just do not see why the haul out Euclid, when the books could have just gone with the flow of the dot product, if the book is going to remain a dynamics book for engineers in the classical sense.

 

[fresh_42]

The point is the flatness. Parallel transport in Euclidean spaces is especially easy, as it's along straight lines, vector addition if you like. However, parallel transport of tangent vectors on a curved surface are no longer trivial and easy, not even unambiguous, because the tangent space at the pole is different from that at the equator, so how can we compare two vectors placed at different locations? But to understand the more complicated problem, you first have to understand the easy one, which´ is why those statements are at the beginning of the books. It is flat versus curved, not Euclid versus Bolyai and Lobachevsky.

 

[Trying2Learn]

THANK YOU!

That was what I need to hear. I just need to hear someone who I respect, tell me that.

>> It is flat versus curved, not Euclid versus Bolyai and Lobachevsky.

And your comment here, solidified it. Thanks again!

 

[mathwonk]

In rerefence to your original question, the dot product as derived from Euclid's postulates, and the answer provided by fresh 42, that it is contained in the law of cosines, generalizing pythagoras, here is a specific reference to Euclid for that fact. In Book II, Proposition 13, Euclid states that:

"In an acute angled triangle, the square on the side of the side subtending the acute angle, is less than the squares on the sides containing the acute angle, by twice the rectangle contained by one of the sides about the acute angle, namely by that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle."

In fact the rectangle mentioned has area equal to the dot product of the vectors spanned by the sides containing the acute angle. I.e. if the acute angle is t radians, then the length of the straight line cut off by the perpendicular mentioned is equal to c.cos(t), where c is the length of the side which is being projected onto the other side containing the angle. So even though there is no mention of the abstract quantity cos(t), still it appears as the ratio of the projected side to the original side being projected. I.e. if the two sides containing the acute angle have lengths c and a, then the area of the rectangle mentioned is a.c.cos(t), the dot product you asked about. If the third side has length b, then the proposition states that b^2 = a^2 + c^2 -2a.c.cos(t), exactly the law of cosines.

By the way, since either side containing the acute angle can be projected onto the other in this argument, this also proves the dot product is commutative, i.e. the two different rectangles have the same area.

In regard to the meaning of the term "Eucldean space", there is no doubt most people mean by that simply the space R^n as fresh 42 has said, essentially the coordinate space defined over the real numbers, which its usual analytic properties. It can however be characterized, (at least in dimension 2), by axioms which are built upon the 5 axioms of Euclid you mention. Recall that already in 1899 Hilbert made clear that Euclid's original axioms were not quite adequate even for the proof of the theorems he stated. Hilbert gave a careful enhancement of Euclid's own axioms, leading to axioms of incidence, betweenness, parallelism and congruence. A further axiom of completeness, built on Dedekind's characterization of the real numbers, when added as well, describes a geometry that is exactly the plane R^2.

A very clear and precise development of Hilbert's system is given by Hartshorne in his beautiful book Geometry: Euclid and beyond. So the upshot is that if we use as our axioms for a Euclidean plane, the more precise and thorough ones of Hilbert, and if we add the Dedekind axiom (essentially that every "separation" of the line, of a certain natural sort, is caused by removing a point), then indeed those axioms do yield the same plane geometry as is given by the coordinate plane R^2 built on the real numbers.

In spite of this, if one omits the completeness axiom of Dedekind, then there are many more than the one model of a "Euclidean plane". Without that axiom one requires an axiom insuring circles to intersect, and one may or may not want also the Archimedean axiom. Either way, one has a Euclidean plane in which essentially all Euclid's arguments make sense (strictly speaking, his own argument for similarity requires the Archimedean axiom, but similarity can be treated otherwise). There is a number system associated to each Euclidean plane, where a "number" is the ratio of two line segments. One can define arithmetic operations geometrically, and one obtains a certain "Euclidean" field of numbers, where for example one can take square roots, but which may not be the real numbers. Hartshorne calls this "segment arithmetic".

So each Euclidean plane, in the sense of one satisfying the modern enhanced "Euclidean" axioms of Hilbert, is associated to a certain field of numbers, and although there is only one "real" Euclidean plane, essentially the usual R^2, there are many others associated to different fields. E.g. the simplest seems to be the rational numbers but with all possible square roots, and square roots of square roots etc,,... added in, i.e. the smallest field containing the rationals and in which one can always take a square root, (but I have not checked this).

 

[lavinia]

In rerefence to your original question, the dot product as derived from Euclid's postulates, and the answer provided by fresh 42, that it is contained in the law of cosines, generalizing pythagoras, here is a specific reference to Euclid for that fact. In Book II, Proposition 13, Euclid states that:

"In an acute angled triangle, the square on the side of the side subtending the acute angle, is less than the squares on the sides containing the acute angle, by twice the rectangle contained by one of the sides about the acute angle, namely by that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle."

In fact the rectangle mentioned has area equal to the dot product of the vectors spanned by the sides containing the acute angle. I.e. if the acute angle is t radians, then the length of the straight line cut off by the perpendicular mentioned is equal to c.cos(t), where c is the length of the side which is being projected onto the other side containing the angle. So even though there is no mention of the abstract quantity cos(t), still it appears as the ratio of the projected side to the original side being projected. I.e. if the two sides containing the acute angle have lengths c and a, then the area of the rectangle mentioned is a.c.cos(t), the dot product you asked about. If the third side has length b, then the proposition states that b^2 = a^2 + c^2 -2a.c.cos(t), exactly the law of cosines.

By the way, since either side containing the acute angle can be projected onto the other in this argument, this also proves the dot product is commutative, i.e. the two different rectangles have the same area.

In regard to the meaning of the term "Eucldean space", there is no doubt most people mean by that simply the space R^n as fresh 42 has said, essentially the coordinate space defined over the real numbers, which its usual analytic properties. It can however be characterized, (at least in dimension 2), by axioms which are built upon the 5 axioms of Euclid you mention. Recall that already in 1899 Hilbert made clear that Euclid's original axioms were not quite adequate even for the proof of the theorems he stated. Hilbert gave a careful enhancement of Euclid's own axioms, leading to axioms of incidence, betweenness, parallelism and congruence. A further axiom of completeness, built on Dedekind's characterization of the real numbers, when added as well, describes a geometry that is exactly the plane R^2.

A very clear and precise development of Hilbert's system is given by Hartshorne in his beautiful book Geometry: Euclid and beyond. So the upshot is that if we use as our axioms for a Euclidean plane, the more precise and thorough ones of Hilbert, and if we add the Dedekind axiom (essentially that every "separation" of the line, of a certain natural sort, is caused by removing a point), then indeed those axioms do yield the same plane geometry as is given by the coordinate plane R^2 built on the real numbers.

In spite of this, if one omits the completeness axiom of Dedekind, then there are many more than the one model of a "Euclidean plane". Without that axiom one requires an axiom insuring circles to intersect, and one may or may not want also the Archimedean axiom. Either way, one has a Euclidean plane in which essentially all Euclid's arguments make sense (strictly speaking, his own argument for similarity requires the Archimedean axiom, but similarity can be treated otherwise). There is a number system associated to each Euclidean plane, where a "number" is the ratio of two line segments. One can define arithmetic operations geometrically, and one obtains a certain "Euclidean" field of numbers, where for example one can take square roots, but which may not be the real numbers. Hartshorne calls this "segment arithmetic".

So each Euclidean plane, in the sense of one satisfying the modern enhanced "Euclidean" axioms of Hilbert, is associated to a certain field of numbers, and although there is only one "real" Euclidean plane, essentially the usual R^2, there are many others associated to different fields. E.g. the simplest seems to be the rational numbers but with all possible square roots, and square roots of square roots etc,,... added in, i.e. the smallest field containing the rationals and in which one can always take a square root, (but I have not checked this).

I wonder if the extension of the rationals to all square roots is the points in the plane that are constructible with ruler and compass.

 

[Infrared]

@lavinia Yes, it is known that a complex number is constructible if and only if it's in a field extension of $\mathbb{Q}$ generated by iterated square roots. See https://en.wikipedia.org/wiki/Constructible_number#Transformation_into_algebra

 

[mathwonk]

You guys are quite right, (see Hartshorne chapter 3, sections 14-16). It seems that this is also the smallest (Euclidean) field whose associated plane satisfies all the axioms of incidence, betweenness, congruence, parallelism, and circle -circle intersection. If we omit that last axiom, then the smaller (Pythagorean) field generated from the rationals by square roots of just those elements of form (1 + a^2), gives a plane with those previous axioms.

 

[Trying2Learn]

In rerefence to your original question, the dot product as derived from Euclid's postulates, and the answer provided by fresh 42, that it is contained in the law of cosines, generalizing pythagoras, here is a specific reference to Euclid for that fact. In Book II, Proposition 13, Euclid states that:

"In an acute angled triangle, the square on the side of the side subtending the acute angle, is less than the squares on the sides containing the acute angle, by twice the rectangle contained by one of the sides about the acute angle, namely by that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle."

In fact the rectangle mentioned has area equal to the dot product of the vectors spanned by the sides containing the acute angle. I.e. if the acute angle is t radians, then the length of the straight line cut off by the perpendicular mentioned is equal to c.cos(t), where c is the length of the side which is being projected onto the other side containing the angle. So even though there is no mention of the abstract quantity cos(t), still it appears as the ratio of the projected side to the original side being projected. I.e. if the two sides containing the acute angle have lengths c and a, then the area of the rectangle mentioned is a.c.cos(t), the dot product you asked about. If the third side has length b, then the proposition states that b^2 = a^2 + c^2 -2a.c.cos(t), exactly the law of cosines.

By the way, since either side containing the acute angle can be projected onto the other in this argument, this also proves the dot product is commutative, i.e. the two different rectangles have the same area.

In regard to the meaning of the term "Eucldean space", there is no doubt most people mean by that simply the space R^n as fresh 42 has said, essentially the coordinate space defined over the real numbers, which its usual analytic properties. It can however be characterized, (at least in dimension 2), by axioms which are built upon the 5 axioms of Euclid you mention. Recall that already in 1899 Hilbert made clear that Euclid's original axioms were not quite adequate even for the proof of the theorems he stated. Hilbert gave a careful enhancement of Euclid's own axioms, leading to axioms of incidence, betweenness, parallelism and congruence. A further axiom of completeness, built on Dedekind's characterization of the real numbers, when added as well, describes a geometry that is exactly the plane R^2.

A very clear and precise development of Hilbert's system is given by Hartshorne in his beautiful book Geometry: Euclid and beyond. So the upshot is that if we use as our axioms for a Euclidean plane, the more precise and thorough ones of Hilbert, and if we add the Dedekind axiom (essentially that every "separation" of the line, of a certain natural sort, is caused by removing a point), then indeed those axioms do yield the same plane geometry as is given by the coordinate plane R^2 built on the real numbers.

In spite of this, if one omits the completeness axiom of Dedekind, then there are many more than the one model of a "Euclidean plane". Without that axiom one requires an axiom insuring circles to intersect, and one may or may not want also the Archimedean axiom. Either way, one has a Euclidean plane in which essentially all Euclid's arguments make sense (strictly speaking, his own argument for similarity requires the Archimedean axiom, but similarity can be treated otherwise). There is a number system associated to each Euclidean plane, where a "number" is the ratio of two line segments. One can define arithmetic operations geometrically, and one obtains a certain "Euclidean" field of numbers, where for example one can take square roots, but which may not be the real numbers. Hartshorne calls this "segment arithmetic".

So each Euclidean plane, in the sense of one satisfying the modern enhanced "Euclidean" axioms of Hilbert, is associated to a certain field of numbers, and although there is only one "real" Euclidean plane, essentially the usual R^2, there are many others associated to different fields. E.g. the simplest seems to be the rational numbers but with all possible square roots, and square roots of square roots etc,,... added in, i.e. the smallest field containing the rationals and in which one can always take a square root, (but I have not checked this).

THank you!