Talk:Cross product

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expanding or moving at a constant velocity in two dimensions simultaneously, like a 2d simplification of ripples on a pond - in which case, would it be more accurate to depict the cross-product with arrows radiating outward in all directions from the surface formed by the two vectors in the cross-product, in the plane of the surface. And in the case of pond ripples, you could even replace the parallelogram that's normally used to depict the area of a cross-product with a circle. Perhaps an acceleration vector cross-product could correspond to something like the motion of the perimeter of something that's growing at a linearly increasing rate simultaneously in two different directions on a 2d surface.

211: 106: 8739:, with an arbitrary-sized arrow (or arrows) emanating from the surface (rather than the origin) indicating the direction, or alternatively, do away with the arrow for the cross-product, and instead just labelling the surfaces top and bottom appropriately. Also, the article states: 'Due to the dependence on handedness, the cross product is said to be a pseudovector', but surely the main reason why it's a pseudovector (ie not a real vector/not like common vectors) is that its units are squared. 96: 75: 42: 6264: 292: 201: 180: 5778: 6957:, this distance. And this example shows that we can find in the coordinate-vector form a torque from the force and distance, which is in the plane, where it should be. And we can solve the inverse problem, find the force from the torque, again in a coordinate-vector form. I emphasize that we could not find the last task before. And she decided because of the inverse vector and its projections.-- 33: 8684: 1407: 3149: 2420: 5773: 6259:{\displaystyle =\left({\dfrac {b_{z}d_{zx}-b_{y}d_{xy}}{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}}\right){\bar {i}}+\left({\dfrac {b_{x}d_{xy}-b_{z}d_{yz}}{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}}\right){\bar {j}}+\left({\dfrac {b_{y}d_{yz}-b_{x}d_{zx}}{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}}\right){\bar {k}}=c_{x}{\bar {i}}+c_{y}{\bar {j}}+c_{z}{\bar {k}}} 3373: 2640: 8814:

Nope, not at all. Acceleration has units of meters times hertz squared; does that suddenly mean acceleration must be an "area vector" as well, in the "hertz space"? What about a partial antiderivative of the position vector with respect to position? You cannot make arguments from units alone. This is

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must be considered as a vector. It is an error to consider one as a vector and the other as a pseudovector. In other words, as soon as an inner product has been defined, the distinction between vectors and pseudovectors is only a question of context or point of view. So the "must be" of the deleted

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The cross product can also be described in terms of quaternions, and this is why the letters i, j, k are a convention for the standard basis on R3. The unit vectors i, j, k correspond to "binary" (180 deg) rotations about their respective axes, said rotations being represented by "pure" quaternions

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No worries buddy, I'm just a tourist. Thanks for all your efforts to explain and the links to related topics in particular. I have no intention of touching the article, but I hope you at least consider the different perspective I've put forward as worthy of staying up on the talk page for a while.

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In response let me say that the above mentioned preprint has not been vetted by any academic community and as such remains in the limbo of unreliable sources. Knowledge does not publish "nice new ideas", only those things that have been accepted by the wider scientific community (in this case). If

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The result of a dot product is not a vector, but a scalar. Your question may be restated as "why some definitions have been chosen as they are?". The general answer is that the most useful definition are generally the best choice. Here, the usefulness can be measured by the number of applications

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It's hard to imagine what squared units could correspond to in the real world in the case of some units. I guess that not every type of real-world vector/unit will have a corresponding real-world cross product? Perhaps a cross-product of velocity vectors could correspond to something that was

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again and compare it with the original vector. For this we use the inverse vector. Thanks to it vectors can be transferred through the sign =, as in arithmetic. But the choice of a sequence of vectors in a cross product changes the results of calculations, and is always chosen

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What do you think, could this be missleading? I also wanted to ask wether this information is relevant, since it is mentioned right at the beginning of the section and I think the key information is on how the quaternion product involves the cross product mentioned below:

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In general, if a vector is represented as the quaternion a1i + a2j + a3k, the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. The real part will be the negative of the dot product of the two

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The article states: 'The units of the cross-product are the product of the units of each vector.' And that makes sense, because if it weren't invariant to a change of units, then it would be absurd. I suggest watching this quick video on area-vectors:

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There are also a special class of vectors that have different properties to normal Euclidian vectors, in particular under reflection. These arise in many circumstances, not just from cross products, and have many names: polar vector, axial vector or

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Nothing is mixed here. The cross product does not agree with the addition of vectors (or the decomposition of the vector into a sum of vectors). If you do not like this proof, consider example №4. It is based only on the logic of the cross

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don't depend on the orientation of the space. They can even be defined in the absence of any orientation. They cannot therefore be pseudovectors. Moreover, if this were true, then all vectors would be pseudovectors. Is the position vector

3169: 7203:. There is no reason that the cross product of the first two component vectors should equal the third. The paper is terribly written and was not worth our time to consider, as it does not discuss these concepts in a sufficiently 5148: 2436: 8912:

In the text I spot non-linear algebra vectors with a superscript 'T' attached to them, looking very much like the notation for transpose of linear algebra vectors and matrices. This "notation" looks very much home made to me.

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induces an isomorphism between a vector space and its dual, that allows identifying vectors of the dual (pseudo-vectors) with normal vectors. Thus any vector can be considered either as a vector or as a pseudovector. So, in

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I can’t see how a vector of the dual space (that is a linear form) may be a pseudo-vector. Linear forms don't depend on the orientation, they aren't related to the cross product, they don’t transform in an unusual

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These sentences seem false for the above definition of the cross product is intrinsic. It doesn’t use any basis and therefore the handedness of any basis. What is used in the definition is the handedness of the

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Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector

7481:. Previously, this product was the only one, and they tried to write it into the theory of vectors. Now the cross product of four. And only now there is a choice. Of these four, we must choose a cross product 7050: 1706:

You are confusing two different things. In vector algebra the cross product, the one described in this article, is a product of two vectors with a vector result. It is well defined and has many applications.

651: 1402:{\displaystyle {\vec {a}}'_{z}={\vec {a}}_{x}\times {\vec {a}}_{y}=(a_{x}{\vec {i}}+0+0)\times (0+a_{y}{\vec {j}}+0)=0{\vec {i}}+0{\vec {j}}+(a_{x}a_{y}-0){\vec {k}}=0{\vec {i}}+0{\vec {j}}+a_{z}{\vec {k}}} 8725:

It's misleading to depict a cross-product as a one-dimensional arrow eminating from the origin, since it has squared units. It's an area-vector in the case of the cross-product of position or displacement

7193: 6474: 3144:{\displaystyle {\vec {b}}={\vec {c}}\times {\vec {d}}=(c_{y}d_{z}-c_{z}d_{y}){\vec {i}}+(c_{z}d_{x}-c_{x}d_{z}){\vec {j}}+(c_{x}d_{y}-c_{y}d_{x}){\vec {k}}=b_{x}{\vec {i}}+b_{y}{\vec {j}}+b_{z}{\vec {k}}} 2415:{\displaystyle {\vec {d}}={\vec {b}}\times {\vec {c}}=(b_{y}c_{z}-b_{z}c_{y}){\vec {i}}+(b_{z}c_{x}-b_{x}c_{z}){\vec {j}}+(b_{x}c_{y}-b_{y}c_{x}){\vec {k}}=d_{x}{\vec {i}}+d_{y}{\vec {j}}+d_{z}{\vec {k}}} 643: 6269: 8706:

The generalized cross product of n vectors in n+1-dimensional space satisfies the nth Filippov identity and so endows the space with a structure of n-Lie algebra. I have added this fact to the section

4978: 6679: 5248: 7588: 7479: 1989: 1930: 1871: 1800: 1678: 1037: 3722: 2084: 8924: 5768:{\displaystyle \quad b'_{x}={\dfrac {b_{x}}{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}};\quad b'_{y}={\dfrac {b_{y}}{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}}\quad b'_{z}={\dfrac {b_{z}}{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}}} 4884: 4827: 4770: 4716: 4662: 7292: 1478: 358:

The result of the cross product of two rectilinear vectors can not be a third rectilinear vector. I will prove it to you in examples from the article "Angular Vectors in the Theory of Vectors"

8102:... But coordinates and coordinate axes (invented by Descartes) predate the concept of quaternions. One has to check the usual notation (in the first halve of the 19th century) for the points 162: 7814:

introduces a new, visual notation for many vector operations such as the cross product. It might be a nice addition to this page. But I'd first like to have some feedback about this idea.

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product to simplify the algebraic approach to geometry for engineering students at the dawn of the twentieth century. The slight modification of the cross product with jk = −i occurs in

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Why is it so that a cross product of two vectors results in a resultant vector in a third dimension and a dot product of two vectors results in a resultant vector in the same plane?

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The unit quaternions correspond to 90° rotations in 4D space. I am unsure wether this quote in the section about quaternions says that they correspond to 180° rotations.

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Okay, if it's allright I would like to delete that part of the text, that will also make it more straightforward and readable, I think. Feel free to revert and discuss.

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for presenting your original ideas. There is no nontrivial vector-valued bilinear product of vectors in any dimension but 3 or 7. I sense you advocate the use of

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This is blatantly incorrect; the vector cross product is not even associative, and is anticommutative rather than commutative. The existence of a multitude of

2635:{\displaystyle {\vec {d}}=(0\cdot 0-0\cdot 10){\vec {i}}+(0\cdot 0-10\cdot 0){\vec {j}}+(10\cdot 10-0\cdot 0){\vec {k}}=0{\vec {i}}+0{\vec {j}}+100{\vec {k}}} 8710:, since the generalized cross product is defined there. Please, consider wether this is the right place to include it; perhaps it merits its own subsection? 46: 8981: 152: 7796:

We typically put new material at the end of the talk page (unlike other places on the web), so I moved your contribution here. I hope you don't mind. --

5027: 1991:. That is, knowing any two vectors, one could find a third vector. But this is not observed (observed only if one presents the vectors as unit vectors). 8012:

I think heard that Quaternions where used before Vectors where a thing, but I'll have a read into that. Is it really a 180° degree roation not a 90°?

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It is impossible to say more exactly in what side the vector projection is directed on an axis Oz. Now present our vector as sum of coordinate vectors.

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In other places I find vectors and tensors on the left hand side of an equation, but linear algebra matrices and vectors on the right hand side.

128: 210: 8976: 7055: 1725: 853:{\displaystyle {\vec {a}}(a_{x},a_{y},a_{z})=a_{x}{\vec {i}}+a_{y}{\vec {j}}+a_{z}{\vec {k}}={\vec {a}}_{x}+{\vec {a}}_{y}+{\vec {a}}_{z}} 8869:'s direction has a clear physical meaning and is defined using a cross product. Continue your discussion at the reference desk, please.-- 3731: 472: 8928: 6669:{\displaystyle c_{y}={\dfrac {b_{x}d_{xy}-b_{z}d_{yz}}{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}}={\dfrac {10\cdot 100-0\cdot 0}{10^{2}+0+0}}=10,} 1615:

3. specifies only a positive direction, although the direction is not precisely defined and can be both positive and negative in the O

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are mathematical vectors, neither axial nor polar. In mathematics, the cross-product of two vectors is a vector. It's that simple!

7008: 119: 80: 6871:{\displaystyle c_{z}={\dfrac {b_{y}d_{yz}-b_{x}d_{zx}}{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}}={\dfrac {0\cdot 0-10\cdot 0}{10^{2}+0+0}}=0} 5486:{\displaystyle =(b'_{z}d_{zx}-b'_{y}d_{xy}){\bar {i}}+(b'_{x}d_{xy}-b'_{z}d_{yz}){\bar {j}}+(b'_{y}d_{yz}-b'_{x}d_{zx}){\bar {k}}=} 3568:

Example: Proposed changes to the cross product. When at the cross product of two rectilinear vectors, a angular vector is created

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With the cross product of two rectilinear vectors, a new rectilinear vector is created which is perpendicular to both original

4929: 3636:. This vector is in the plane formed by rectilinear vectors. Its magnitude is equal to the area of some figure in this plane. 8736: 1718:

in 3D. You have just rediscovered pseudovectors but as you can see from our article it is already a well understood topic.--

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I have also tagged the part of the sentence beginning by "why", as I am pretty sure (I have no source under hand) that

7237: 343: 1412: 8793:, it seems to me that depicting a quantity that in fact has squared units as a simple arrow is far more conflating. 1743:

Example 4: One should pay attention to another problem - the logical inconsistency of the cross product of vectors

7484: 4357: 3592: 322: 8692: 3393: 2656: 8105: 7695:, will receive the status of uncoordinated cross product. And will be used only in geometric constructions.-- 3903: 3832: 8296: 8271:

This shows that the idea of the cross and dot product were known a long time before the quaternion product.

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a long-settled question; you're not going to be changing the consensus here. If you still have questions,

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is not consistent with the other properties of the rectilinear vectors, and has no right to exist.

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Remember, this is an article about math, and the notation must live up to that level of precision.

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Thats right, however the idea of the cross and dot product both stem from the Quaternion product.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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on Knowledge. If you would like to participate, please visit the project page, where you can join

8587:, do you want to delete this talk section, since the words you wanted deleted are now long-gone? 8574: 8276: 8232: 8194: 7999: 7849: 7801: 7409: 7204: 7138: 6973: 328: 111: 7666: 95: 74: 8372:

all be pseudovectors (if a basis of mixed vector types is disallowed, as it normally is) since

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2. is not consistent with the modulus of the specified vector, if one will find modulus using

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the base vectors are left out from the notation, the reader should be informed about this.

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https://sites.science.oregonstate.edu/~hadlekat/COURSES/ph213/gauss/images/areaVector.JPG

7362:, plain and simple, so your earlier notion of an "inverse vector" is complete nonsense.-- 4265: 8857:

Ultimately, the current contents of the article are what the community has deemed to be

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introduced the component form of both the dot and cross products in order to study the

7195:). You will not get that by simply relabeling the result as an "angular" vector as all 5225: 4315: 4290: 216: 7618: 8940: 8570: 8526: 8272: 8256: 8228: 7995: 7845: 7797: 7405: 7002: 4340: 362:. The result of the cross product of two rectilinear vectors is the angular vector. 8620:

In three dimensions, a pseudo-vector is associated with the curl of a polar vector

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I agree with the deletion, although the error in the sentence is not exactly what

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this preprint gets published and reviewed, then this may be looked at again. --

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See how a similar example is solved using the angular and inverse vectors.

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or with the cross-product of two polar vectors (Knowledge: Pseudovector)

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Who told you that it should be equal to the original vector? Not at all.

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is given by the left-hand rule and points in the opposite direction.

3820:{\displaystyle ={\bar {b}}\times {\bar {c}}=(b_{x}c_{y}-b_{y}c_{x})} 559:{\displaystyle |{\vec {a}}|={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}} 8457:

quantities such as the velocity or the magnetic field. The vectors

4496:- the projections of the angular vector onto the coordinate planes. 7052:) precludes any meaningful notion of an "inverse" in the sense of 5494:

Further, we use the projections of the inverse rectilinear vector

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For the solution, we need the cross products of the basis vectors

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Example 3: Let us assume that the modulus of rectilinear vector

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of an orthonormal, right-handed (Cartesian) coordinate frame

7994:) that correspond to rotations, but the "unit quaternions". 7188:{\displaystyle \mathbf {a} \times \mathbf {b} =\mathbf {c} } 7862:

Potentially missleading and/or irrelevant part of the text?

638:{\displaystyle a_{z}={\sqrt {15^{2}-10^{2}-10^{2}}}=\pm 5} 7732:, which however are outside the scope of this article.-- 4973:{\displaystyle {\bar {c}}={\dfrac {1}{\bar {b}}}\times } 1513:

1. is not consistent with the projection of the vector

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I came to wikipedia to find the most common practice.

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So by the above relationships, the unit basis vectors

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a long time before the introduction of the quaternion.

7583:{\displaystyle {\bar {a}}\times {\bar {b}}={\bar {c}}} 7474:{\displaystyle {\bar {a}}\times {\bar {b}}={\bar {c}}} 1984:{\displaystyle {\vec {c}}={\vec {d}}\times {\vec {b}}} 1925:{\displaystyle {\vec {b}}={\vec {c}}\times {\vec {d}}} 1866:{\displaystyle {\vec {d}}={\vec {b}}\times {\vec {c}}} 1795:{\displaystyle {\vec {i}}\times {\vec {j}}={\vec {k}}} 1673:{\displaystyle {\vec {i}}\times {\vec {j}}={\vec {k}}} 1032:{\displaystyle {\vec {i}}\times {\vec {j}}={\vec {k}}} 8702:

Generalized cross product and n-Lie algebra structure

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Hence, the end of the paragraph had to be modified.--

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while the square of a 180° rotation is the identity.

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and two of its projections onto the coordinate axes

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Cross product and handedness: deletion of few lines

7762:to see that the two "products" were taken from the 7531:consistent with vector theory. And the remaining 3 7329:Don't see how this contradicts what I said at all. 4765:{\displaystyle ={\bar {i}},\quad {\bar {j}}\times } 4711:{\displaystyle ={\bar {k}},\quad {\bar {k}}\times } 4657:{\displaystyle ={\bar {j}},\quad {\bar {j}}\times } 8664:Consistent hyphenation needed within the same page 8631: 8612: 8219: 8183: 7986: 7954: 7925: 7687: 7653: 7627: 7605: 7582: 7520: 7473: 7350: 7287:{\displaystyle {\bar {a}}\times {\bar {a}}^{-1}=0} 7286: 7187: 7149: 7127: 7099: 7044: 6923: 6870: 6668: 6463: 6258: 5767: 5485: 5237: 5212: 5177: 5142: 5014: 4972: 4917: 4878: 4821: 4764: 4710: 4656: 4602: 4566: 4531: 4488: 4426: 4346: 4327: 4302: 4277: 4252: 4203: 4154: 4087: 4066: 4031: 3996: 3961: 3890: 3819: 3716: 3628: 3579: 3503: 3453: 3367: 3143: 2805: 2766: 2716: 2634: 2414: 2078: 2000:For example, if there are two rectilinear vectors 1983: 1924: 1865: 1794: 1672: 1606: 1575: 1532: 1503: 1472: 1401: 1088: 1058: 1031: 972: 852: 637: 558: 459: 402: 7091: 8453:vectors, that is to say vectors which represent 1473:{\displaystyle a'_{z}=a_{x}a_{y}=10\cdot 10=100} 8967:Knowledge level-4 vital articles in Mathematics 8735:, and I suggest using a graphic like this one: 7422:In fact, there was no choice for cross-product 8761:No. You are conflating the cross product with 7358:with the cross product is not and cannot be a 3638:Suppose we are given two rectilinear vectors 3511:, and is not equal to the original vector. -- 2813:again and compare it with the original vector 337:This page has archives. Sections older than 8: 7521:{\displaystyle {\bar {a}}\times {\bar {b}}=} 4427:{\displaystyle d_{xy}=100,d_{yz}=0,d_{zx}=0} 3629:{\displaystyle ={\bar {b}}\times {\bar {c}}} 8733:https://www.youtube.com/watch?v=9GOoiaQq4GY 8472:So, we can say that the position vector is 8997:B-Class physics articles of Mid-importance 8908:Vector Algebra vs Linear Algebra Confusion 7815: 3454:{\displaystyle b_{x}=1000,b_{y}=0,b_{z}=0} 1681: 566:, the projection of the vector is equal 174: 69: 8625: 8606: 8556:must be considered as a pseudovector. If 8202: 8196: 8107: 7978: 7974: 7973: 7970: 7946: 7942: 7941: 7938: 7906: 7674: 7673: 7668: 7646: 7620: 7598: 7569: 7568: 7554: 7553: 7539: 7538: 7536: 7504: 7503: 7489: 7488: 7486: 7460: 7459: 7445: 7444: 7430: 7429: 7427: 7342: 7338: 7337: 7334: 7269: 7258: 7257: 7242: 7241: 7239: 7180: 7172: 7164: 7162: 7142: 7140: 7114: 7112: 7086: 7085: 7073: 7068: 7059: 7057: 7037: 7029: 7018: 7010: 6889: 6888: 6886: 6840: 6809: 6796: 6791: 6778: 6773: 6760: 6755: 6740: 6730: 6714: 6704: 6696: 6687: 6681: 6635: 6604: 6591: 6586: 6573: 6568: 6555: 6550: 6535: 6525: 6509: 6499: 6491: 6482: 6476: 6430: 6399: 6386: 6381: 6368: 6363: 6350: 6345: 6330: 6320: 6304: 6294: 6286: 6277: 6271: 6245: 6244: 6238: 6220: 6219: 6213: 6195: 6194: 6188: 6170: 6169: 6155: 6150: 6137: 6132: 6119: 6114: 6099: 6089: 6073: 6063: 6055: 6037: 6036: 6022: 6017: 6004: 5999: 5986: 5981: 5966: 5956: 5940: 5930: 5922: 5904: 5903: 5889: 5884: 5871: 5866: 5853: 5848: 5833: 5823: 5807: 5797: 5789: 5780: 5755: 5750: 5737: 5732: 5719: 5714: 5703: 5696: 5684: 5668: 5663: 5650: 5645: 5632: 5627: 5616: 5609: 5597: 5578: 5573: 5560: 5555: 5542: 5537: 5526: 5519: 5507: 5499: 5469: 5468: 5456: 5443: 5427: 5414: 5393: 5392: 5380: 5367: 5351: 5338: 5317: 5316: 5304: 5291: 5275: 5262: 5250: 5227: 5201: 5192: 5166: 5157: 5131: 5107: 5106: 5097: 5079: 5078: 5069: 5051: 5050: 5041: 5029: 4993: 4992: 4987: 4954: 4948: 4934: 4933: 4931: 4902: 4901: 4893: 4862: 4861: 4845: 4844: 4836: 4805: 4804: 4788: 4787: 4779: 4748: 4747: 4731: 4730: 4725: 4694: 4693: 4677: 4676: 4671: 4640: 4639: 4623: 4622: 4617: 4586: 4585: 4583: 4551: 4550: 4548: 4506: 4477: 4461: 4445: 4439: 4409: 4387: 4365: 4359: 4342: 4317: 4292: 4267: 4218: 4169: 4120: 4081: 4055: 4046: 4020: 4011: 3985: 3976: 3950: 3940: 3927: 3917: 3905: 3879: 3869: 3856: 3846: 3834: 3808: 3798: 3785: 3775: 3754: 3753: 3739: 3738: 3733: 3682: 3681: 3646: 3645: 3643: 3615: 3614: 3600: 3599: 3594: 3573: 3469: 3468: 3466: 3439: 3420: 3401: 3395: 3354: 3353: 3336: 3335: 3318: 3317: 3300: 3299: 3258: 3257: 3216: 3215: 3174: 3173: 3171: 3130: 3129: 3123: 3105: 3104: 3098: 3080: 3079: 3073: 3055: 3054: 3045: 3035: 3022: 3012: 2991: 2990: 2981: 2971: 2958: 2948: 2927: 2926: 2917: 2907: 2894: 2884: 2863: 2862: 2848: 2847: 2833: 2832: 2830: 2792: 2791: 2789: 2732: 2731: 2729: 2717:{\displaystyle d_{x}=0,d_{y}=0,d_{z}=100} 2702: 2683: 2664: 2658: 2621: 2620: 2603: 2602: 2585: 2584: 2567: 2566: 2525: 2524: 2483: 2482: 2441: 2440: 2438: 2401: 2400: 2394: 2376: 2375: 2369: 2351: 2350: 2344: 2326: 2325: 2316: 2306: 2293: 2283: 2262: 2261: 2252: 2242: 2229: 2219: 2198: 2197: 2188: 2178: 2165: 2155: 2134: 2133: 2119: 2118: 2104: 2103: 2101: 2044: 2043: 2008: 2007: 2005: 1970: 1969: 1955: 1954: 1940: 1939: 1937: 1911: 1910: 1896: 1895: 1881: 1880: 1878: 1852: 1851: 1837: 1836: 1822: 1821: 1819: 1781: 1780: 1766: 1765: 1751: 1750: 1748: 1659: 1658: 1644: 1643: 1629: 1628: 1626: 1595: 1589: 1564: 1551: 1545: 1524: 1518: 1492: 1486: 1446: 1436: 1420: 1414: 1388: 1387: 1381: 1363: 1362: 1345: 1344: 1327: 1326: 1311: 1301: 1280: 1279: 1262: 1261: 1235: 1234: 1228: 1186: 1185: 1179: 1163: 1152: 1151: 1141: 1130: 1129: 1116: 1105: 1104: 1101: 1077: 1071: 1050: 1044: 1018: 1017: 1003: 1002: 988: 987: 985: 964: 951: 940: 939: 929: 916: 905: 904: 894: 881: 870: 869: 866: 844: 833: 832: 822: 811: 810: 800: 789: 788: 773: 772: 766: 748: 747: 741: 723: 722: 716: 700: 687: 674: 656: 655: 653: 618: 605: 592: 586: 577: 571: 548: 543: 530: 525: 512: 507: 501: 493: 482: 481: 476: 474: 448: 418: 417: 415: 389: 378: 377: 372: 370: 8184:{\displaystyle (1,0,0),(0,1,0),(0,0,1).} 7135:, there are multiple orthogonal vectors 3962:{\displaystyle +(b_{z}c_{x}-b_{x}c_{z})} 3891:{\displaystyle +(b_{y}c_{z}-b_{z}c_{y})} 8957:Knowledge vital articles in Mathematics 8925:2A02:1660:69AA:E500:BEAE:C5FF:FE26:2574 8865:or simply would not be due weight. The 7965:Also, it is not the "unit vectors" (of 1511:projection found by the second method: 460:{\displaystyle {\vec {a}}(10,10,a_{z})} 347:when more than 10 sections are present. 176: 71: 30: 8564:are considered as pseudovectors, then 8260: 8191:About the rotation, you are right, as 6931:, and is equal to the original vector. 4155:{\displaystyle =(10\cdot 10-0\cdot 0)} 8972:B-Class vital articles in Mathematics 8311:Definition: deletion of two sentences 7933:where used for the standard basis of 6949:is a force, and a rectilinear vector 4253:{\displaystyle +(0\cdot 0-10\cdot 0)} 4204:{\displaystyle +(0\cdot 0-0\cdot 10)} 467:. One must find its third projection. 7: 8765:, of which the cross product is the 6941:is the torque. A rectilinear vector 6933:Now imagine that the angular vector 4489:{\displaystyle d_{xy},d_{yz},d_{zx}} 3504:{\displaystyle {\vec {b}}(1000,0,0)} 222:This article is within the scope of 117:This article is within the scope of 8569:sentence is definitively an error. 5015:{\displaystyle ={\bar {b'}}\times } 4543:Now we find the rectilinear vector 2767:{\displaystyle {\vec {d}}(0,0,100)} 360:https://doi.org/10.5539/jmr.v9n5p71 60:It is of interest to the following 8861:. Any of what you're proposing is 6924:{\displaystyle {\vec {c}}(0,10,0)} 4918:{\displaystyle =-{\bar {k}}\quad } 25: 8982:Mid-priority mathematics articles 8255:Not sure. The history section of 7871:(zero real part) with unit norms. 4603:{\displaystyle {\bar {i}}\times } 341:may be automatically archived by 137:Knowledge:WikiProject Mathematics 8952:Knowledge level-4 vital articles 8552:are considered as vectors, then 7987:{\displaystyle \mathbb {R} ^{3}} 7955:{\displaystyle \mathbb {R} ^{3}} 7351:{\displaystyle \mathbb {R} ^{3}} 7181: 7173: 7165: 7143: 7121: 7118: 7115: 7088: 7069: 7060: 7038: 7030: 7019: 4567:{\displaystyle {\bar {c}}\quad } 1714:. They are also identified with 1576:{\displaystyle a_{z}\neq a'_{z}} 1481:Conclusions: Rectilinear vector 290: 209: 199: 178: 140:Template:WikiProject Mathematics 104: 94: 73: 40: 31: 8992:Mid-importance physics articles 403:{\displaystyle |{\vec {a}}|=15} 262:This article has been rated as 157:This article has been rated as 8962:B-Class level-4 vital articles 8175: 8157: 8151: 8133: 8127: 8109: 7679: 7574: 7559: 7544: 7509: 7494: 7465: 7450: 7435: 7263: 7247: 7128:{\displaystyle \mathbf {a,c} } 7023: 7012: 6918: 6900: 6894: 6250: 6225: 6200: 6175: 6042: 5909: 5474: 5465: 5407: 5398: 5389: 5331: 5322: 5313: 5255: 5229: 5124: 5118: 5112: 5084: 5056: 5034: 5003: 4959: 4939: 4907: 4867: 4850: 4810: 4793: 4753: 4736: 4699: 4682: 4645: 4628: 4591: 4556: 4526: 4508: 4247: 4223: 4198: 4174: 4149: 4125: 3956: 3910: 3885: 3839: 3814: 3768: 3759: 3744: 3711: 3693: 3687: 3675: 3657: 3651: 3620: 3605: 3521:11:39, 18 September 2017 (UTC) 3498: 3480: 3474: 3359: 3341: 3323: 3305: 3296: 3272: 3263: 3254: 3230: 3221: 3212: 3188: 3179: 3135: 3110: 3085: 3060: 3051: 3005: 2996: 2987: 2941: 2932: 2923: 2877: 2868: 2853: 2838: 2797: 2761: 2743: 2737: 2626: 2608: 2590: 2572: 2563: 2539: 2530: 2521: 2497: 2488: 2479: 2455: 2446: 2406: 2381: 2356: 2331: 2322: 2276: 2267: 2258: 2212: 2203: 2194: 2148: 2139: 2124: 2109: 2073: 2055: 2049: 2037: 2019: 2013: 1975: 1960: 1945: 1916: 1901: 1886: 1857: 1842: 1827: 1786: 1771: 1756: 1730:15:00, 17 September 2017 (UTC) 1700:13:54, 17 September 2017 (UTC) 1664: 1649: 1634: 1393: 1368: 1350: 1332: 1323: 1294: 1285: 1267: 1252: 1240: 1215: 1209: 1191: 1172: 1157: 1135: 1110: 1023: 1008: 993: 945: 910: 875: 838: 816: 794: 778: 753: 728: 706: 667: 661: 494: 487: 477: 454: 429: 423: 390: 383: 373: 1: 8892:08:32, 15 February 2024 (UTC) 8877:08:21, 15 February 2024 (UTC) 8846:08:18, 15 February 2024 (UTC) 8827:06:33, 15 February 2024 (UTC) 8803:06:18, 15 February 2024 (UTC) 8785:05:41, 15 February 2024 (UTC) 8749:05:39, 15 February 2024 (UTC) 8720:10:22, 15 November 2022 (UTC) 8627: 8608: 8597:08:57, 15 February 2024 (UTC) 8525:wrote. The truth is that the 7854:22:35, 13 November 2019 (UTC) 7834:20:20, 13 November 2019 (UTC) 7806:22:35, 13 November 2019 (UTC) 5678: 5591: 5501: 4913: 4859: 4802: 4745: 4691: 4637: 4562: 4083: 3575: 469:Solution: Using the formula 242:Knowledge:WikiProject Physics 236:and see a list of open tasks. 131:and see a list of open tasks. 8977:B-Class mathematics articles 8315:I delete the two sentences 8220:{\displaystyle i^{2}\neq 1,} 7759:A History of Vector Analysis 7404:that use these definitions. 7150:{\displaystyle \mathbf {b} } 6986:09:40, 14 October 2017 (UTC) 6967:08:34, 14 October 2017 (UTC) 4498:That is, the angular vector 3551:15:21, 12 October 2017 (UTC) 3461:, that is there is a vector 354:Cross product does not exist 245:Template:WikiProject Physics 8305:20:37, 16 August 2020 (UTC) 8281:21:01, 16 August 2020 (UTC) 8251:20:37, 16 August 2020 (UTC) 8237:19:53, 16 August 2020 (UTC) 8098:19:43, 16 August 2020 (UTC) 8022:19:36, 16 August 2020 (UTC) 8004:11:12, 16 August 2020 (UTC) 7896:10:49, 16 August 2020 (UTC) 7792:Alternative visual notation 7688:{\displaystyle ={\bar {c}}} 7107:(and even given orthogonal 4111:- the basic angular vectors 1540:found by the first method, 18:Talk:Cross product/Comments 9013: 8687:is giving me what I want. 8504:without any contradiction. 8488:and the magnetic field is 8034:Found it: On the site for 6881:that is there is a vector 2806:{\displaystyle {\vec {b}}} 268:project's importance scale 8933:20:09, 29 June 2024 (UTC) 7780:22:56, 23 June 2019 (UTC) 7740:10:46, 23 June 2019 (UTC) 7705:10:42, 19 June 2019 (UTC) 7414:09:36, 11 June 2019 (UTC) 7398:07:13, 11 June 2019 (UTC) 7370:10:39, 23 June 2019 (UTC) 7304:10:33, 23 June 2019 (UTC) 4532:{\displaystyle (100,0,0)} 1873:. It would be logical if 261: 194: 156: 89: 68: 8987:B-Class physics articles 8697:09:44, 23 May 2022 (UTC) 8656:15:05, 9 June 2021 (UTC) 8579:17:22, 8 June 2021 (UTC) 8516:15:10, 8 June 2021 (UTC) 8445:"Axial" and "polar" are 8344:15:56, 6 June 2021 (UTC) 7215:18:28, 4 July 2018 (UTC) 1621:Thus, the cross product 163:project's priority scale 8632:{\displaystyle \qquad } 8613:{\displaystyle \qquad } 7654:{\displaystyle \times } 7606:{\displaystyle \times } 5213:{\displaystyle +d_{zx}} 5178:{\displaystyle +d_{yz}} 4067:{\displaystyle +d_{zx}} 4032:{\displaystyle +d_{yz}} 3997:{\displaystyle =d_{xy}} 2784:Now we find the vector 120:WikiProject Mathematics 8947:B-Class vital articles 8633: 8614: 8221: 8185: 8080:) and a 3-dimensional 8047:William Rowan Hamilton 7988: 7956: 7927: 7689: 7655: 7629: 7607: 7584: 7522: 7475: 7352: 7288: 7189: 7151: 7129: 7101: 7046: 6925: 6872: 6670: 6465: 6260: 5769: 5487: 5239: 5214: 5179: 5144: 5016: 4974: 4919: 4880: 4823: 4766: 4712: 4658: 4604: 4568: 4533: 4490: 4428: 4348: 4329: 4304: 4279: 4254: 4205: 4156: 4089: 4088:{\displaystyle \quad } 4068: 4033: 3998: 3963: 3892: 3821: 3718: 3630: 3581: 3580:{\displaystyle \quad } 3505: 3455: 3369: 3145: 2807: 2768: 2718: 2636: 2416: 2080: 1985: 1926: 1867: 1796: 1674: 1608: 1607:{\displaystyle a'_{z}} 1577: 1534: 1505: 1504:{\displaystyle a'_{z}} 1474: 1403: 1090: 1089:{\displaystyle a'_{z}} 1060: 1033: 980:and using the formula 974: 854: 639: 560: 461: 404: 344:Lowercase sigmabot III 8634: 8615: 8263:Joseph-Louis Lagrange 8222: 8186: 7989: 7957: 7928: 7926:{\displaystyle i,j,k} 7690: 7656: 7630: 7608: 7585: 7523: 7476: 7353: 7289: 7190: 7152: 7130: 7102: 7047: 6926: 6873: 6671: 6466: 6261: 5770: 5488: 5240: 5215: 5180: 5145: 5017: 4975: 4920: 4881: 4824: 4767: 4713: 4659: 4605: 4569: 4534: 4491: 4429: 4349: 4330: 4305: 4280: 4255: 4206: 4157: 4090: 4069: 4034: 3999: 3964: 3893: 3822: 3719: 3631: 3582: 3506: 3456: 3370: 3146: 2808: 2769: 2719: 2637: 2417: 2081: 1986: 1927: 1868: 1797: 1675: 1609: 1578: 1535: 1533:{\displaystyle a_{z}} 1506: 1475: 1404: 1091: 1061: 1059:{\displaystyle a_{z}} 1034: 975: 855: 640: 561: 462: 405: 47:level-4 vital article 8624: 8605: 8269:in three dimensions. 8195: 8106: 7969: 7937: 7905: 7667: 7645: 7619: 7597: 7535: 7485: 7426: 7333: 7238: 7161: 7139: 7111: 7056: 7009: 6885: 6680: 6475: 6270: 5779: 5498: 5249: 5226: 5191: 5156: 5028: 4986: 4930: 4892: 4835: 4778: 4724: 4670: 4616: 4582: 4547: 4539:is in the plane OXY 4505: 4438: 4358: 4341: 4316: 4291: 4278:{\displaystyle =100} 4266: 4217: 4168: 4119: 4080: 4045: 4010: 3975: 3904: 3833: 3732: 3642: 3593: 3572: 3465: 3394: 3170: 2829: 2788: 2728: 2724:that is, the vector 2657: 2437: 2100: 2004: 1936: 1877: 1818: 1747: 1625: 1588: 1544: 1517: 1485: 1413: 1100: 1070: 1043: 984: 865: 652: 570: 473: 414: 369: 143:mathematics articles 8708:Multilinear algebra 8671:appears 159 times. 7234:You have a mistake 7199:of dimension 3 are 6801: 6783: 6765: 6596: 6578: 6560: 6391: 6373: 6355: 6160: 6142: 6124: 6027: 6009: 5991: 5894: 5876: 5858: 5760: 5742: 5724: 5692: 5673: 5655: 5637: 5605: 5583: 5565: 5547: 5515: 5451: 5422: 5375: 5346: 5299: 5270: 5105: 5077: 5049: 1603: 1572: 1500: 1428: 1124: 1085: 553: 535: 517: 225:WikiProject Physics 8773:that appears in a 8629: 8628: 8610: 8609: 8442:a pseudovector? 8297:TheFibonacciEffect 8243:TheFibonacciEffect 8217: 8181: 8090:TheFibonacciEffect 8045:was introduced by 8014:TheFibonacciEffect 7984: 7952: 7923: 7888:TheFibonacciEffect 7724:This talk page is 7685: 7651: 7625: 7603: 7580: 7518: 7471: 7348: 7284: 7185: 7147: 7125: 7097: 7042: 6921: 6868: 6860: 6804: 6787: 6769: 6751: 6666: 6655: 6599: 6582: 6564: 6546: 6461: 6450: 6394: 6377: 6359: 6341: 6256: 6163: 6146: 6128: 6110: 6030: 6013: 5995: 5977: 5897: 5880: 5862: 5844: 5765: 5763: 5746: 5728: 5710: 5680: 5679: 5676: 5659: 5641: 5623: 5593: 5592: 5586: 5569: 5551: 5533: 5503: 5502: 5483: 5439: 5410: 5363: 5334: 5287: 5258: 5238:{\displaystyle )=} 5235: 5210: 5175: 5140: 5093: 5065: 5037: 5012: 4970: 4965: 4915: 4914: 4876: 4860: 4819: 4803: 4762: 4746: 4708: 4692: 4654: 4638: 4600: 4564: 4563: 4529: 4486: 4424: 4344: 4328:{\displaystyle +0} 4325: 4303:{\displaystyle +0} 4300: 4275: 4250: 4201: 4152: 4085: 4084: 4064: 4029: 3994: 3959: 3888: 3817: 3714: 3626: 3577: 3576: 3501: 3451: 3365: 3141: 2803: 2764: 2714: 2632: 2412: 2076: 1981: 1922: 1863: 1792: 1670: 1604: 1591: 1573: 1560: 1530: 1501: 1488: 1470: 1416: 1399: 1103: 1086: 1073: 1056: 1029: 970: 850: 635: 556: 539: 521: 503: 457: 400: 112:Mathematics portal 56:content assessment 8677:appears 5 times. 8053:, which is a sum 7836: 7820:comment added by 7682: 7628:{\displaystyle =} 7577: 7562: 7547: 7512: 7497: 7468: 7453: 7438: 7266: 7250: 7094: 6897: 6859: 6803: 6654: 6598: 6449: 6393: 6253: 6228: 6203: 6178: 6162: 6045: 6029: 5912: 5896: 5762: 5675: 5585: 5477: 5401: 5325: 5115: 5087: 5059: 5006: 4964: 4962: 4942: 4910: 4870: 4853: 4813: 4796: 4756: 4739: 4702: 4685: 4648: 4631: 4594: 4559: 3762: 3747: 3690: 3654: 3623: 3608: 3477: 3362: 3344: 3326: 3308: 3266: 3224: 3182: 3138: 3113: 3088: 3063: 2999: 2935: 2871: 2856: 2841: 2800: 2740: 2629: 2611: 2593: 2575: 2533: 2491: 2449: 2409: 2384: 2359: 2334: 2270: 2206: 2142: 2127: 2112: 2052: 2016: 1978: 1963: 1948: 1919: 1904: 1889: 1860: 1845: 1830: 1789: 1774: 1759: 1723: 1702: 1686:comment added by 1667: 1652: 1637: 1396: 1371: 1353: 1335: 1288: 1270: 1243: 1194: 1160: 1138: 1113: 1026: 1011: 996: 948: 913: 878: 841: 819: 797: 781: 756: 731: 664: 624: 554: 490: 426: 386: 351: 350: 282: 281: 278: 277: 274: 273: 173: 172: 169: 168: 16:(Redirected from 9004: 8856: 8813: 8775:surface integral 8760: 8638: 8636: 8635: 8630: 8619: 8617: 8616: 8611: 8543: 8413: 8399: 8385: 8226: 8224: 8223: 8218: 8207: 8206: 8190: 8188: 8187: 8182: 8075: 8066: 8036:Euclidean vector 7993: 7991: 7990: 7985: 7983: 7982: 7977: 7961: 7959: 7958: 7953: 7951: 7950: 7945: 7932: 7930: 7929: 7924: 7846:Bill Cherowitzo 7798:Bill Cherowitzo 7723: 7694: 7692: 7691: 7686: 7684: 7683: 7675: 7663: 7660: 7658: 7657: 7652: 7641: 7637: 7634: 7632: 7631: 7626: 7615: 7612: 7610: 7609: 7604: 7593: 7589: 7587: 7586: 7581: 7579: 7578: 7570: 7564: 7563: 7555: 7549: 7548: 7540: 7530: 7527: 7525: 7524: 7519: 7514: 7513: 7505: 7499: 7498: 7490: 7480: 7478: 7477: 7472: 7470: 7469: 7461: 7455: 7454: 7446: 7440: 7439: 7431: 7360:division algebra 7357: 7355: 7354: 7349: 7347: 7346: 7341: 7328: 7293: 7291: 7290: 7285: 7277: 7276: 7268: 7267: 7259: 7252: 7251: 7243: 7233: 7194: 7192: 7191: 7186: 7184: 7176: 7168: 7156: 7154: 7153: 7148: 7146: 7134: 7132: 7131: 7126: 7124: 7106: 7104: 7103: 7098: 7096: 7095: 7087: 7081: 7080: 7072: 7063: 7051: 7049: 7048: 7043: 7041: 7033: 7022: 7000: 6954: 6946: 6938: 6930: 6928: 6927: 6922: 6899: 6898: 6890: 6877: 6875: 6874: 6869: 6861: 6858: 6845: 6844: 6834: 6811: 6805: 6802: 6800: 6795: 6782: 6777: 6764: 6759: 6749: 6748: 6747: 6735: 6734: 6722: 6721: 6709: 6708: 6698: 6692: 6691: 6675: 6673: 6672: 6667: 6656: 6653: 6640: 6639: 6629: 6606: 6600: 6597: 6595: 6590: 6577: 6572: 6559: 6554: 6544: 6543: 6542: 6530: 6529: 6517: 6516: 6504: 6503: 6493: 6487: 6486: 6470: 6468: 6467: 6462: 6451: 6448: 6435: 6434: 6424: 6401: 6395: 6392: 6390: 6385: 6372: 6367: 6354: 6349: 6339: 6338: 6337: 6325: 6324: 6312: 6311: 6299: 6298: 6288: 6282: 6281: 6265: 6263: 6262: 6257: 6255: 6254: 6246: 6243: 6242: 6230: 6229: 6221: 6218: 6217: 6205: 6204: 6196: 6193: 6192: 6180: 6179: 6171: 6168: 6164: 6161: 6159: 6154: 6141: 6136: 6123: 6118: 6108: 6107: 6106: 6094: 6093: 6081: 6080: 6068: 6067: 6057: 6047: 6046: 6038: 6035: 6031: 6028: 6026: 6021: 6008: 6003: 5990: 5985: 5975: 5974: 5973: 5961: 5960: 5948: 5947: 5935: 5934: 5924: 5914: 5913: 5905: 5902: 5898: 5895: 5893: 5888: 5875: 5870: 5857: 5852: 5842: 5841: 5840: 5828: 5827: 5815: 5814: 5802: 5801: 5791: 5774: 5772: 5771: 5766: 5764: 5761: 5759: 5754: 5741: 5736: 5723: 5718: 5708: 5707: 5698: 5688: 5677: 5674: 5672: 5667: 5654: 5649: 5636: 5631: 5621: 5620: 5611: 5601: 5587: 5584: 5582: 5577: 5564: 5559: 5546: 5541: 5531: 5530: 5521: 5511: 5492: 5490: 5489: 5484: 5479: 5478: 5470: 5464: 5463: 5447: 5435: 5434: 5418: 5403: 5402: 5394: 5388: 5387: 5371: 5359: 5358: 5342: 5327: 5326: 5318: 5312: 5311: 5295: 5283: 5282: 5266: 5244: 5242: 5241: 5236: 5222: 5219: 5217: 5216: 5211: 5209: 5208: 5187: 5184: 5182: 5181: 5176: 5174: 5173: 5152: 5149: 5147: 5146: 5141: 5139: 5138: 5117: 5116: 5108: 5101: 5089: 5088: 5080: 5073: 5061: 5060: 5052: 5045: 5024: 5021: 5019: 5018: 5013: 5008: 5007: 5002: 4994: 4982: 4979: 4977: 4976: 4971: 4966: 4963: 4955: 4950: 4944: 4943: 4935: 4924: 4922: 4921: 4916: 4912: 4911: 4903: 4888: 4885: 4883: 4882: 4877: 4872: 4871: 4863: 4855: 4854: 4846: 4831: 4828: 4826: 4825: 4820: 4815: 4814: 4806: 4798: 4797: 4789: 4774: 4771: 4769: 4768: 4763: 4758: 4757: 4749: 4741: 4740: 4732: 4720: 4717: 4715: 4714: 4709: 4704: 4703: 4695: 4687: 4686: 4678: 4666: 4663: 4661: 4660: 4655: 4650: 4649: 4641: 4633: 4632: 4624: 4612: 4609: 4607: 4606: 4601: 4596: 4595: 4587: 4573: 4571: 4570: 4565: 4561: 4560: 4552: 4538: 4536: 4535: 4530: 4501: 4495: 4493: 4492: 4487: 4485: 4484: 4469: 4468: 4453: 4452: 4433: 4431: 4430: 4425: 4417: 4416: 4395: 4394: 4373: 4372: 4353: 4351: 4350: 4347:{\displaystyle } 4345: 4337: 4334: 4332: 4331: 4326: 4312: 4309: 4307: 4306: 4301: 4287: 4284: 4282: 4281: 4276: 4262: 4259: 4257: 4256: 4251: 4213: 4210: 4208: 4207: 4202: 4164: 4161: 4159: 4158: 4153: 4115: 4108: 4104: 4100: 4094: 4092: 4091: 4086: 4076: 4073: 4071: 4070: 4065: 4063: 4062: 4041: 4038: 4036: 4035: 4030: 4028: 4027: 4006: 4003: 4001: 4000: 3995: 3993: 3992: 3971: 3968: 3966: 3965: 3960: 3955: 3954: 3945: 3944: 3932: 3931: 3922: 3921: 3900: 3897: 3895: 3894: 3889: 3884: 3883: 3874: 3873: 3861: 3860: 3851: 3850: 3829: 3826: 3824: 3823: 3818: 3813: 3812: 3803: 3802: 3790: 3789: 3780: 3779: 3764: 3763: 3755: 3749: 3748: 3740: 3728: 3723: 3721: 3720: 3715: 3692: 3691: 3683: 3656: 3655: 3647: 3635: 3633: 3632: 3627: 3625: 3624: 3616: 3610: 3609: 3601: 3589: 3586: 3584: 3583: 3578: 3510: 3508: 3507: 3502: 3479: 3478: 3470: 3460: 3458: 3457: 3452: 3444: 3443: 3425: 3424: 3406: 3405: 3374: 3372: 3371: 3366: 3364: 3363: 3355: 3346: 3345: 3337: 3328: 3327: 3319: 3310: 3309: 3301: 3268: 3267: 3259: 3226: 3225: 3217: 3184: 3183: 3175: 3150: 3148: 3147: 3142: 3140: 3139: 3131: 3128: 3127: 3115: 3114: 3106: 3103: 3102: 3090: 3089: 3081: 3078: 3077: 3065: 3064: 3056: 3050: 3049: 3040: 3039: 3027: 3026: 3017: 3016: 3001: 3000: 2992: 2986: 2985: 2976: 2975: 2963: 2962: 2953: 2952: 2937: 2936: 2928: 2922: 2921: 2912: 2911: 2899: 2898: 2889: 2888: 2873: 2872: 2864: 2858: 2857: 2849: 2843: 2842: 2834: 2812: 2810: 2809: 2804: 2802: 2801: 2793: 2773: 2771: 2770: 2765: 2742: 2741: 2733: 2723: 2721: 2720: 2715: 2707: 2706: 2688: 2687: 2669: 2668: 2641: 2639: 2638: 2633: 2631: 2630: 2622: 2613: 2612: 2604: 2595: 2594: 2586: 2577: 2576: 2568: 2535: 2534: 2526: 2493: 2492: 2484: 2451: 2450: 2442: 2421: 2419: 2418: 2413: 2411: 2410: 2402: 2399: 2398: 2386: 2385: 2377: 2374: 2373: 2361: 2360: 2352: 2349: 2348: 2336: 2335: 2327: 2321: 2320: 2311: 2310: 2298: 2297: 2288: 2287: 2272: 2271: 2263: 2257: 2256: 2247: 2246: 2234: 2233: 2224: 2223: 2208: 2207: 2199: 2193: 2192: 2183: 2182: 2170: 2169: 2160: 2159: 2144: 2143: 2135: 2129: 2128: 2120: 2114: 2113: 2105: 2085: 2083: 2082: 2077: 2054: 2053: 2045: 2018: 2017: 2009: 1990: 1988: 1987: 1982: 1980: 1979: 1971: 1965: 1964: 1956: 1950: 1949: 1941: 1931: 1929: 1928: 1923: 1921: 1920: 1912: 1906: 1905: 1897: 1891: 1890: 1882: 1872: 1870: 1869: 1864: 1862: 1861: 1853: 1847: 1846: 1838: 1832: 1831: 1823: 1801: 1799: 1798: 1793: 1791: 1790: 1782: 1776: 1775: 1767: 1761: 1760: 1752: 1719: 1679: 1677: 1676: 1671: 1669: 1668: 1660: 1654: 1653: 1645: 1639: 1638: 1630: 1613: 1611: 1610: 1605: 1599: 1582: 1580: 1579: 1574: 1568: 1556: 1555: 1539: 1537: 1536: 1531: 1529: 1528: 1510: 1508: 1507: 1502: 1496: 1479: 1477: 1476: 1471: 1451: 1450: 1441: 1440: 1424: 1408: 1406: 1405: 1400: 1398: 1397: 1389: 1386: 1385: 1373: 1372: 1364: 1355: 1354: 1346: 1337: 1336: 1328: 1316: 1315: 1306: 1305: 1290: 1289: 1281: 1272: 1271: 1263: 1245: 1244: 1236: 1233: 1232: 1196: 1195: 1187: 1184: 1183: 1168: 1167: 1162: 1161: 1153: 1146: 1145: 1140: 1139: 1131: 1120: 1115: 1114: 1106: 1095: 1093: 1092: 1087: 1081: 1065: 1063: 1062: 1057: 1055: 1054: 1038: 1036: 1035: 1030: 1028: 1027: 1019: 1013: 1012: 1004: 998: 997: 989: 979: 977: 976: 971: 969: 968: 956: 955: 950: 949: 941: 934: 933: 921: 920: 915: 914: 906: 899: 898: 886: 885: 880: 879: 871: 859: 857: 856: 851: 849: 848: 843: 842: 834: 827: 826: 821: 820: 812: 805: 804: 799: 798: 790: 783: 782: 774: 771: 770: 758: 757: 749: 746: 745: 733: 732: 724: 721: 720: 705: 704: 692: 691: 679: 678: 666: 665: 657: 644: 642: 641: 636: 625: 623: 622: 610: 609: 597: 596: 587: 582: 581: 565: 563: 562: 557: 555: 552: 547: 534: 529: 516: 511: 502: 497: 492: 491: 483: 480: 466: 464: 463: 458: 453: 452: 428: 427: 419: 409: 407: 406: 401: 393: 388: 387: 379: 376: 346: 330: 294: 286: 250: 249: 248:physics articles 246: 243: 240: 219: 214: 213: 203: 196: 195: 190: 182: 175: 145: 144: 141: 138: 135: 114: 109: 108: 98: 91: 90: 85: 77: 70: 53: 44: 43: 36: 35: 27: 21: 9012: 9011: 9007: 9006: 9005: 9003: 9002: 9001: 8937: 8936: 8910: 8867:Poynting vector 8850: 8807: 8771:surface element 8754: 8728: 8704: 8689:Universemaster1 8666: 8640: 8622: 8621: 8603: 8602: 8531: 8449:qualifiers for 8415: 8401: 8387: 8373: 8351: 8324: 8313: 8198: 8193: 8192: 8104: 8103: 8071: 8054: 7972: 7967: 7966: 7940: 7935: 7934: 7903: 7902: 7864: 7794: 7717: 7665: 7664: 7661: 7643: 7642: 7639: 7635: 7617: 7616: 7613: 7595: 7594: 7591: 7533: 7532: 7528: 7483: 7482: 7424: 7423: 7386: 7336: 7331: 7330: 7322: 7256: 7236: 7235: 7227: 7159: 7158: 7137: 7136: 7109: 7108: 7067: 7054: 7053: 7007: 7006: 6994: 6978:Boris Tsirelson 6952: 6944: 6936: 6883: 6882: 6836: 6835: 6812: 6750: 6736: 6726: 6710: 6700: 6699: 6683: 6678: 6677: 6631: 6630: 6607: 6545: 6531: 6521: 6505: 6495: 6494: 6478: 6473: 6472: 6426: 6425: 6402: 6340: 6326: 6316: 6300: 6290: 6289: 6273: 6268: 6267: 6234: 6209: 6184: 6109: 6095: 6085: 6069: 6059: 6058: 6051: 5976: 5962: 5952: 5936: 5926: 5925: 5918: 5843: 5829: 5819: 5803: 5793: 5792: 5785: 5777: 5776: 5709: 5699: 5622: 5612: 5532: 5522: 5496: 5495: 5452: 5423: 5376: 5347: 5300: 5271: 5247: 5246: 5224: 5223: 5220: 5197: 5189: 5188: 5185: 5162: 5154: 5153: 5150: 5127: 5026: 5025: 5022: 4995: 4984: 4983: 4980: 4928: 4927: 4890: 4889: 4886: 4833: 4832: 4829: 4776: 4775: 4772: 4722: 4721: 4718: 4668: 4667: 4664: 4614: 4613: 4610: 4580: 4579: 4545: 4544: 4503: 4502: 4499: 4473: 4457: 4441: 4436: 4435: 4405: 4383: 4361: 4356: 4355: 4339: 4338: 4335: 4314: 4313: 4310: 4289: 4288: 4285: 4264: 4263: 4260: 4215: 4214: 4211: 4166: 4165: 4162: 4117: 4116: 4113: 4106: 4102: 4098: 4078: 4077: 4074: 4051: 4043: 4042: 4039: 4016: 4008: 4007: 4004: 3981: 3973: 3972: 3969: 3946: 3936: 3923: 3913: 3902: 3901: 3898: 3875: 3865: 3852: 3842: 3831: 3830: 3827: 3804: 3794: 3781: 3771: 3730: 3729: 3726: 3640: 3639: 3591: 3590: 3587: 3570: 3569: 3543:Boris Tsirelson 3463: 3462: 3435: 3416: 3397: 3392: 3391: 3168: 3167: 3119: 3094: 3069: 3041: 3031: 3018: 3008: 2977: 2967: 2954: 2944: 2913: 2903: 2890: 2880: 2827: 2826: 2786: 2785: 2726: 2725: 2698: 2679: 2660: 2655: 2654: 2435: 2434: 2390: 2365: 2340: 2312: 2302: 2289: 2279: 2248: 2238: 2225: 2215: 2184: 2174: 2161: 2151: 2098: 2097: 2002: 2001: 1934: 1933: 1875: 1874: 1816: 1815: 1745: 1744: 1728: 1623: 1622: 1618: 1586: 1585: 1547: 1542: 1541: 1520: 1515: 1514: 1483: 1482: 1442: 1432: 1411: 1410: 1377: 1307: 1297: 1224: 1175: 1150: 1128: 1098: 1097: 1068: 1067: 1046: 1041: 1040: 982: 981: 960: 938: 925: 903: 890: 868: 863: 862: 831: 809: 787: 762: 737: 712: 696: 683: 670: 650: 649: 614: 601: 588: 573: 568: 567: 471: 470: 444: 412: 411: 367: 366: 356: 342: 331: 325: 299: 247: 244: 241: 238: 237: 215: 208: 188: 142: 139: 136: 133: 132: 110: 103: 83: 54:on Knowledge's 51: 41: 23: 22: 15: 12: 11: 5: 9010: 9008: 9000: 8999: 8994: 8989: 8984: 8979: 8974: 8969: 8964: 8959: 8954: 8949: 8939: 8938: 8909: 8906: 8905: 8904: 8903: 8902: 8901: 8900: 8899: 8898: 8897: 8896: 8895: 8894: 8819:is that way.-- 8727: 8723: 8703: 8700: 8665: 8662: 8661: 8660: 8659: 8658: 8601: 8600: 8599: 8581: 8505: 8355: 8350: 8347: 8317: 8312: 8309: 8308: 8307: 8293: 8292: 8291: 8290: 8289: 8288: 8287: 8286: 8285: 8284: 8283: 8216: 8213: 8210: 8205: 8201: 8180: 8177: 8174: 8171: 8168: 8165: 8162: 8159: 8156: 8153: 8150: 8147: 8144: 8141: 8138: 8135: 8132: 8129: 8126: 8123: 8120: 8117: 8114: 8111: 8087: 8086: 8085: 8038:it says that: 8027: 8026: 8025: 8024: 8007: 8006: 7981: 7976: 7963: 7949: 7944: 7922: 7919: 7916: 7913: 7910: 7885: 7883: 7882: 7873: 7872: 7863: 7860: 7859: 7858: 7857: 7856: 7838: 7837: 7793: 7790: 7789: 7788: 7787: 7786: 7785: 7784: 7783: 7782: 7747: 7746: 7745: 7744: 7743: 7742: 7710: 7709: 7708: 7707: 7681: 7678: 7672: 7650: 7624: 7602: 7576: 7573: 7567: 7561: 7558: 7552: 7546: 7543: 7517: 7511: 7508: 7502: 7496: 7493: 7467: 7464: 7458: 7452: 7449: 7443: 7437: 7434: 7417: 7416: 7385: 7382: 7381: 7380: 7379: 7378: 7377: 7376: 7375: 7374: 7373: 7372: 7345: 7340: 7311: 7310: 7309: 7308: 7307: 7306: 7283: 7280: 7275: 7272: 7265: 7262: 7255: 7249: 7246: 7220: 7219: 7218: 7217: 7183: 7179: 7175: 7171: 7167: 7145: 7123: 7120: 7117: 7093: 7090: 7084: 7079: 7076: 7071: 7066: 7062: 7040: 7036: 7032: 7028: 7025: 7021: 7017: 7014: 6989: 6988: 6932: 6920: 6917: 6914: 6911: 6908: 6905: 6902: 6896: 6893: 6878: 6867: 6864: 6857: 6854: 6851: 6848: 6843: 6839: 6833: 6830: 6827: 6824: 6821: 6818: 6815: 6808: 6799: 6794: 6790: 6786: 6781: 6776: 6772: 6768: 6763: 6758: 6754: 6746: 6743: 6739: 6733: 6729: 6725: 6720: 6717: 6713: 6707: 6703: 6695: 6690: 6686: 6676: 6665: 6662: 6659: 6652: 6649: 6646: 6643: 6638: 6634: 6628: 6625: 6622: 6619: 6616: 6613: 6610: 6603: 6594: 6589: 6585: 6581: 6576: 6571: 6567: 6563: 6558: 6553: 6549: 6541: 6538: 6534: 6528: 6524: 6520: 6515: 6512: 6508: 6502: 6498: 6490: 6485: 6481: 6471: 6460: 6457: 6454: 6447: 6444: 6441: 6438: 6433: 6429: 6423: 6420: 6417: 6414: 6411: 6408: 6405: 6398: 6389: 6384: 6380: 6376: 6371: 6366: 6362: 6358: 6353: 6348: 6344: 6336: 6333: 6329: 6323: 6319: 6315: 6310: 6307: 6303: 6297: 6293: 6285: 6280: 6276: 6266: 6252: 6249: 6241: 6237: 6233: 6227: 6224: 6216: 6212: 6208: 6202: 6199: 6191: 6187: 6183: 6177: 6174: 6167: 6158: 6153: 6149: 6145: 6140: 6135: 6131: 6127: 6122: 6117: 6113: 6105: 6102: 6098: 6092: 6088: 6084: 6079: 6076: 6072: 6066: 6062: 6054: 6050: 6044: 6041: 6034: 6025: 6020: 6016: 6012: 6007: 6002: 5998: 5994: 5989: 5984: 5980: 5972: 5969: 5965: 5959: 5955: 5951: 5946: 5943: 5939: 5933: 5929: 5921: 5917: 5911: 5908: 5901: 5892: 5887: 5883: 5879: 5874: 5869: 5865: 5861: 5856: 5851: 5847: 5839: 5836: 5832: 5826: 5822: 5818: 5813: 5810: 5806: 5800: 5796: 5788: 5784: 5775: 5758: 5753: 5749: 5745: 5740: 5735: 5731: 5727: 5722: 5717: 5713: 5706: 5702: 5695: 5691: 5687: 5683: 5671: 5666: 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2164: 2158: 2154: 2150: 2147: 2141: 2138: 2132: 2126: 2123: 2117: 2111: 2108: 2090: 2089: 2088: 2087: 2075: 2072: 2069: 2066: 2063: 2060: 2057: 2051: 2048: 2042: 2039: 2036: 2033: 2030: 2027: 2024: 2021: 2015: 2012: 1995: 1994: 1993: 1992: 1977: 1974: 1968: 1962: 1959: 1953: 1947: 1944: 1918: 1915: 1909: 1903: 1900: 1894: 1888: 1885: 1859: 1856: 1850: 1844: 1841: 1835: 1829: 1826: 1812: 1806: 1805: 1804: 1803: 1788: 1785: 1779: 1773: 1770: 1764: 1758: 1755: 1741: 1740: 1733: 1732: 1724: 1721:JohnBlackburne 1707: 1666: 1663: 1657: 1651: 1648: 1642: 1636: 1633: 1620: 1616: 1614: 1602: 1598: 1594: 1583: 1571: 1567: 1563: 1559: 1554: 1550: 1527: 1523: 1512: 1499: 1495: 1491: 1480: 1469: 1466: 1463: 1460: 1457: 1454: 1449: 1445: 1439: 1435: 1431: 1427: 1423: 1419: 1409: 1395: 1392: 1384: 1380: 1376: 1370: 1367: 1361: 1358: 1352: 1349: 1343: 1340: 1334: 1331: 1325: 1322: 1319: 1314: 1310: 1304: 1300: 1296: 1293: 1287: 1284: 1278: 1275: 1269: 1266: 1260: 1257: 1254: 1251: 1248: 1242: 1239: 1231: 1227: 1223: 1220: 1217: 1214: 1211: 1208: 1205: 1202: 1199: 1193: 1190: 1182: 1178: 1174: 1171: 1166: 1159: 1156: 1149: 1144: 1137: 1134: 1127: 1123: 1119: 1112: 1109: 1096: 1084: 1080: 1076: 1053: 1049: 1025: 1022: 1016: 1010: 1007: 1001: 995: 992: 967: 963: 959: 954: 947: 944: 937: 932: 928: 924: 919: 912: 909: 902: 897: 893: 889: 884: 877: 874: 861: 847: 840: 837: 830: 825: 818: 815: 808: 803: 796: 793: 786: 780: 777: 769: 765: 761: 755: 752: 744: 740: 736: 730: 727: 719: 715: 711: 708: 703: 699: 695: 690: 686: 682: 677: 673: 669: 663: 660: 648: 634: 631: 628: 621: 617: 613: 608: 604: 600: 595: 591: 585: 580: 576: 551: 546: 542: 538: 533: 528: 524: 520: 515: 510: 506: 500: 496: 489: 486: 479: 468: 456: 451: 447: 443: 440: 437: 434: 431: 425: 422: 399: 396: 392: 385: 382: 375: 355: 352: 349: 348: 336: 333: 332: 327: 323: 321: 318: 317: 301: 300: 295: 289: 280: 279: 276: 275: 272: 271: 264:Mid-importance 260: 254: 253: 251: 234:the discussion 221: 220: 217:Physics portal 204: 192: 191: 189:Mid‑importance 183: 171: 170: 167: 166: 155: 149: 148: 146: 129:the discussion 116: 115: 99: 87: 86: 78: 66: 65: 59: 37: 24: 14: 13: 10: 9: 6: 4: 3: 2: 9009: 8998: 8995: 8993: 8990: 8988: 8985: 8983: 8980: 8978: 8975: 8973: 8970: 8968: 8965: 8963: 8960: 8958: 8955: 8953: 8950: 8948: 8945: 8944: 8942: 8935: 8934: 8930: 8926: 8921: 8919: 8914: 8907: 8893: 8889: 8885: 8880: 8879: 8878: 8875: 8872: 8868: 8864: 8860: 8854: 8849: 8848: 8847: 8843: 8839: 8834: 8830: 8829: 8828: 8825: 8822: 8818: 8811: 8806: 8805: 8804: 8800: 8796: 8792: 8788: 8787: 8786: 8783: 8780: 8776: 8772: 8768: 8764: 8758: 8753: 8752: 8751: 8750: 8746: 8742: 8738: 8734: 8722: 8721: 8717: 8713: 8709: 8701: 8699: 8698: 8694: 8690: 8686: 8685:google ngrams 8683:I'm not sure 8681: 8678: 8676: 8675:Cross-product 8672: 8670: 8669:Cross product 8663: 8657: 8653: 8649: 8644: 8643: 8642: 8641: 8598: 8594: 8590: 8586: 8582: 8580: 8576: 8572: 8567: 8563: 8559: 8555: 8551: 8547: 8542: 8538: 8534: 8528: 8527:inner product 8524: 8520: 8519: 8518: 8517: 8513: 8509: 8503: 8499: 8495: 8491: 8487: 8483: 8479: 8475: 8470: 8468: 8464: 8460: 8456: 8452: 8448: 8443: 8441: 8437: 8433: 8428: 8424: 8420: 8412: 8408: 8404: 8398: 8394: 8390: 8384: 8380: 8376: 8371: 8367: 8363: 8359: 8354: 8348: 8346: 8345: 8341: 8337: 8332: 8330: 8322: 8316: 8310: 8306: 8302: 8298: 8294: 8282: 8278: 8274: 8270: 8268: 8264: 8258: 8257:Cross product 8254: 8253: 8252: 8248: 8244: 8240: 8239: 8238: 8234: 8230: 8214: 8211: 8208: 8203: 8199: 8178: 8172: 8169: 8166: 8163: 8160: 8154: 8148: 8145: 8142: 8139: 8136: 8130: 8124: 8121: 8118: 8115: 8112: 8101: 8100: 8099: 8095: 8091: 8088: 8083: 8079: 8076:(also called 8074: 8070: 8065: 8061: 8057: 8052: 8049:as part of a 8048: 8044: 8040: 8039: 8037: 8033: 8032: 8031: 8030: 8029: 8028: 8023: 8019: 8015: 8011: 8010: 8009: 8008: 8005: 8001: 7997: 7979: 7964: 7947: 7920: 7917: 7914: 7911: 7908: 7900: 7899: 7898: 7897: 7893: 7889: 7886:Kind regards 7879: 7878: 7877: 7869: 7868: 7867: 7861: 7855: 7851: 7847: 7842: 7841: 7840: 7839: 7835: 7831: 7827: 7823: 7819: 7813: 7810: 7809: 7808: 7807: 7803: 7799: 7791: 7781: 7777: 7773: 7769: 7768:coquaternions 7765: 7761: 7760: 7755: 7754: 7753: 7752: 7751: 7750: 7749: 7748: 7741: 7738: 7735: 7731: 7727: 7721: 7716: 7715: 7714: 7713: 7712: 7711: 7706: 7702: 7698: 7676: 7670: 7648: 7622: 7600: 7571: 7565: 7556: 7550: 7541: 7515: 7506: 7500: 7491: 7462: 7456: 7447: 7441: 7432: 7421: 7420: 7419: 7418: 7415: 7411: 7407: 7402: 7401: 7400: 7399: 7395: 7391: 7383: 7371: 7368: 7365: 7361: 7343: 7326: 7321: 7320: 7319: 7318: 7317: 7316: 7315: 7314: 7313: 7312: 7305: 7301: 7297: 7281: 7278: 7273: 7270: 7260: 7253: 7244: 7231: 7226: 7225: 7224: 7223: 7222: 7221: 7216: 7213: 7210: 7206: 7202: 7198: 7197:vector spaces 7177: 7169: 7082: 7077: 7074: 7064: 7034: 7026: 7015: 7004: 7003:zero divisors 6998: 6993: 6992: 6991: 6990: 6987: 6983: 6979: 6975: 6971: 6970: 6969: 6968: 6964: 6960: 6956: 6955: 6948: 6947: 6940: 6939: 6915: 6912: 6909: 6906: 6903: 6891: 6879: 6865: 6862: 6855: 6852: 6849: 6846: 6841: 6837: 6831: 6828: 6825: 6822: 6819: 6816: 6813: 6806: 6797: 6792: 6788: 6784: 6779: 6774: 6770: 6766: 6761: 6756: 6752: 6744: 6741: 6737: 6731: 6727: 6723: 6718: 6715: 6711: 6705: 6701: 6693: 6688: 6684: 6663: 6660: 6657: 6650: 6647: 6644: 6641: 6636: 6632: 6626: 6623: 6620: 6617: 6614: 6611: 6608: 6601: 6592: 6587: 6583: 6579: 6574: 6569: 6565: 6561: 6556: 6551: 6547: 6539: 6536: 6532: 6526: 6522: 6518: 6513: 6510: 6506: 6500: 6496: 6488: 6483: 6479: 6458: 6455: 6452: 6445: 6442: 6439: 6436: 6431: 6427: 6421: 6418: 6415: 6412: 6409: 6406: 6403: 6396: 6387: 6382: 6378: 6374: 6369: 6364: 6360: 6356: 6351: 6346: 6342: 6334: 6331: 6327: 6321: 6317: 6313: 6308: 6305: 6301: 6295: 6291: 6283: 6278: 6274: 6247: 6239: 6235: 6231: 6222: 6214: 6210: 6206: 6197: 6189: 6185: 6181: 6172: 6165: 6156: 6151: 6147: 6143: 6138: 6133: 6129: 6125: 6120: 6115: 6111: 6103: 6100: 6096: 6090: 6086: 6082: 6077: 6074: 6070: 6064: 6060: 6052: 6048: 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1809: 1808: 1807: 1783: 1777: 1768: 1762: 1753: 1742: 1737: 1736: 1735: 1734: 1731: 1727: 1722: 1717: 1713: 1708: 1705: 1704: 1703: 1701: 1697: 1693: 1689: 1685: 1661: 1655: 1646: 1640: 1631: 1600: 1596: 1592: 1569: 1565: 1561: 1557: 1552: 1548: 1525: 1521: 1497: 1493: 1489: 1467: 1464: 1461: 1458: 1455: 1452: 1447: 1443: 1437: 1433: 1429: 1425: 1421: 1417: 1390: 1382: 1378: 1374: 1365: 1359: 1356: 1347: 1341: 1338: 1329: 1320: 1317: 1312: 1308: 1302: 1298: 1291: 1282: 1276: 1273: 1264: 1258: 1255: 1249: 1246: 1237: 1229: 1225: 1221: 1218: 1212: 1206: 1203: 1200: 1197: 1188: 1180: 1176: 1169: 1164: 1154: 1147: 1142: 1132: 1125: 1121: 1117: 1107: 1082: 1078: 1074: 1051: 1047: 1020: 1014: 1005: 999: 990: 965: 961: 957: 952: 942: 935: 930: 926: 922: 917: 907: 900: 895: 891: 887: 882: 872: 845: 835: 828: 823: 813: 806: 801: 791: 784: 775: 767: 763: 759: 750: 742: 738: 734: 725: 717: 713: 709: 701: 697: 693: 688: 684: 680: 675: 671: 658: 645: 632: 629: 626: 619: 615: 611: 606: 602: 598: 593: 589: 583: 578: 574: 549: 544: 540: 536: 531: 526: 522: 518: 513: 508: 504: 498: 484: 449: 445: 441: 438: 435: 432: 420: 397: 394: 380: 363: 361: 353: 345: 340: 335: 334: 320: 319: 316: 315: 311: 307: 303: 302: 298: 293: 288: 287: 284: 269: 265: 259: 256: 255: 252: 235: 231: 227: 226: 218: 212: 207: 205: 202: 198: 197: 193: 187: 184: 181: 177: 164: 160: 154: 151: 150: 147: 130: 126: 122: 121: 113: 107: 102: 100: 97: 93: 92: 88: 82: 79: 76: 72: 67: 63: 57: 49: 48: 38: 34: 29: 28: 19: 8922: 8917: 8915: 8911: 8859:WP:DUEWEIGHT 8853:MatthewMunro 8729: 8707: 8705: 8682: 8679: 8674: 8673: 8668: 8667: 8648:KharanteDeux 8585:KharanteDeux 8565: 8561: 8557: 8553: 8549: 8545: 8540: 8536: 8532: 8523:KharanteDeux 8508:KharanteDeux 8501: 8497: 8493: 8489: 8485: 8481: 8477: 8473: 8471: 8466: 8462: 8458: 8454: 8450: 8446: 8444: 8439: 8435: 8431: 8426: 8422: 8418: 8417:The vectors 8416: 8410: 8406: 8402: 8396: 8392: 8388: 8382: 8378: 8374: 8369: 8365: 8361: 8357: 8352: 8336:KharanteDeux 8333: 8328: 8325: 8320: 8314: 8081: 8077: 8072: 8063: 8059: 8055: 8042: 7884: 7874: 7865: 7816:— Preceding 7795: 7757: 7726:WP:NOTAFORUM 7390:ABCDSocrates 7387: 6972:Irrelevant; 6951: 6950: 6943: 6942: 6935: 6934: 6880: 4542: 4097: 4096: 3567: 3563: 1712:pseudovector 1682:— Preceding 1066:, naming it 646: 364: 357: 338: 304: 296: 283: 263: 223: 159:Mid-priority 158: 118: 84:Mid‑priority 62:WikiProjects 45: 8884:MathewMunro 8871:Jasper Deng 8838:MathewMunro 8833:Jasper Deng 8821:Jasper Deng 8810:MathewMunro 8795:MathewMunro 8791:Jasper Deng 8779:Jasper Deng 8757:MathewMunro 8741:MathewMunro 8589:MathewMunro 8267:tetrahedron 8259:begin with 8069:Real number 7734:Jasper Deng 7364:Jasper Deng 7230:Jasper Deng 7209:Jasper Deng 6974:WP:NOTFORUM 134:Mathematics 125:mathematics 81:Mathematics 8941:Categories 8767:Hodge dual 8051:quaternion 7822:Juliacubed 7812:This paper 7764:quaternion 7384:Dimensions 7201:isomorphic 7157:such that 8763:bivectors 8712:Jose Brox 8353:I delete 8261:In 1773, 8041:The term 7730:bivectors 7207:manner.-- 1716:bivectors 50:is rated 8726:vectors. 8571:D.Lazard 8455:physical 8451:physical 8447:physical 8273:D.Lazard 8229:D.Lazard 7996:D.Lazard 7881:vectors. 7830:contribs 7818:unsigned 7406:D.Lazard 7205:rigorous 4434:, where 4095:, where 1814:vectors 1739:product. 1696:contribs 1684:unsigned 339:365 days 297:Archives 8817:WP:RDMA 7772:Rgdboer 860:, then 266:on the 239:Physics 230:Physics 186:Physics 161:on the 52:B-class 8874:(talk) 8824:(talk) 8782:(talk) 8646:way.-- 8082:vector 8078:scalar 8043:vector 7737:(talk) 7720:Ujin-X 7697:Ujin-X 7367:(talk) 7325:Ujin-X 7296:Ujin-X 7212:(talk) 7005:(e.g. 6997:Ujin-X 6959:Ujin-X 3724:, then 3513:Ujin-X 2086:, then 1688:Ujin-X 58:scale. 8863:WP:OR 8544:, if 8329:space 8067:of a 7756:Read 4925:,then 1726:deeds 1619:axis. 39:This 8929:talk 8888:talk 8842:talk 8799:talk 8745:talk 8716:talk 8693:talk 8652:talk 8593:talk 8575:talk 8560:and 8548:and 8512:talk 8465:and 8425:and 8400:and 8370:must 8364:and 8340:talk 8331:. 8301:talk 8277:talk 8247:talk 8233:talk 8094:talk 8018:talk 8000:talk 7892:talk 7850:talk 7826:talk 7802:talk 7776:talk 7701:talk 7410:talk 7394:talk 7300:talk 6982:talk 6963:talk 3547:talk 3517:talk 3484:1000 3411:1000 3315:1000 1932:and 1692:talk 8777:.-- 8500:– 5 8496:+ 3 8492:= 2 8484:– 5 8480:+ 3 8476:= 2 8438:– 5 8434:+ 3 8414:. 6615:100 6422:100 4512:100 4378:100 4273:100 3252:100 3198:100 2759:100 2712:100 2618:100 1468:100 258:Mid 153:Mid 8943:: 8931:) 8918:If 8890:) 8844:) 8801:) 8747:) 8718:) 8695:) 8654:) 8595:) 8577:) 8539:= 8535:× 8514:) 8506:-- 8461:, 8421:, 8409:= 8405:× 8395:= 8391:× 8386:, 8381:= 8377:× 8360:, 8342:) 8303:) 8279:) 8249:) 8235:) 8209:≠ 8096:) 8062:+ 8058:= 8020:) 8002:) 7894:) 7852:) 7832:) 7828:• 7804:) 7778:) 7703:) 7680:¯ 7662:b̑ 7649:× 7640:ȃ 7638:, 7636:c̑ 7614:b̑ 7601:× 7592:ȃ 7590:, 7575:¯ 7560:¯ 7551:× 7545:¯ 7529:c̑ 7510:¯ 7501:× 7495:¯ 7466:¯ 7451:¯ 7442:× 7436:¯ 7412:) 7396:) 7302:) 7294:-- 7271:− 7264:¯ 7254:× 7248:¯ 7170:× 7092:^ 7075:− 7065:× 7027:× 6984:) 6976:. 6965:) 6953:b̄ 6945:c̄ 6937:d̑ 6910:10 6895:→ 6838:10 6829:⋅ 6826:10 6823:− 6817:⋅ 6724:− 6661:10 6633:10 6624:⋅ 6618:− 6612:⋅ 6609:10 6519:− 6428:10 6419:⋅ 6413:− 6407:⋅ 6314:− 6251:¯ 6226:¯ 6201:¯ 6176:¯ 6083:− 6043:¯ 5950:− 5910:¯ 5817:− 5475:¯ 5437:− 5399:¯ 5361:− 5323:¯ 5285:− 5221:n̑ 5186:m̑ 5151:l̑ 5122:× 5113:¯ 5085:¯ 5057:¯ 5023:d̑ 5010:× 5004:¯ 4981:d̑ 4968:× 4960:¯ 4940:¯ 4908:¯ 4899:− 4887:n̑ 4874:× 4868:¯ 4851:¯ 4842:− 4830:m̑ 4817:× 4811:¯ 4794:¯ 4785:− 4773:l̑ 4760:× 4754:¯ 4737:¯ 4719:n̑ 4706:× 4700:¯ 4683:¯ 4665:m̑ 4652:× 4646:¯ 4629:¯ 4611:l̑ 4598:× 4592:¯ 4557:¯ 4500:d̑ 4336:n̑ 4311:m̑ 4286:l̑ 4261:n̑ 4242:⋅ 4239:10 4236:− 4230:⋅ 4212:m̑ 4196:10 4193:⋅ 4187:− 4181:⋅ 4163:l̑ 4144:⋅ 4138:− 4135:10 4132:⋅ 4129:10 4114:d̑ 4107:n̑ 4103:m̑ 4099:l̑ 4075:n̑ 4040:m̑ 4005:l̑ 3970:n̑ 3934:− 3899:m̑ 3863:− 3828:l̑ 3792:− 3760:¯ 3751:× 3745:¯ 3727:d̑ 3703:10 3688:¯ 3661:10 3652:¯ 3621:¯ 3612:× 3606:¯ 3588:d̑ 3549:) 3519:) 3475:→ 3360:→ 3342:→ 3324:→ 3306:→ 3291:⋅ 3288:10 3285:− 3279:⋅ 3264:→ 3249:⋅ 3243:− 3237:⋅ 3222:→ 3207:⋅ 3201:− 3195:⋅ 3192:10 3180:→ 3136:→ 3111:→ 3086:→ 3061:→ 3029:− 2997:→ 2965:− 2933:→ 2901:− 2869:→ 2860:× 2854:→ 2839:→ 2798:→ 2738:→ 2627:→ 2609:→ 2591:→ 2573:→ 2558:⋅ 2552:− 2549:10 2546:⋅ 2543:10 2531:→ 2516:⋅ 2513:10 2510:− 2504:⋅ 2489:→ 2477:10 2474:⋅ 2468:− 2462:⋅ 2447:→ 2407:→ 2382:→ 2357:→ 2332:→ 2300:− 2268:→ 2236:− 2204:→ 2172:− 2140:→ 2131:× 2125:→ 2110:→ 2065:10 2050:→ 2023:10 2014:→ 1976:→ 1967:× 1961:→ 1946:→ 1917:→ 1908:× 1902:→ 1887:→ 1858:→ 1849:× 1843:→ 1828:→ 1787:→ 1772:→ 1763:× 1757:→ 1698:) 1694:• 1665:→ 1650:→ 1641:× 1635:→ 1558:≠ 1462:10 1459:⋅ 1456:10 1394:→ 1369:→ 1351:→ 1333:→ 1318:− 1286:→ 1268:→ 1241:→ 1213:× 1192:→ 1158:→ 1148:× 1136:→ 1111:→ 1024:→ 1009:→ 1000:× 994:→ 946:→ 911:→ 876:→ 839:→ 817:→ 795:→ 779:→ 754:→ 729:→ 662:→ 630:± 616:10 612:− 603:10 599:− 590:15 488:→ 439:10 433:10 424:→ 398:15 384:→ 312:, 308:, 8927:( 8886:( 8855:: 8851:@ 8840:( 8831:@ 8812:: 8808:@ 8797:( 8789:@ 8759:: 8755:@ 8743:( 8714:( 8691:( 8650:( 8591:( 8583:@ 8573:( 8566:k 8562:j 8558:i 8554:k 8550:j 8546:i 8541:k 8537:j 8533:i 8510:( 8502:k 8498:j 8494:i 8490:B 8486:k 8482:j 8478:i 8474:r 8467:k 8463:j 8459:i 8440:k 8436:j 8432:i 8430:2 8427:k 8423:j 8419:i 8411:j 8407:i 8403:k 8397:i 8393:k 8389:j 8383:k 8379:j 8375:i 8366:k 8362:j 8358:i 8338:( 8321:n 8299:( 8275:( 8245:( 8231:( 8215:, 8212:1 8204:2 8200:i 8179:. 8176:) 8173:1 8170:, 8167:0 8164:, 8161:0 8158:( 8155:, 8152:) 8149:0 8146:, 8143:1 8140:, 8137:0 8134:( 8131:, 8128:) 8125:0 8122:, 8119:0 8116:, 8113:1 8110:( 8092:( 8084:. 8073:s 8064:v 8060:s 8056:q 8016:( 7998:( 7980:3 7975:R 7948:3 7943:R 7921:k 7918:, 7915:j 7912:, 7909:i 7890:( 7848:( 7824:( 7800:( 7774:( 7722:: 7718:@ 7699:( 7677:c 7671:= 7623:= 7572:c 7566:= 7557:b 7542:a 7516:= 7507:b 7492:a 7463:c 7457:= 7448:b 7433:a 7408:( 7392:( 7344:3 7339:R 7327:: 7323:@ 7298:( 7282:0 7279:= 7274:1 7261:a 7245:a 7232:: 7228:@ 7182:c 7178:= 7174:b 7166:a 7144:b 7122:c 7119:, 7116:a 7089:a 7083:= 7078:1 7070:a 7061:a 7039:0 7035:= 7031:a 7024:) 7020:a 7016:k 7013:( 6999:: 6995:@ 6980:( 6961:( 6919:) 6916:0 6913:, 6907:, 6904:0 6901:( 6892:c 6866:0 6863:= 6856:0 6853:+ 6850:0 6847:+ 6842:2 6832:0 6820:0 6814:0 6807:= 6798:2 6793:z 6789:b 6785:+ 6780:2 6775:y 6771:b 6767:+ 6762:2 6757:x 6753:b 6745:x 6742:z 6738:d 6732:x 6728:b 6719:z 6716:y 6712:d 6706:y 6702:b 6694:= 6689:z 6685:c 6664:, 6658:= 6651:0 6648:+ 6645:0 6642:+ 6637:2 6627:0 6621:0 6602:= 6593:2 6588:z 6584:b 6580:+ 6575:2 6570:y 6566:b 6562:+ 6557:2 6552:x 6548:b 6540:z 6537:y 6533:d 6527:z 6523:b 6514:y 6511:x 6507:d 6501:x 6497:b 6489:= 6484:y 6480:c 6459:, 6456:0 6453:= 6446:0 6443:+ 6440:0 6437:+ 6432:2 6416:0 6410:0 6404:0 6397:= 6388:2 6383:z 6379:b 6375:+ 6370:2 6365:y 6361:b 6357:+ 6352:2 6347:x 6343:b 6335:y 6332:x 6328:d 6322:y 6318:b 6309:x 6306:z 6302:d 6296:z 6292:b 6284:= 6279:x 6275:c 6248:k 6240:z 6236:c 6232:+ 6223:j 6215:y 6211:c 6207:+ 6198:i 6190:x 6186:c 6182:= 6173:k 6166:) 6157:2 6152:z 6148:b 6144:+ 6139:2 6134:y 6130:b 6126:+ 6121:2 6116:x 6112:b 6104:x 6101:z 6097:d 6091:x 6087:b 6078:z 6075:y 6071:d 6065:y 6061:b 6053:( 6049:+ 6040:j 6033:) 6024:2 6019:z 6015:b 6011:+ 6006:2 6001:y 5997:b 5993:+ 5988:2 5983:x 5979:b 5971:z 5968:y 5964:d 5958:z 5954:b 5945:y 5942:x 5938:d 5932:x 5928:b 5920:( 5916:+ 5907:i 5900:) 5891:2 5886:z 5882:b 5878:+ 5873:2 5868:y 5864:b 5860:+ 5855:2 5850:x 5846:b 5838:y 5835:x 5831:d 5825:y 5821:b 5812:x 5809:z 5805:d 5799:z 5795:b 5787:( 5783:= 5757:2 5752:z 5748:b 5744:+ 5739:2 5734:y 5730:b 5726:+ 5721:2 5716:x 5712:b 5705:z 5701:b 5694:= 5690:′ 5686:z 5682:b 5670:2 5665:z 5661:b 5657:+ 5652:2 5647:y 5643:b 5639:+ 5634:2 5629:x 5625:b 5618:y 5614:b 5607:= 5603:′ 5599:y 5595:b 5589:; 5580:2 5575:z 5571:b 5567:+ 5562:2 5557:y 5553:b 5549:+ 5544:2 5539:x 5535:b 5528:x 5524:b 5517:= 5513:′ 5509:x 5505:b 5481:= 5472:k 5466:) 5461:x 5458:z 5454:d 5449:′ 5445:x 5441:b 5432:z 5429:y 5425:d 5420:′ 5416:y 5412:b 5408:( 5405:+ 5396:j 5390:) 5385:z 5382:y 5378:d 5373:′ 5369:z 5365:b 5356:y 5353:x 5349:d 5344:′ 5340:x 5336:b 5332:( 5329:+ 5320:i 5314:) 5309:y 5306:x 5302:d 5297:′ 5293:y 5289:b 5280:x 5277:z 5273:d 5268:′ 5264:z 5260:b 5256:( 5253:= 5233:= 5230:) 5206:x 5203:z 5199:d 5195:+ 5171:z 5168:y 5164:d 5160:+ 5136:y 5133:x 5129:d 5125:( 5119:) 5110:k 5103:′ 5099:z 5095:b 5091:+ 5082:j 5075:′ 5071:y 5067:b 5063:+ 5054:i 5047:′ 5043:x 5039:b 5035:( 5032:= 5000:′ 4997:b 4990:= 4957:b 4952:1 4946:= 4937:c 4905:k 4896:= 4865:i 4857:, 4848:j 4839:= 4808:k 4800:, 4791:i 4782:= 4751:j 4743:, 4734:i 4728:= 4697:k 4689:, 4680:k 4674:= 4643:j 4635:, 4626:j 4620:= 4589:i 4554:c 4527:) 4524:0 4521:, 4518:0 4515:, 4509:( 4482:x 4479:z 4475:d 4471:, 4466:z 4463:y 4459:d 4455:, 4450:y 4447:x 4443:d 4422:0 4419:= 4414:x 4411:z 4407:d 4403:, 4400:0 4397:= 4392:z 4389:y 4385:d 4381:, 4375:= 4370:y 4367:x 4363:d 4323:0 4320:+ 4298:0 4295:+ 4270:= 4248:) 4245:0 4233:0 4227:0 4224:( 4221:+ 4199:) 4190:0 4184:0 4178:0 4175:( 4172:+ 4150:) 4147:0 4141:0 4126:( 4123:= 4105:, 4101:, 4060:x 4057:z 4053:d 4049:+ 4025:z 4022:y 4018:d 4014:+ 3990:y 3987:x 3983:d 3979:= 3957:) 3952:z 3948:c 3942:x 3938:b 3929:x 3925:c 3919:z 3915:b 3911:( 3908:+ 3886:) 3881:y 3877:c 3871:z 3867:b 3858:z 3854:c 3848:y 3844:b 3840:( 3837:+ 3815:) 3810:x 3806:c 3800:y 3796:b 3787:y 3783:c 3777:x 3773:b 3769:( 3766:= 3757:c 3742:b 3736:= 3712:) 3709:0 3706:, 3700:, 3697:0 3694:( 3685:c 3679:, 3676:) 3673:0 3670:, 3667:0 3664:, 3658:( 3649:b 3618:c 3603:b 3597:= 3545:( 3515:( 3499:) 3496:0 3493:, 3490:0 3487:, 3481:( 3472:b 3449:0 3446:= 3441:z 3437:b 3433:, 3430:0 3427:= 3422:y 3418:b 3414:, 3408:= 3403:x 3399:b 3357:k 3351:0 3348:+ 3339:j 3333:0 3330:+ 3321:i 3312:= 3303:k 3297:) 3294:0 3282:0 3276:0 3273:( 3270:+ 3261:j 3255:) 3246:0 3240:0 3234:0 3231:( 3228:+ 3219:i 3213:) 3210:0 3204:0 3189:( 3186:= 3177:b 3133:k 3125:z 3121:b 3117:+ 3108:j 3100:y 3096:b 3092:+ 3083:i 3075:x 3071:b 3067:= 3058:k 3052:) 3047:x 3043:d 3037:y 3033:c 3024:y 3020:d 3014:x 3010:c 3006:( 3003:+ 2994:j 2988:) 2983:z 2979:d 2973:x 2969:c 2960:x 2956:d 2950:z 2946:c 2942:( 2939:+ 2930:i 2924:) 2919:y 2915:d 2909:z 2905:c 2896:z 2892:d 2886:y 2882:c 2878:( 2875:= 2866:d 2851:c 2845:= 2836:b 2795:b 2762:) 2756:, 2753:0 2750:, 2747:0 2744:( 2735:d 2709:= 2704:z 2700:d 2696:, 2693:0 2690:= 2685:y 2681:d 2677:, 2674:0 2671:= 2666:x 2662:d 2624:k 2615:+ 2606:j 2600:0 2597:+ 2588:i 2582:0 2579:= 2570:k 2564:) 2561:0 2555:0 2540:( 2537:+ 2528:j 2522:) 2519:0 2507:0 2501:0 2498:( 2495:+ 2486:i 2480:) 2471:0 2465:0 2459:0 2456:( 2453:= 2444:d 2404:k 2396:z 2392:d 2388:+ 2379:j 2371:y 2367:d 2363:+ 2354:i 2346:x 2342:d 2338:= 2329:k 2323:) 2318:x 2314:c 2308:y 2304:b 2295:y 2291:c 2285:x 2281:b 2277:( 2274:+ 2265:j 2259:) 2254:z 2250:c 2244:x 2240:b 2231:x 2227:c 2221:z 2217:b 2213:( 2210:+ 2201:i 2195:) 2190:y 2186:c 2180:z 2176:b 2167:z 2163:c 2157:y 2153:b 2149:( 2146:= 2137:c 2122:b 2116:= 2107:d 2074:) 2071:0 2068:, 2062:, 2059:0 2056:( 2047:c 2041:, 2038:) 2035:0 2032:, 2029:0 2026:, 2020:( 2011:b 1973:b 1958:d 1952:= 1943:c 1914:d 1899:c 1893:= 1884:b 1855:c 1840:b 1834:= 1825:d 1802:. 1784:k 1778:= 1769:j 1754:i 1690:( 1662:k 1656:= 1647:j 1632:i 1617:z 1601:′ 1597:z 1593:a 1570:′ 1566:z 1562:a 1553:z 1549:a 1526:z 1522:a 1498:′ 1494:z 1490:a 1465:= 1453:= 1448:y 1444:a 1438:x 1434:a 1430:= 1426:′ 1422:z 1418:a 1391:k 1383:z 1379:a 1375:+ 1366:j 1360:0 1357:+ 1348:i 1342:0 1339:= 1330:k 1324:) 1321:0 1313:y 1309:a 1303:x 1299:a 1295:( 1292:+ 1283:j 1277:0 1274:+ 1265:i 1259:0 1256:= 1253:) 1250:0 1247:+ 1238:j 1230:y 1226:a 1222:+ 1219:0 1216:( 1210:) 1207:0 1204:+ 1201:0 1198:+ 1189:i 1181:x 1177:a 1173:( 1170:= 1165:y 1155:a 1143:x 1133:a 1126:= 1122:′ 1118:z 1108:a 1083:′ 1079:z 1075:a 1052:z 1048:a 1021:k 1015:= 1006:j 991:i 966:z 962:a 958:= 953:z 943:a 936:, 931:y 927:a 923:= 918:y 908:a 901:, 896:x 892:a 888:= 883:x 873:a 846:z 836:a 829:+ 824:y 814:a 807:+ 802:x 792:a 785:= 776:k 768:z 764:a 760:+ 751:j 743:y 739:a 735:+ 726:i 718:x 714:a 710:= 707:) 702:z 698:a 694:, 689:y 685:a 681:, 676:x 672:a 668:( 659:a 633:5 627:= 620:2 607:2 594:2 584:= 579:z 575:a 550:2 545:z 541:a 537:+ 532:2 527:y 523:a 519:+ 514:2 509:x 505:a 499:= 495:| 485:a 478:| 455:) 450:z 446:a 442:, 436:, 430:( 421:a 395:= 391:| 381:a 374:| 314:3 310:2 306:1 270:. 165:. 64:: 20:)

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Index