1903:
are all agreed that the 3D binary operation should dominate the article. However, there is a section, and rightly so, which deals with the varoius ways of extending this concept to higher dimensions. The note at the top of the article is therefore inaccurate. The problem is not so much a problem with the article itself as a problem with this inaccurate and restrictive note which has been added at the top of the page. In my opinion, the article is about 'cross product'. It should deal with the 3D case first, and then discuss the extensions to higher dimensions further down. That is more or less exactly as it is at the moment.
474:
degrees in there? I know a pure maths professor who was in my form at school. He is a specialist in 'space in n dimensions'. He couldn't answer when I asked him to explain the meaning of angle in 7 dimensions. He also just happens to be the one who taught me the importance of the distributive law in making a cross product useful. Everything to do with the concept of angle originates within the context of our perception of 3D space. I have often believed that when pure mathematicians extrapolate things to their favourite 'n dimensions' that it is an extrapolation beyond reality. It is were maths leaves the real world.
1266:
familiar 3D cross product. If a special relationship holds in both 3D and 7D, we cannot state that this relationship is a special 3D case without also mentioning that it is a 7D case as well, because to do so would be misleading. And in that respect, the article as it now stands after your revert, is misleading, because it makes readers think that this is a relationship which is exclusively restricted to 3D. The chances are that whoever first wrote this article didn't think about the 7D case. But we have now established beyond any doubt over at the talk page of
2842:
Clifford's geometric algebra, newly discovered in 1878. Geometric algebra includes both
Grassmann's algebra and quaternions so can seem more complex, though it can also used to define both from very simple axioms. Gibbs' cross product though was derived from the quite different quaternion product, by taking just the imaginary part, ignoring the insights available with the full quaternion algebra, seemingly unaware of Clifford's deeper discoveries. But that paragraph does seem out of place, as Clifford is mentioned later.--
166:. When mathematicians and non-mathematicians use the name "cross product" they are almost always referring to this product, not anything else. For most people it's the only one they know. The seven dimensional one is pretty obscure and most people never come across it (I learned of it after finishing my degree). Including it in here would just confuse people, as it's not a mainstream part of vector algebra, which is largely concerned with 3D, and it is also not as well defined as in 3D. --
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1324:
spread misinformation. So I will add the information in in brackets. Having said all that, the article shouldn't be only about the 3D cross product. It should be about the 'vector cross product' in general, with an emphasis on the more familiar 3D case. There is really no need to segregate the two articles. They can be written together coherently, pointing out the commonality and then the differences in the introduction, and then doing separate treatments of each.
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2823:: Did Clifford, in 1878, 3 years before the publication of the notes in which Gibbs described for the first time its cross product, really define a "product of two vectors" which returned another vector (i.e. a directed line segment) in the same vector space? This would not be an exterior product or a Clifford product, but a vector product identical to what we now call a cross product! The answer to this question is crucial.
283:|sinθ holds in either one or seven dimensions, I'd be fascinated if you could explain to me the meaning of the angle θ in either one or seven dimensions. Could we have a seven dimensional square? I read something in one of these two articles about seven dimensional Euclidean geometry. Now there is a concept I had never heard of before! Can you please explain to me what angle θ means in seven dimensions?
914:
how only 1D, 3D, and 7D satisfy the basic axioms. One of those basic axioms is the distributive law, and that happens to be the crucial axiom in this issue. But perhaps then you would prefer to put something more vague and general into the main article along the lines of 'the 5D and the 15D cross products don't work because they fail to satisfy the basic axioms'.
2110:—they look the same, so with the exception of algebraic restrictions (such as adding two points being meaningless), the space of points is isomorphic to the space of vectors, so the distiction is usually glossed over in introductory treatments. I find the affine-space treatment more pedantically-satisfying. In more-general spaces such as on
724:
identically to 3D (as an angle is a 2D thing it needs defining in more general dimensions). Apart from the two cross product pages there are the links we've given above and the various wikillinks and references on the pages, where it is explained in much more detail at all sorts of levels of expertise. --
2769:
This sentence is very interesting, but it puzzles me. The problem is: who is the father of the cross product? I always thought that
Grassmann, Hamilton and Clifford were not able to define a vector product which yielded another vector perpendicular to their plane (what we now call the cross product).
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Although it does not provide references, the statement is not superfluous in my opinion. It is useful to see that mathematics, at that time, followed two paths, one of which brought the cross product. The other path produced an exterior product which, dualized in 3D, produces a cross product. There's
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If the source is to be used, the notation of the source is preferable to an unsourced (invented?) notation. I am a bit surprised that the simple substitution A=C and B=D which makes the cited form of
Lagrange's identity exactly that of the link happened to escape your notice. This extra generality is
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is the "left" direction, then would be "up". Is it the assumption of handedness that you are wondering about? You are right that if we don't assume a right-handed coordinate system that we would need to define which way is up versus down in addition to left versus right. The assumption is completely
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John, Even if the article is only about the 3D cross product, we still need to make any general information unambiguous and not have readers thinking that a particular result is exclusive to 3D. Adding the information about 7D does not detract from the purpose of the article, whereas ommitting it can
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is only equivalent to the above ("Pythagorean") identity in 3D, as its article says. Further this article is about the cross product in 3D, so should not have higher dimensional results inserted at arbitrary points. It links to the 7D version in the lede, has a section on higher dimensions with links
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John, You were reading the wrong bit. The bit that I was referring to is the quote that I made in italics above, and the sentence immediately following it. I was not talking about any proofs. You are the one who keeps changing the subject to proofs. I was talking about the fact that it is easy to see
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David, I advise you to avoid this kind of alley. Again, your lack of expertise is not the root problem here but it isn't really helpful, given your obvious failing to know how angles are defined in n-dimensional vector spaces yet attempting to reorganise articles about cross products in 7-dim spaces,
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While on the subject, let's imagine a 3D rotation. We'll fix the k axis in space and rotate about the k axis in the ij plane. The concept of angle will be understood in that context. Now let's imagine a 7D situation. Once again, we'll fix k in space. But this time we could define rotations in quite a
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DVdm, With cosine involved, I think we're back to tautologies again. Everything to do with cosine begins on a 3D premises. I don't think that the physical concept of 3D space can be extrapolated to 7D that easily. Do we have the same π? Try to imagine seven mutually orthogonal axes. How do we fit 360
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This seems to be a rather unnecessary argument in my opinion, and it stems from the note at the top of the article which restricts the article to the 3D binary operation. I think that we are all agreed that the 3D binary operation is by far the most important and well known case. And I think that we
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I just restored the previous version of the definition, which is the generally accepted and most widely understood one. It is also the one given by
Lounesto, on page 93. On 94 he gives an alternate definition in terms of purely vector products, so he can do algebra with it, but it is not a generally
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The derivation of the formula for the cross product depends on the distributive property, but I'm not aware of a simple proof of distributivity. The article should either include a simple proof, or briefly sketch a hard one, or give a reference. But it should at least acknowledge that there's some
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Your edit is very interesting. It seems that 3 years before Gibbs published his notes, Clifford already defined the cross product (although he did not call it cross product). Does it mean that
Clifford is the father of cross product? This is weird, as I have read everywhere that Gibbs and Heaviside
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John, There is clearly a problem with this article as regards terminologies and disambiguation. The statement at the top that the article is exclusively about the binary operation in 3D is not true. As you have already said yourself, the multilinear algebra section deals with that other concept of
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are the unit vectors in the respective directions) should suffice. But that's a different article. The sense in which there are cross products only in dimensions 3 and 7 (with reference) might have a definition which could be extended to multilinear cross products, but I don't have a copy of any
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John, You have just reverted an edit and referred me to the talk page. I'm now looking at your explanation and I can't see anything which remotely addresses the issue of why you reverted the edit. The edit in question did two unrelated things. First of all, it segregated the issue of the "Lagrange
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John, the reason which you gave for reverting that latest edit doesn't wash. You went to the WikiProject mathematics and you had it confirmed there that the relationship in question applies in seven dimensions. This article is about the cross product in general, even if it concentrates on the more
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It appears to me that you are both overlooking something very fundamental. The 3D cross product involves 3 components in any operation, and it can be used to describe realities in 3D space. The same analogy does not exist with the 7D cross product. In any given operation, we are still only using 3
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For two vectors, the vector part of their quaternion product is the cross product. So
Hamilton's earlier product includes the idea. As for Clifford, he represents area, and its orientation, by a vector. This notion he may have gotten from Grassmann, but generally Clifford doesn't give references.
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it is a volume. Geometric algbra addresses this type system. Tensor variance and related concepts address the issues of how these sorts of quantites transform as you change coordinate systems. (For example, a distance gets numerically bigger if you use kilometers rather than miles, but a gradient
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holds in both 3 and 7 dimensions. As the article stands now, it is misleading by its omission of any mention of the 7D case. It doesn't matter whether the article is exclusively about the 3D cross product or not. The information regarding the applicability in 7D is so closely related to the topic
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This article is a about the product in 3D, as the introduction says, and everything from sections 1 to 7 is written with the assumption the maths is in 3D. The reason for this is the cross product, to mathematicians, means the product in 3D. Most don't know about the product in 7D, and if they do
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Further to that and knowing that in
Phyisics where we can visualize qualities and quantities the representation of a triple cross rpoduct could be a sphere with three radiuses not necessarily connected in right angle. In fact we need four vectors and the fourth could be the unit of measure as no
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I would not be surprised if he did (though the mentioned source seems not to exist) – the cross product can be derived from the exterior product of
Grassmann within geometric algebra by taking the straigtforward dual of the bivector result of the exterior product, and this is only possible with
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Also, it is possible to rotate such a 2D representation or to mirror it when you lose left or right directions again. What I am also after is the visual representation of mathematical symbols, including numbers. I do not like that in writing for instance 12345, we do not "see, because it is not
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It's the 67th most popular maths article, and the vast majority of those readers will I think be after the cross product as used in vector algebra, i.e. the 3D one, it's definition and basic properties. The more advanced material should come later, as it now does. Moving it to the top will just
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I've merged the two definitions, preferring the notation which was there before as it's more informative and there's more of it. At the same time I moved the other
Lagrange's identity up, so it's clearer how it fits in, using consistent notation for that too, and saying a bit more how they're
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As has come up before the place for extending it to 7D, the only dimension it extends to, is the 7D cross product page. That's already linked in the introduction and has a section in
Generalizations, so does not need anything else adding. This is a prominent article on an elementary topic, so
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in this plane gives the direction, by the right hand rule, as given e.g. by the diagram at right. The right hand rule is a convention: it's not something that can be proved, we just chose from the two possibilities the result given by the right hand rule, so it matches the convention for the
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Not much more I can say. The cross product is defined in 3D and 7D. It can be proven to exist in only those dimensions, subject to the restrictions in the definition. All this is on the 7D cross product page. There's no problem with doing it 7D as vectors, angles, etc. are all defined in 7D,
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cross product which involves multiple vectors in a single operation and extends to the corresponding multiple dimensions. It's a different subject, except in the 3D case, and it is covered in this article. The scope of this article needs to be clarified and the headings amended accordingly.
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Towards the end it mentions the different generalisations of the cross product - there are more than just the seven dimensional one, depending on how you relax the conditions on the cross product. But the article is not also about them, or any one of them, it is just about the product in
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The note suggests the proof is "elementary" for "students" but does not say how elementary. It also does not give the proof, but "indicates" them. Lastly it says nothing about the distributive law, saying instead that the proofs will need "knowledge of" various things, mostly related to
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was defined by Hamilton before the cross product, but quaternions are 4-element vectors. Although the vector part of the quaternion product, which involves only 3 elements, is what now we would call a cross product, Hamilton failed to see it. So, Hamilton is not the father of the cross
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John: The template claims that the article is about the cross-product in 3-space. However, it has a section on "Generalizations" which has little if anything to do with the 3-D cross product. It mentions normed division algebras, exterior algebra, and multilinear algebra including
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but it was deleted without a reply, so I'll raise it here. The just added material on the matrix representation duplicates what's already there, except less clearly (it's not clear what the significance of the last two lines of formulae is) and with one error (the link to
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And since the three vectors are used to represnet directions in space, the fourth should autimatically be time with the difference that it has just one direction and it is positive. So the visualisation could or should be someting similar to what you get with seven cross
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of the Lagrange identity to 3D. After working it out from first principles I finally conceded that you were correct and that it holds in 7D also. I came here to fix this article in that respect. You have now reversed your position. You had better explain yourself.
2304:, but I think it provides the framework you are craving.) As you have seen, in the classical treatment of 3-vectors and cross products, the algebra doesn't mirror the physics: if a quantity is a cross product of vectors, it is an oriented area, and if it is a
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that its omission is positively misleading. Anybody reading the article as it stands now will easily believe that the above equation only holds in 3D. The purpose of wikipedia is to supply information and not to hold back little secrets for another day.
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John, The source talks about axioms. The distributive law is one of the axioms of the 'cross product'. A 5D cross product does not satisfy that axiom. That is why we cannot have a 5D cross product. I am very surprised that you didn't know all this.
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Mentioned later, but for a different reason. The inserted sentence does not say that you need a dual to obtain a vector. It seems to say that the "product of two vectors" defined by Clifford in 1878 is already a vector (not a bivector).
1957:≥1 dimensions, I can define "left" to be any direction and "right" to be the opposite direction. There are issues with the cross product in any other than three dimensions, but the page discusses this. Can you rephrase your question?
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Yes, I think that educating an interested lay person in elementary aspects of vector spaces is a bit off-topic on this talk page. O.t.o.h referring to the reference desk can only point back to where we have been pointing already.
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hold, then we will only have a result in 1, 3,and 7 dimensions. As I have been repeatedly saying, this fact is too basic to require a source. But although it is basic,it seems to have been overlooked by some of the editors here.
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The article is about the cross product in 3D, i.e. what mathematicians generally understand by the name "cross product". It does not describe the generalisations: it provides links summary-style to most of them, and only the
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Yes, as the article says the angle is always measured so it is between 0 and 180 degrees. The angles 0 and 180 can be ignored as the vectors are then parallel and the cross product is zero, so we need only consider 0° <
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define the cross product entirely. It's not so important in 3D as for most people it's just the product given by the formula, it could be clearer though in 3D so something about the conditions should probably be included.
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An angle between two vectors in 7 dimensions means the arccos of the dot product of the vectors divided by the norms of the vectors. That is what it means in n dimensions for any n. I'm really amazed you didn't know that.
2802:(a 2-D surface) with an area, not as a directed line segment with a length. I guess that this also means that, if a vector is measured in meters, the exterior product is in square meters. So, Grassmann was not the father.
1979:≥1 dimensions, where you say you can define "left" to be any direction and "right" to be the opposite direction in 2D and I need to have up or down defined, otherwise I have a problem (rotate by 180 degrees to check).
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A truly 7D cross product which would link to a real 7D space by the same analogy with the 3D case, would have to involve all 7 components in any given operation. We do not have a truly 7D cross product in that sense.
674:|sinθ was a general result for cross products in 1,3, or 7 dimensions. This was of course no doubt due to the fact that whoever wrote the article was clearly writing it exclusively with the 3D cross product in mind.
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is similar to, but even more complex than the exterior product, as it takes into account an additional argument, called Q. So, Clifford was going away from the cross product, rather than toward its discovery.
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are defined in terms of all seven components, then we will be completely lost. Because once we have established the rotation axis, we will have a headache trying to decide which plane to take our angle from.
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The 7D cross product that we do have, is somewhat of a misnomer. It is 7D only in the sense that it involves 7 components. But these seven components do not actually represent unit vectors in any space.
642:|sinθ will be explicitly in relation to the 3D cross product's linkage to 3D space. There is no proof in the literature that the 7D cross product has got any corresponding equivalent to the equation |
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should at least get a more explicit link. As it stands now, we have to guess that the link to 'seven dimensions' will open out to 'seven dimensional cross product'. Anyway, I note that you say that
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John, As regards your answer about the angle θ in seven dimensions, we'll just have to agree to differ. I cannot remotely imagine what an angle means in seven dimensions. Neither can I imagine
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for vectors has a few variations. They aren't exactly the same, but are all interrelated: some involve two vectors, some three and some four. It may be too much trouble to sort them all out.
2114:, the distinction between points and tangent vectors becomes critical because points are not in general in a vector space, and each tangent space can look different (e.g., having a different
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which more or less backs up my viewpoint that no source is necessary to back up the view that the reason why cross product doesn't work in 5D is because it doesn't obey the distributive law.
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components, and the linkage to 3D space has been lost. As regards 7D space, if such a thing actually exists, the analogy with the 3D cross product's relationship to 3D space does not exist.
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should be brought over to this article, and the latter article deleted. The material from the latter article should be put in a special section entitled 'Seven Dimensional Cross Product'.
2066:
Yes. Thank you very much. What do you think of the fact that in space there is symmetry, so it does not matter where you put the origo of a co-ordinate system or where you start time from?
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In one it's a bit less obvious, until you realise that all lines in one dimension are parallel, so θ is zero or π, and so sin θ is always zero. The product is zero always, i.e. trivial.--
2209:, etc.) as physical things, not just symbols. However, "...an individual digit is a cross product..." doesn't make any sense to me. I don't know what you mean by "hidden cross-product".
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there" that an individual digit is a cross product of the value of that position (power) and the number iself. I would like to have a notation that reveals such a hidden cross-product.
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That's already in the article, under Multilinear algebra. The restriction to 3 and 7 dimensions is for binary products satisfying the criteria, so if you say the crosss product is
2671:. The above statement has no reference. Further, the allusion to Clifford Algebras does not improve the discription of cross product's history. The statement appears superfluous.
1841:-1)-ary products. There are a few choices: change the template and introduction to describe the article as it actually is, or rewrite the article to fit the present description.
1306:
To settle this matter, I have added a reference template to the 7D case at the top of this article, and added the 7D case to the disambiguation page for "cross product".
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and have created a deletion discussion for it, hopefully to get a consensus over what if anything is the best title for an article on such an identity or identities.--
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identity" into a separate sub-section from the "vector triple product". Secondly, it made an important clarification that this form of the Lagrange identity,
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you are saying that it is both 3D and 7D. Your argument about this being the 3D page doesn't wash because you also did the same reversion on the more general
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Anything to do with the equation |a×b| = |a|×|b|sinθ only relates to the 3D case, and hence should only appear in the sections about the 3D cross product.
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a section in this article about "Cross product as an exterior product", and I think it is nice to have a reference to that section in the history section.
959:, which earned you a year long ban on physics related articles. What you are doing could lead to an extension to mathematics related articles, or worse.
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I fully agree with JohnB. The seven dimensional obscurity deserves an honorable mention under the generalisations section. It already has the mention.
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orthonormality. As even if you pay $ 12 you don't get a proof, just an indication of one, it's a lot less useful as a source than the two proofs on
658:|sinθ in relation to 7D space. It was an over extrapolation on the part of whoever wrote the article, to write it in such a way as to imply that |
2659:, where in the case of three-dimensional vectors the bivector produced from two vectors dualizes to a vector, thus reproducing the cross product.
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In fact the concept of spacetime is also a cross product with motion (and speed to be also considered), if I am not mistaken. What do you think?
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The source states that students may enquire as to whether or not a cross product exists in dimensions other than 3. The source states,
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It's just an algebraic theorem which can be applied here, using fairly elementary algebra. I suggest you try it before dismissing it.--
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article, which is more concerned with that particular generalisation. That together with the orthogonality and the mangnitude being
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Hi, you cannot define right or left without defining up and down, and it is not in 3D space anyway, but in 2D. Can you see that?
1490:{\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\ }
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This note points out that by proving the elementary propositions and theorem below, students may answer this question themselves.
2090:. In a vector space, there is one origin and all vectors have their tail there; in affine space, there are points and so-called
1190:{\displaystyle |\mathbf {a} \times \mathbf {b} |^{2}+|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} |^{2}|\mathbf {b} |^{2}.}
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of a few sources. This is the kind of talk page disruption that brought you in trouble before, so perhaps you shoud be careful.
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is zero. The vectors must form two sides of a non-trivial triangle which defines a plane. And the sense of the rotation from
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There is nothing wrong with either article that needs 'fixing', it is only that you (still, clearly) do not understand it.
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John, I think you had better explain your reversion. We have just spent a few days arguing about this on the talk page of
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Or more precisely I think we should see the cross product of base x power x count explicitely for visualization purposes.
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tag you've added is nonsensical as it's the same content you added - you've just added a perfectly good source for it.--
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of points). Subtracting one point from another gives a vector from one to the other. To rephrase your observation, in
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which brought together many mathematical ideas. He defined the product of two vectors to have magnitude equal to the
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1599:-space is well-defined; and the properties on p.779 of the Mathematica Book, together with the "right-hand" rule
2786:(or wedge procuct) was defined even earlier by Grassmann. In R it is the same as a cross product as far as its "
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What needs to be done it is to make a clear segregation in the article between the 3D case and the 7D case.
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but as for merging, no that would not be appropriate. This article is on the common vector operation, as per
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confuse many readers, as e.g. those who do maths at school won't have the math background to deal with it.--
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This article includes short sections on octonion, quaternion, wedge products etc. So it is not strictly 3-D
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John, Are you sure you understand the subject? On this page, you are saying that it is 3D only. But over at
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were the fathers of cross and dot product. Does Clifford refer to Gibbs or Heaviside, in his 1878 text?
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David, you might consider stopping point-push-disrupting this talk page. You are behaving here (and on
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In matters to do with maths, first principle arguments are often sufficient in the absence of sources.
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I have ever seen here. That phrase should be taken as an example in various policies and guidelines.
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it's given a separate name, so the 3D version can be called "The Cross Product" without ambiguity.
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Does anybody object if I go ahead and bring the seven dimensional material over to this article?
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the vector quadruple product is (AxB)x(CxD), not (AxB).(CxD). The latter is called a "quadruple
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It's a different Lagrange's identity. I've seen it before, on WP I think, but you provided a
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Hi John: I'm OK with that. What I was trying for was a definition that is easily extended to
129:|sinθ applies in both three and seven dimensions. It's equivalent to one of the conditions, |
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261:
1558:
This article is about the cross product of two vectors in three-dimensional Euclidean space
2893:
2892:, it says the direction of the cross product by the right-hand rule. But the wiki page of
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including more content from the 7D cross product article would I think be inappropriate.--
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focus the relation of x,y,z-axis of 3-D coordinates. When we consider the cross product
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The added material has a plus: it's sourced. The pre-existing material is not. The
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It doesn't need to be proved as it's part of the definition: it's required to be a
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The angle is defined in seven dimensions as in two and three. You can use e.g. the
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If you wish to start a new discussion or revive an old one, please do so on the
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2798:. And a bivector, as far as I know, was described and interpreted as a directed
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If you wish to start a new discussion or revive an old one, please do so on the
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Huh? Don't you know that in n dimensions, the angle θ between two vectors is
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But the matrix content is just a copy of what's already there: The matrices
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In the sources, any proof that the 3D cross product obeys the relationship |
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Oh I see. This is disruption is it? In that case I better leave you to it.
2791:
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This article is not intended to be exclusively about the 3D cross product
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of which they are two sides, and direction perpendicular to their plane.
2197:
I am not sure quite what you mean. I agree that it is very important in
1718:
it is only is sensibly and non-trivially defined in those dimensions.--
1339:
No, for the reasons above and earlier on this talk page. I recall that
2206:
2202:
2565:
product", whereas the former is called "a quadruple vector product"
1675:{\displaystyle e_{1}\times e_{2}\times \cdots \times e_{n-1}=e_{n},}
504:
DVdm, You can have the cosine. But what does the angle itself mean?
1918:
The note at the top of the article looks perfectly accurate to me.
812:
2942:
825:
In other words, if the basic axioms such as distributive law, and
358:
to calculate it. This can be used to relate the above formula to |
301:
2441:"which is sometimes known as Lagrange's identity" so the link to
2757:
2346:
is wrong - the formula is a different one with the same name).--
2011:, I'm assuming we have a coordinate system defined so a vector,
750:
The 7D cross product that we do have, is somewhat of a misnomer.
521:
I didn't say anything about cosines. I said something about the
1025:
which implies distributivity. This is stated explicitly in the
2989:
25:
2040:
standard, though, so it is usually omitted. Does that help?
2655:
combined the algebras of Hamilton and Grassmann to produce
2884:
the direction of the cross product by the right-hand rule
2817:
in the history section seems to say something different.
2770:
Here's why they, as far as I know, were not the fathers:
2624:
Knowledge:Articles_for_deletion/Vector_quadruple_product
1340:
883:
This is probably the best example of a combined case of
2790:" are concerned. However, its "vector components" are
2648:
Currently the History section contains this statement:
2570:
2338:
1880:
1872:
807:
A source to back up the idea that no source is required
760:
and at the same using the talk page to complain about "
2494:
sometimes used in naming this identity, sometimes not.
2602:
Being introduced to it by that change I've looked at
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are identical and they are used the same way, so the
2082:
I think you've stumbled on the distinction between a
1607:
1374:
1073:
681:
few planes. And if we take the general cross product
2663:
Today I added the reference to Clifford's 1878 text
2035:= . If we assume the space is right-handed, then if
1347:
as there is really no point reopening this debate.--
929:
The source says nothing about the distributive law--
533:" in the 2-dim subspace spanned by the two vectors.
2813:However, the above mentioned sentence inserted by
1674:
1489:
1189:
1975:I am trying to visualize what you claim to be in
1863:The article is not about the cross product in 3D
552:Inner product space#Norms on inner product spaces
336:| ) ) )? Not all that fascinating if you ask me.
264:. The proof involved three dimensional geometry.
755:The speed of light is somewhat of a tautology...
227:|sinθ applies in both three and seven dimensions
1746:-dimensions should be brought up, for example.
232:The way I was taught it was that the equation |
320:| ), which you can write as arcsin( √( 1 - ( (
382:) – just replace the dot product and use the
8:
2587:) and which was corrected for caps by John.
2240:count makes sense without a unit of measure!
1521:accepted or readily understood definition.--
1343:, with no better arguments than now. Please
2956:< 180°. We can also ignore cases where
256:satisfying certain conditions such as the
157:. So it's true in both 3 and 7 dimensions.
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1056:The Lagrange Identity in seven dimensions
772:resulting from what looks like a shallow
768:". You seem to be insisting to push some
84:This article needs to be merged with the
2298:covariance and contravariance of tensors
525:of a real number. But anyway, tell your
2735:Who is the father of the cross product?
3008:Do not edit the contents of this page.
2385:which made it clear: it's the formula
2300:interesting. (Be warned, it is a huge
1556:No need for that as long as the text "
44:Do not edit the contents of this page.
2904:, how to determine the rotation from
7:
2622:Heh. I have added a delete !vote at
1560:" is sitting on top of the article.
953:Talk:Seven-dimensional cross product
2912:? It seems the rotation angle from
529:that it produces for instance the "
80:The seven dimensional cross product
384:Pythagorean trigonometric identity
248:|sinθ follows as a consequence of
155:Pythagorean trigonometric identity
24:
2201:to think of quantities (numbers,
1800:dimensions is described at all.--
766:where maths leaves the real world
489:You really continue to amaze me.
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1875:was clearly making a disruptive
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267:If you think that the equation |
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1268:seven dimensional cross product
1240:Seven dimensional cross product
1062:Seven dimensional cross product
1027:Seven-dimensional cross product
853:which are complete and free. --
851:Seven-dimensional cross product
205:seven dimensional cross product
90:seven dimensional cross product
86:seven dimensional cross product
2739:This sentence was inserted by
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2597:14:53, 12 September 2010 (UTC)
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2854:16:35, 10 December 2010 (UTC)
2836:15:58, 10 December 2010 (UTC)
2730:02:02, 10 December 2010 (UTC)
762:extrapolations beyond reality
2712:09:28, 9 December 2010 (UTC)
2681:03:46, 9 December 2010 (UTC)
2569:. I have removed the remark
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2272:07:08, 28 October 2010 (UTC)
2256:07:20, 28 October 2010 (UTC)
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1995:21:55, 25 October 2010 (UTC)
1967:17:46, 25 October 2010 (UTC)
1948:15:26, 25 October 2010 (UTC)
1934:It does not make sense to me
1050:20:25, 29 January 2010 (UTC)
955:) like you were behaving on
2015:can be distinguished from −
1694:such reference with me. —
1542:-dimensions. Can you help?
1341:you tried to do this before
986:11:57, 4 January 2010 (UTC)
969:11:49, 4 January 2010 (UTC)
945:11:38, 4 January 2010 (UTC)
924:11:27, 4 January 2010 (UTC)
907:10:19, 4 January 2010 (UTC)
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843:05:47, 4 January 2010 (UTC)
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293:11:40, 2 January 2010 (UTC)
203:John, Fair enough. But the
196:11:03, 2 January 2010 (UTC)
182:10:09, 2 January 2010 (UTC)
108:09:47, 2 January 2010 (UTC)
3054:
2940:) 21:48 May 13,2011 (UTC)
2029:Without loss of generality
2007:I'm missing something. In
1511:14:47, 24 April 2010 (UTC)
1359:08:40, 24 April 2010 (UTC)
1334:02:43, 24 April 2010 (UTC)
1316:16:31, 23 April 2010 (UTC)
1302:15:42, 23 April 2010 (UTC)
1280:15:20, 23 April 2010 (UTC)
1256:12:47, 20 April 2010 (UTC)
1233:12:01, 20 April 2010 (UTC)
1211:10:54, 20 April 2010 (UTC)
2920:in the plane containing
2667:which is explicit on the
2333:Duplicate matrix material
1865:"and its generalisations"
2985:22:06, 13 May 2011 (UTC)
2750:William Kingdon Clifford
2743:in the history section:
2653:William Kingdon Clifford
2551:Vector quadruple product
2545:13:32, 2 June 2010 (UTC)
2527:00:48, 1 June 2010 (UTC)
2504:00:34, 1 June 2010 (UTC)
2489:21:50, 31 May 2010 (UTC)
2377:21:23, 31 May 2010 (UTC)
2358:20:16, 31 May 2010 (UTC)
1928:12:30, 24 May 2010 (UTC)
1913:12:04, 24 May 2010 (UTC)
1893:09:03, 24 May 2010 (UTC)
1883:Please stop doing that.
1851:02:48, 24 May 2010 (UTC)
1828:00:47, 24 May 2010 (UTC)
1812:22:26, 23 May 2010 (UTC)
1773:22:18, 23 May 2010 (UTC)
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1533:20:53, 23 May 2010 (UTC)
1015:) 20:09, 29 January 2010
88:. The material from the
2754:Introduction to Dynamic
2665:Introduction to Dynamic
2517:, a more general form.
2292:I think you would find
2106:all tangent spaces are
1591:The "cross product" of
811:I've got a source here
2947:
2928:must be smaller (0° ≤
2805:As far as I know, the
1871:some generalisations.
1676:
1491:
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3006:of past discussions.
2946:
2515:Binet–Cauchy identity
1677:
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416:in seven dimensions.
42:of past discussions.
2513:I added the link to
1953:I don't follow. In
1605:
1516:Incorrect definition
1372:
1071:
753:This reminds me of "
527:pure maths professor
414:Pythagoras's theorem
2443:Lagrange's identity
2344:Lagrange's identity
1982:Is that any better?
1796:− 1 way product in
1218:Lagrange's identity
2973:coordinate axes.--
2948:
2888:In the section of
2776:quaternion product
1672:
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1187:
18:Talk:Cross product
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3012:current talk page
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2788:scalar components
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2365:Lagrange identity
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2027:rotated by 180°.
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2311:—Ben FrantzDale
2211:—Ben FrantzDale
2120:—Ben FrantzDale
2104:Euclidean space
2096:tangent vectors
2042:—Ben FrantzDale
1959:—Ben FrantzDale
1936:
1879:in response to
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1221:further down.--
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856:John Blackburne
827:Jacobi identity
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258:Jacobi identity
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2838:
2837:
2833:
2829:
2824:
2822:
2818:
2816:
2808:
2804:
2801:
2800:parallelogram
2797:
2793:
2789:
2785:
2781:
2777:
2773:
2772:
2771:
2765:
2763:
2762:parallelogram
2759:
2755:
2751:
2746:
2745:
2744:
2742:
2734:
2732:
2731:
2727:
2723:
2713:
2709:
2705:
2701:
2697:
2696:
2695:
2694:
2687:
2686:
2685:
2684:
2683:
2682:
2678:
2674:
2670:
2666:
2658:
2654:
2651:
2650:
2649:
2643:
2637:
2633:
2629:
2625:
2621:
2620:
2619:
2615:
2610:
2605:
2601:
2600:
2599:
2598:
2594:
2590:
2586:
2583:
2580:
2576:
2572:
2568:
2564:
2560:
2557:
2555:According to
2550:
2546:
2542:
2537:
2531:
2530:
2529:
2528:
2524:
2520:
2516:
2505:
2501:
2497:
2492:
2491:
2490:
2486:
2481:
2473:
2461:
2457:
2453:
2452:
2451:
2450:
2445:is incorrect.
2444:
2440:
2435:
2431:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2391:
2387:
2386:
2384:
2380:
2379:
2378:
2374:
2370:
2366:
2362:
2361:
2360:
2359:
2355:
2350:
2345:
2340:
2332:
2320:
2316:
2312:
2307:
2303:
2299:
2295:
2291:
2290:
2289:
2288:
2287:
2286:
2285:
2284:
2283:
2282:
2273:
2269:
2265:
2262:
2259:
2257:
2253:
2249:
2246:
2242:
2238:
2237:
2236:
2235:
2234:
2233:
2232:
2231:
2220:
2216:
2212:
2208:
2204:
2200:
2196:
2195:
2194:
2193:
2192:
2191:
2190:
2189:
2188:
2187:
2178:
2174:
2170:
2167:
2164:
2162:
2158:
2154:
2151:
2147:
2146:
2145:
2144:
2143:
2142:
2141:
2140:
2129:
2125:
2121:
2117:
2116:metric tensor
2113:
2109:
2105:
2101:
2100:tangent space
2097:
2093:
2092:bound vectors
2089:
2085:
2081:
2080:
2079:
2075:
2071:
2068:
2065:
2064:
2063:
2062:
2061:
2060:
2059:
2058:
2051:
2047:
2043:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2005:
2004:
2003:
2002:
2001:
1996:
1992:
1988:
1984:
1981:
1978:
1974:
1973:
1972:
1971:
1968:
1964:
1960:
1956:
1952:
1951:
1950:
1949:
1945:
1941:
1933:
1929:
1925:
1921:
1917:
1916:
1915:
1914:
1910:
1906:
1894:
1890:
1886:
1882:
1878:
1874:
1870:
1867:. It briefly
1866:
1862:
1861:
1860:
1859:
1858:
1857:
1852:
1848:
1844:
1840:
1835:
1834:
1833:
1832:
1829:
1825:
1821:
1816:
1815:
1814:
1813:
1809:
1804:
1799:
1795:
1787:Article topic
1786:
1774:
1770:
1765:
1759:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1732:
1731:
1727:
1722:
1717:
1713:
1709:
1705:
1704:
1703:
1700:
1697:
1692:
1685:
1669:
1664:
1660:
1656:
1651:
1648:
1645:
1641:
1637:
1634:
1631:
1626:
1622:
1618:
1613:
1609:
1601:
1600:
1598:
1594:
1590:
1589:
1588:
1584:
1579:
1573:
1571:
1567:
1563:
1559:
1555:
1554:
1553:
1549:
1545:
1541:
1537:
1536:
1535:
1534:
1530:
1525:
1515:
1513:
1512:
1508:
1504:
1480:
1465:
1452:
1447:
1432:
1419:
1414:
1392:
1368:
1367:
1366:
1360:
1356:
1351:
1346:
1342:
1338:
1337:
1336:
1335:
1331:
1327:
1317:
1313:
1309:
1305:
1304:
1303:
1299:
1294:
1288:
1284:
1283:
1282:
1281:
1277:
1273:
1269:
1260:
1258:
1257:
1253:
1249:
1245:
1241:
1234:
1230:
1225:
1219:
1215:
1214:
1213:
1212:
1208:
1204:
1184:
1179:
1157:
1137:
1132:
1117:
1104:
1099:
1084:
1067:
1066:
1065:
1063:
1055:
1051:
1047:
1042:
1036:
1032:
1028:
1024:
1020:
1019:
1018:
1014:
1010:
1006:
1005:129.89.14.247
1002:
990:
988:
987:
983:
979:
970:
966:
962:
958:
954:
950:
949:
946:
941:
937:
933:
928:
927:
926:
925:
921:
917:
908:
904:
900:
896:
893:
890:
886:
881:
877:
876:
875:
870:
865:
861:
857:
852:
847:
846:
845:
844:
840:
836:
831:
828:
823:
822:
818:
815:
813:
806:
804:
803:
799:
795:
788:
787:
783:
779:
775:
771:
767:
763:
756:
751:
747:
746:
741:
736:
732:
728:
722:
721:
720:
719:
718:
717:
713:
709:
703:
699:
696:
692:
688:
684:
678:
675:
673:
669:
665:
661:
657:
653:
649:
645:
641:
637:
633:
629:
624:
616:
612:
608:
603:
602:
601:
600:
597:
592:
588:
584:
579:
575:
574:
571:
566:
562:
558:
553:
550:
549:
544:
540:
536:
532:
528:
524:
520:
519:
518:
517:
516:
515:
511:
507:
500:
496:
492:
488:
487:
486:
485:
481:
477:
469:
464:
460:
456:
451:
450:
445:
441:
437:
432:
431:
430:
429:
428:
427:
423:
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415:
408:
403:
399:
395:
390:
389:
385:
381:
377:
373:
369:
365:
361:
357:
353:
352:
347:
343:
339:
335:
331:
327:
323:
319:
315:
311:
307:
303:
299:
298:
297:
296:
295:
294:
290:
286:
282:
278:
274:
270:
265:
263:
259:
255:
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247:
243:
239:
235:
230:
228:
226:
222:
218:
214:
208:
206:
197:
193:
189:
185:
184:
183:
178:
174:
170:
165:
164:WP:COMMONNAME
161:
160:
156:
152:
148:
144:
140:
136:
132:
128:
124:
120:
116:
112:
111:
110:
109:
105:
101:
96:
93:
91:
87:
79:
73:
70:
67:
63:
61:
58:
57:
49:
45:
41:
40:
35:
28:
27:
19:
3029:
3007:
3001:
2969:
2965:
2961:
2957:
2953:
2929:
2925:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2889:
2887:
2879:
2825:
2820:
2819:
2812:
2796:unit vectors
2768:
2753:
2747:
2738:
2718:
2668:
2664:
2662:
2647:
2581:
2562:
2554:
2512:
2459:
2455:
2433:
2429:
2425:
2421:
2417:
2413:
2409:
2405:
2401:
2397:
2393:
2389:
2336:
2302:can of worms
2199:applied math
2088:affine space
2084:vector space
2036:
2032:
2024:
2023:is equal to
2020:
2016:
2012:
2008:
1976:
1954:
1937:
1901:
1881:this remark.
1868:
1864:
1838:
1797:
1793:
1790:
1743:
1739:
1735:
1715:
1711:
1707:
1696:Arthur Rubin
1687:
1596:
1592:
1557:
1539:
1519:
1499:
1363:
1322:
1264:
1237:
1199:
1059:
1034:
1030:
994:
973:
911:
891:
879:
873:
832:
824:
820:
819:
816:
810:
791:
765:
761:
758:
754:
749:
704:
700:
694:
690:
686:
682:
679:
676:
671:
667:
663:
659:
655:
651:
647:
643:
639:
635:
631:
627:
625:
621:
530:
526:
522:
503:
472:
411:
379:
375:
371:
367:
363:
359:
333:
329:
325:
321:
317:
313:
309:
305:
304:as arccos( (
280:
276:
272:
268:
266:
253:
249:
245:
241:
237:
233:
231:
224:
220:
216:
212:
210:
209:
202:
150:
146:
142:
138:
134:
130:
126:
122:
118:
114:
97:
94:
83:
65:
43:
37:
3000:This is an
2604:the article
2519:Brews ohare
2496:Brews ohare
2369:Brews ohare
2019:and where −
1905:David Tombe
1843:Brews ohare
1820:David Tombe
1748:Brews ohare
1544:Brews ohare
1503:David Tombe
1326:David Tombe
1308:Brews ohare
1272:David Tombe
1248:David Tombe
1203:David Tombe
999:—Preceding
978:David Tombe
916:David Tombe
835:David Tombe
794:David Tombe
708:David Tombe
506:David Tombe
476:David Tombe
418:David Tombe
356:dot product
285:David Tombe
153:) from the
100:David Tombe
36:This is an
2932:≤ 180°)? -
2890:Definition
2533:related.--
2264:Genezistan
2248:Genezistan
2169:Genezistan
2153:Genezistan
2108:isomorphic
2070:Genezistan
1987:Genezistan
1940:Genezistan
1023:Linear map
3036:Archive 3
3030:Archive 2
3024:Archive 1
2792:bivectors
2244:products.
2112:manifolds
2031:, assume
1985:Regards,
1873:This edit
895:wp:SOURCE
882:" ==: -->
774:synthesis
752:" ==: -->
72:Archive 3
66:Archive 2
60:Archive 1
2865:Paolo.dL
2828:Paolo.dL
2821:Question
2779:product.
2748:In 1878
2704:Paolo.dL
2585:contribs
1877:WP:POINT
1869:mentions
1013:contribs
1001:unsigned
578:WP:RD/MA
260:and the
3003:archive
2815:Rgdboer
2760:of the
2741:Rgdboer
2722:Rgdboer
2673:Rgdboer
2207:tensors
2203:vectors
2098:in the
2086:and an
1686:(where
1345:drop it
885:wp:NPOV
764:" and "
302:defined
39:archive
2934:ligand
2794:, not
2563:scalar
1699:(talk)
1246:page.
889:wp:SYN
689:where
523:arccos
2981:deeds
2850:deeds
2614:deeds
2571:added
2541:deeds
2485:deeds
2420:) - (
2404:) = (
2354:deeds
1808:deeds
1769:deeds
1736:a × b
1726:deeds
1583:deeds
1529:deeds
1355:deeds
1298:deeds
1290:3D.--
1229:deeds
1046:deeds
940:deeds
936:words
864:deeds
860:words
770:point
735:deeds
731:words
666:| = |
650:| = |
634:| = |
591:deeds
587:words
565:deeds
561:words
463:deeds
459:words
402:deeds
398:words
374:| - (
366:| = |
328:) / |
312:) / |
275:| = |
240:| = |
219:| = |
177:deeds
173:words
145:| - (
137:| = |
121:| = |
16:<
2938:talk
2924:and
2869:talk
2832:talk
2782:The
2774:The
2758:area
2726:talk
2708:talk
2677:talk
2632:talk
2628:DVdm
2593:talk
2589:DVdm
2579:talk
2567:here
2558:and
2523:talk
2500:talk
2463:and
2383:link
2373:talk
2339:here
2315:talk
2296:and
2268:talk
2252:talk
2215:talk
2173:talk
2157:talk
2124:talk
2074:talk
2046:talk
1991:talk
1963:talk
1944:talk
1924:talk
1920:DVdm
1909:talk
1889:talk
1885:DVdm
1847:talk
1824:talk
1752:talk
1566:talk
1562:DVdm
1548:talk
1507:talk
1330:talk
1312:talk
1276:talk
1252:talk
1207:talk
1033:sin
1009:talk
982:talk
965:talk
961:DVdm
920:talk
903:talk
899:DVdm
839:talk
798:talk
782:talk
778:DVdm
712:talk
693:and
611:talk
607:DVdm
576:And
539:talk
535:DVdm
510:talk
495:talk
491:DVdm
480:talk
440:talk
436:DVdm
422:talk
342:talk
338:DVdm
289:talk
229:.
192:talk
188:DVdm
104:talk
2968:to
2960:or
2916:to
2908:to
2573:by
2396:)⋅(
2118:).
892:and
2900:×
2871:)
2834:)
2826:—
2728:)
2710:)
2702:—
2679:)
2634:)
2626:.
2595:)
2525:)
2502:)
2475:}}
2472:cn
2469:{{
2428:)(
2412:)(
2400:×
2392:×
2375:)
2317:)
2270:)
2254:)
2217:)
2205:,
2175:)
2159:)
2126:)
2076:)
2048:)
1993:)
1965:)
1946:)
1926:)
1911:)
1891:)
1849:)
1826:)
1754:)
1714:=
1710:×
1649:−
1638:×
1635:⋯
1632:×
1619:×
1568:)
1550:)
1509:)
1466:×
1433:⋅
1420:−
1332:)
1314:)
1278:)
1254:)
1209:)
1118:⋅
1085:×
1038:--
1031:ab
1011:•
984:)
967:)
938:‡
922:)
905:)
887:,
862:‡
841:)
800:)
784:)
757:".
733:‡
714:)
670:||
654:||
638:||
613:)
589:‡
580:--
563:‡
554:--
541:)
512:)
497:)
482:)
461:‡
442:)
424:)
400:‡
370:||
344:)
332:||
316:||
291:)
279:||
244:||
223:||
194:)
175:‡
141:||
125:||
106:)
3014:.
2970:b
2966:a
2962:b
2958:a
2954:θ
2936:(
2930:θ
2926:b
2922:a
2918:b
2914:a
2910:b
2906:a
2902:b
2898:a
2867:(
2830:(
2724:(
2706:(
2675:(
2630:(
2591:(
2582:·
2577:(
2521:(
2498:(
2465:×
2460:a
2456:T
2436:)
2434:C
2432:⋅
2430:B
2426:D
2424:⋅
2422:A
2418:D
2416:⋅
2414:B
2410:C
2408:⋅
2406:A
2402:D
2398:C
2394:B
2390:A
2388:(
2371:(
2313:(
2266:(
2250:(
2213:(
2171:(
2155:(
2122:(
2094:(
2072:(
2044:(
2037:x
2033:x
2025:x
2021:x
2017:x
2013:x
2009:R
1989:(
1977:n
1961:(
1955:n
1942:(
1922:(
1907:(
1887:(
1845:(
1839:n
1837:(
1822:(
1798:n
1794:n
1750:(
1744:n
1740:n
1716:c
1712:b
1708:a
1690:i
1688:e
1670:,
1665:n
1661:e
1657:=
1652:1
1646:n
1642:e
1627:2
1623:e
1614:1
1610:e
1597:n
1593:n
1564:(
1546:(
1540:n
1505:(
1481:2
1476:|
1470:b
1462:a
1457:|
1453:=
1448:2
1443:|
1437:b
1429:a
1424:|
1415:2
1410:|
1404:b
1399:|
1393:2
1388:|
1382:a
1377:|
1328:(
1310:(
1274:(
1250:(
1205:(
1185:.
1180:2
1175:|
1169:b
1164:|
1158:2
1153:|
1147:a
1142:|
1138:=
1133:2
1128:|
1122:b
1114:a
1109:|
1105:+
1100:2
1095:|
1089:b
1081:a
1076:|
1035:θ
1007:(
980:(
963:(
942:)
934:(
918:(
901:(
878:"
866:)
858:(
837:(
796:(
780:(
748:"
737:)
729:(
710:(
695:b
691:a
687:b
685:×
683:a
672:b
668:a
664:b
662:×
660:a
656:b
652:a
648:b
646:×
644:a
640:b
636:a
632:b
630:×
628:a
609:(
593:)
585:(
567:)
559:(
537:(
508:(
493:(
478:(
465:)
457:(
438:(
420:(
404:)
396:(
386:.
380:b
378:·
376:a
372:b
368:a
364:b
362:×
360:a
340:(
334:b
330:a
326:b
324:·
322:a
318:b
314:a
310:b
308:·
306:a
287:(
281:b
277:a
273:b
271:×
269:a
254:b
252:×
250:a
246:b
242:a
238:b
236:×
234:a
225:b
221:a
217:b
215:×
213:a
211:|
190:(
179:)
171:(
151:b
149:·
147:a
143:b
139:a
135:b
133:×
131:a
127:b
123:a
119:b
117:×
115:a
113:|
102:(
50:.
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