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.

200: 95: 8728:, 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. 85: 64: 31: 6253: 281: 190: 169: 5767: 6946:, 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.-- 22: 8673: 1396: 3138: 2409: 5762: 6248:{\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}}} 3362: 2629: 8803:

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

3158: 7192:. 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 5137: 2425: 8901:

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

7470:. 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 7039: 1695:

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.

640: 1391:{\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}}} 8714:

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

7182: 6463: 3133:{\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}}} 2404:{\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}}} 632: 6258: 8695:

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

4967: 6668: 5237: 7577: 7468: 1978: 1919: 1860: 1789: 1667: 1026: 3711: 2073: 8913: 5757:{\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}}}} 4873: 4816: 4759: 4705: 4651: 7281: 1467: 347:

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"

8091:... 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 151: 7803:

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

2624:{\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}}} 8699:, 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? 35: 8970: 141: 7785:

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. --

5016: 1980:. 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). 8001:

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.

117: 199: 8965: 7044: 1714: 842:{\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}} 8858:'s direction has a clear physical meaning and is defined using a cross product. Continue your discussion at the reference desk, please.-- 3720: 461: 8917: 6658:{\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,} 1604:

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

222: 6453:{\displaystyle c_{x}={\dfrac {b_{z}d_{zx}-b_{y}d_{xy}}{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}}={\dfrac {0\cdot 0-0\cdot 100}{10^{2}+0+0}}=0,} 8975: 8458:

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

6997: 108: 69: 6860:{\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} 5475:{\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}}=} 3557:

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

4918: 3625:. This vector is in the plane formed by rectilinear vectors. Its magnitude is equal to the area of some figure in this plane. 8725: 1707:

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

7226: 332: 1401: 8782:, it seems to me that depicting a quantity that in fact has squared units as a simple arrow is far more conflating. 1732:

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

7473: 4346: 3581: 311: 8681: 3382: 2645: 8094: 7684:, will receive the status of uncoordinated cross product. And will be used only in geometric constructions.-- 3892: 3821: 8285: 8260:

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

8576:, do you want to delete this talk section, since the words you wanted deleted are now long-gone? 8563: 8265: 8221: 8183: 7988: 7838: 7790: 7398: 7193: 7127: 6962: 317: 100: 7655: 84: 63: 8361:

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

7351:, plain and simple, so your earlier notion of an "inverse vector" is complete nonsense.-- 4254: 8846:

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

7184:). You will not get that by simply relabeling the result as an "angular" vector as all 5214: 4304: 4279: 205: 7607: 8929: 8559: 8515: 8261: 8245: 8217: 7984: 7834: 7786: 7394: 6991: 4329: 351:. The result of the cross product of two rectilinear vectors is the angular vector. 8609:

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.

3809:{\displaystyle ={\bar {b}}\times {\bar {c}}=(b_{x}c_{y}-b_{y}c_{x})} 548:{\displaystyle |{\vec {a}}|={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}} 8446:

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

4485:- the projections of the angular vector onto the coordinate planes. 7041:) precludes any meaningful notion of an "inverse" in the sense of 5483:

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

7983:) that correspond to rotations, but the "unit quaternions". 7177:{\displaystyle \mathbf {a} \times \mathbf {b} =\mathbf {c} } 7851:

Potentially missleading and/or irrelevant part of the text?

627:{\displaystyle a_{z}={\sqrt {15^{2}-10^{2}-10^{2}}}=\pm 5} 7721:, which however are outside the scope of this article.-- 4962:{\displaystyle {\bar {c}}={\dfrac {1}{\bar {b}}}\times } 1502:

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.

7572:{\displaystyle {\bar {a}}\times {\bar {b}}={\bar {c}}} 7463:{\displaystyle {\bar {a}}\times {\bar {b}}={\bar {c}}} 1973:{\displaystyle {\vec {c}}={\vec {d}}\times {\vec {b}}} 1914:{\displaystyle {\vec {b}}={\vec {c}}\times {\vec {d}}} 1855:{\displaystyle {\vec {d}}={\vec {b}}\times {\vec {c}}} 1784:{\displaystyle {\vec {i}}\times {\vec {j}}={\vec {k}}} 1662:{\displaystyle {\vec {i}}\times {\vec {j}}={\vec {k}}} 1021:{\displaystyle {\vec {i}}\times {\vec {j}}={\vec {k}}} 8691:

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

7751:to see that the two "products" were taken from the 7520:consistent with vector theory. And the remaining 3 7318:Don't see how this contradicts what I said at all. 4754:{\displaystyle ={\bar {i}},\quad {\bar {j}}\times } 4700:{\displaystyle ={\bar {k}},\quad {\bar {k}}\times } 4646:{\displaystyle ={\bar {j}},\quad {\bar {j}}\times } 8653:Consistent hyphenation needed within the same page 8620: 8601: 8208: 8172: 7975: 7943: 7914: 7676: 7642: 7616: 7594: 7571: 7509: 7462: 7339: 7276:{\displaystyle {\bar {a}}\times {\bar {a}}^{-1}=0} 7275: 7176: 7138: 7116: 7088: 7033: 6912: 6859: 6657: 6452: 6247: 5756: 5474: 5226: 5201: 5166: 5131: 5003: 4961: 4906: 4867: 4810: 4753: 4699: 4645: 4591: 4555: 4520: 4477: 4415: 4335: 4316: 4291: 4266: 4241: 4192: 4143: 4076: 4055: 4020: 3985: 3950: 3879: 3808: 3705: 3617: 3568: 3492: 3442: 3356: 3132: 2794: 2755: 2705: 2623: 2403: 2067: 1989:For example, if there are two rectilinear vectors 1972: 1913: 1854: 1783: 1661: 1595: 1564: 1521: 1492: 1461: 1390: 1077: 1047: 1020: 961: 841: 626: 547: 448: 391: 7080: 8442:vectors, that is to say vectors which represent 1462:{\displaystyle a'_{z}=a_{x}a_{y}=10\cdot 10=100} 8956:Knowledge level-4 vital articles in Mathematics 8724:, and I suggest using a graphic like this one: 7411:In fact, there was no choice for cross-product 8750:No. You are conflating the cross product with 7347:with the cross product is not and cannot be a 3627:Suppose we are given two rectilinear vectors 3500:, and is not equal to the original vector. -- 2802:again and compare it with the original vector 326:This page has archives. Sections older than 8: 7510:{\displaystyle {\bar {a}}\times {\bar {b}}=} 4416:{\displaystyle d_{xy}=100,d_{yz}=0,d_{zx}=0} 3618:{\displaystyle ={\bar {b}}\times {\bar {c}}} 8722:https://www.youtube.com/watch?v=9GOoiaQq4GY 8461:So, we can say that the position vector is 8986:B-Class physics articles of Mid-importance 8897:Vector Algebra vs Linear Algebra Confusion 7804: 3443:{\displaystyle b_{x}=1000,b_{y}=0,b_{z}=0} 1670: 555:, the projection of the vector is equal 163: 58: 8614: 8595: 8545:must be considered as a pseudovector. If 8191: 8185: 8096: 7967: 7963: 7962: 7959: 7935: 7931: 7930: 7927: 7895: 7663: 7662: 7657: 7635: 7609: 7587: 7558: 7557: 7543: 7542: 7528: 7527: 7525: 7493: 7492: 7478: 7477: 7475: 7449: 7448: 7434: 7433: 7419: 7418: 7416: 7331: 7327: 7326: 7323: 7258: 7247: 7246: 7231: 7230: 7228: 7169: 7161: 7153: 7151: 7131: 7129: 7103: 7101: 7075: 7074: 7062: 7057: 7048: 7046: 7026: 7018: 7007: 6999: 6878: 6877: 6875: 6829: 6798: 6785: 6780: 6767: 6762: 6749: 6744: 6729: 6719: 6703: 6693: 6685: 6676: 6670: 6624: 6593: 6580: 6575: 6562: 6557: 6544: 6539: 6524: 6514: 6498: 6488: 6480: 6471: 6465: 6419: 6388: 6375: 6370: 6357: 6352: 6339: 6334: 6319: 6309: 6293: 6283: 6275: 6266: 6260: 6234: 6233: 6227: 6209: 6208: 6202: 6184: 6183: 6177: 6159: 6158: 6144: 6139: 6126: 6121: 6108: 6103: 6088: 6078: 6062: 6052: 6044: 6026: 6025: 6011: 6006: 5993: 5988: 5975: 5970: 5955: 5945: 5929: 5919: 5911: 5893: 5892: 5878: 5873: 5860: 5855: 5842: 5837: 5822: 5812: 5796: 5786: 5778: 5769: 5744: 5739: 5726: 5721: 5708: 5703: 5692: 5685: 5673: 5657: 5652: 5639: 5634: 5621: 5616: 5605: 5598: 5586: 5567: 5562: 5549: 5544: 5531: 5526: 5515: 5508: 5496: 5488: 5458: 5457: 5445: 5432: 5416: 5403: 5382: 5381: 5369: 5356: 5340: 5327: 5306: 5305: 5293: 5280: 5264: 5251: 5239: 5216: 5190: 5181: 5155: 5146: 5120: 5096: 5095: 5086: 5068: 5067: 5058: 5040: 5039: 5030: 5018: 4982: 4981: 4976: 4943: 4937: 4923: 4922: 4920: 4891: 4890: 4882: 4851: 4850: 4834: 4833: 4825: 4794: 4793: 4777: 4776: 4768: 4737: 4736: 4720: 4719: 4714: 4683: 4682: 4666: 4665: 4660: 4629: 4628: 4612: 4611: 4606: 4575: 4574: 4572: 4540: 4539: 4537: 4495: 4466: 4450: 4434: 4428: 4398: 4376: 4354: 4348: 4331: 4306: 4281: 4256: 4207: 4158: 4109: 4070: 4044: 4035: 4009: 4000: 3974: 3965: 3939: 3929: 3916: 3906: 3894: 3868: 3858: 3845: 3835: 3823: 3797: 3787: 3774: 3764: 3743: 3742: 3728: 3727: 3722: 3671: 3670: 3635: 3634: 3632: 3604: 3603: 3589: 3588: 3583: 3562: 3458: 3457: 3455: 3428: 3409: 3390: 3384: 3343: 3342: 3325: 3324: 3307: 3306: 3289: 3288: 3247: 3246: 3205: 3204: 3163: 3162: 3160: 3119: 3118: 3112: 3094: 3093: 3087: 3069: 3068: 3062: 3044: 3043: 3034: 3024: 3011: 3001: 2980: 2979: 2970: 2960: 2947: 2937: 2916: 2915: 2906: 2896: 2883: 2873: 2852: 2851: 2837: 2836: 2822: 2821: 2819: 2781: 2780: 2778: 2721: 2720: 2718: 2706:{\displaystyle d_{x}=0,d_{y}=0,d_{z}=100} 2691: 2672: 2653: 2647: 2610: 2609: 2592: 2591: 2574: 2573: 2556: 2555: 2514: 2513: 2472: 2471: 2430: 2429: 2427: 2390: 2389: 2383: 2365: 2364: 2358: 2340: 2339: 2333: 2315: 2314: 2305: 2295: 2282: 2272: 2251: 2250: 2241: 2231: 2218: 2208: 2187: 2186: 2177: 2167: 2154: 2144: 2123: 2122: 2108: 2107: 2093: 2092: 2090: 2033: 2032: 1997: 1996: 1994: 1959: 1958: 1944: 1943: 1929: 1928: 1926: 1900: 1899: 1885: 1884: 1870: 1869: 1867: 1841: 1840: 1826: 1825: 1811: 1810: 1808: 1770: 1769: 1755: 1754: 1740: 1739: 1737: 1648: 1647: 1633: 1632: 1618: 1617: 1615: 1584: 1578: 1553: 1540: 1534: 1513: 1507: 1481: 1475: 1435: 1425: 1409: 1403: 1377: 1376: 1370: 1352: 1351: 1334: 1333: 1316: 1315: 1300: 1290: 1269: 1268: 1251: 1250: 1224: 1223: 1217: 1175: 1174: 1168: 1152: 1141: 1140: 1130: 1119: 1118: 1105: 1094: 1093: 1090: 1066: 1060: 1039: 1033: 1007: 1006: 992: 991: 977: 976: 974: 953: 940: 929: 928: 918: 905: 894: 893: 883: 870: 859: 858: 855: 833: 822: 821: 811: 800: 799: 789: 778: 777: 762: 761: 755: 737: 736: 730: 712: 711: 705: 689: 676: 663: 645: 644: 642: 607: 594: 581: 575: 566: 560: 537: 532: 519: 514: 501: 496: 490: 482: 471: 470: 465: 463: 437: 407: 406: 404: 378: 367: 366: 361: 359: 8173:{\displaystyle (1,0,0),(0,1,0),(0,0,1).} 7124:, there are multiple orthogonal vectors 3951:{\displaystyle +(b_{z}c_{x}-b_{x}c_{z})} 3880:{\displaystyle +(b_{y}c_{z}-b_{z}c_{y})} 8946:Knowledge vital articles in Mathematics 8914:2A02:1660:69AA:E500:BEAE:C5FF:FE26:2574 8854:or simply would not be due weight. The 7954:Also, it is not the "unit vectors" (of 1500:projection found by the second method: 449:{\displaystyle {\vec {a}}(10,10,a_{z})} 336:when more than 10 sections are present. 165: 60: 19: 8553:are considered as pseudovectors, then 8249: 8180:About the rotation, you are right, as 6920:, and is equal to the original vector. 4144:{\displaystyle =(10\cdot 10-0\cdot 0)} 8961:B-Class vital articles in Mathematics 8300:Definition: deletion of two sentences 7922:where used for the standard basis of 6938:is a force, and a rectilinear vector 4242:{\displaystyle +(0\cdot 0-10\cdot 0)} 4193:{\displaystyle +(0\cdot 0-0\cdot 10)} 456:. One must find its third projection. 7: 8754:, of which the cross product is the 6930:is the torque. A rectilinear vector 6922:Now imagine that the angular vector 4478:{\displaystyle d_{xy},d_{yz},d_{zx}} 3493:{\displaystyle {\vec {b}}(1000,0,0)} 211:This article is within the scope of 106:This article is within the scope of 8558:sentence is definitively an error. 5004:{\displaystyle ={\bar {b'}}\times } 4532:Now we find the rectilinear vector 2756:{\displaystyle {\vec {d}}(0,0,100)} 349:https://doi.org/10.5539/jmr.v9n5p71 49:It is of interest to the following 8850:. Any of what you're proposing is 6913:{\displaystyle {\vec {c}}(0,10,0)} 4907:{\displaystyle =-{\bar {k}}\quad } 14: 8971:Mid-priority mathematics articles 8244:Not sure. The history section of 7860:(zero real part) with unit norms. 4592:{\displaystyle {\bar {i}}\times } 330:may be automatically archived by 126:Knowledge:WikiProject Mathematics 8941:Knowledge level-4 vital articles 8541:are considered as vectors, then 7976:{\displaystyle \mathbb {R} ^{3}} 7944:{\displaystyle \mathbb {R} ^{3}} 7340:{\displaystyle \mathbb {R} ^{3}} 7170: 7162: 7154: 7132: 7110: 7107: 7104: 7077: 7058: 7049: 7027: 7019: 7008: 4556:{\displaystyle {\bar {c}}\quad } 1703:. They are also identified with 1565:{\displaystyle a_{z}\neq a'_{z}} 1470:Conclusions: Rectilinear vector 279: 198: 188: 167: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 8981:Mid-importance physics articles 392:{\displaystyle |{\vec {a}}|=15} 251:This article has been rated as 146:This article has been rated as 8951:B-Class level-4 vital articles 8164: 8146: 8140: 8122: 8116: 8098: 7668: 7563: 7548: 7533: 7498: 7483: 7454: 7439: 7424: 7252: 7236: 7117:{\displaystyle \mathbf {a,c} } 7012: 7001: 6907: 6889: 6883: 6239: 6214: 6189: 6164: 6031: 5898: 5463: 5454: 5396: 5387: 5378: 5320: 5311: 5302: 5244: 5218: 5113: 5107: 5101: 5073: 5045: 5023: 4992: 4948: 4928: 4896: 4856: 4839: 4799: 4782: 4742: 4725: 4688: 4671: 4634: 4617: 4580: 4545: 4515: 4497: 4236: 4212: 4187: 4163: 4138: 4114: 3945: 3899: 3874: 3828: 3803: 3757: 3748: 3733: 3700: 3682: 3676: 3664: 3646: 3640: 3609: 3594: 3510:11:39, 18 September 2017 (UTC) 3487: 3469: 3463: 3348: 3330: 3312: 3294: 3285: 3261: 3252: 3243: 3219: 3210: 3201: 3177: 3168: 3124: 3099: 3074: 3049: 3040: 2994: 2985: 2976: 2930: 2921: 2912: 2866: 2857: 2842: 2827: 2786: 2750: 2732: 2726: 2615: 2597: 2579: 2561: 2552: 2528: 2519: 2510: 2486: 2477: 2468: 2444: 2435: 2395: 2370: 2345: 2320: 2311: 2265: 2256: 2247: 2201: 2192: 2183: 2137: 2128: 2113: 2098: 2062: 2044: 2038: 2026: 2008: 2002: 1964: 1949: 1934: 1905: 1890: 1875: 1846: 1831: 1816: 1775: 1760: 1745: 1719:15:00, 17 September 2017 (UTC) 1689:13:54, 17 September 2017 (UTC) 1653: 1638: 1623: 1382: 1357: 1339: 1321: 1312: 1283: 1274: 1256: 1241: 1229: 1204: 1198: 1180: 1161: 1146: 1124: 1099: 1012: 997: 982: 934: 899: 864: 827: 805: 783: 767: 742: 717: 695: 656: 650: 483: 476: 466: 443: 418: 412: 379: 372: 362: 1: 8881:08:32, 15 February 2024 (UTC) 8866:08:21, 15 February 2024 (UTC) 8835:08:18, 15 February 2024 (UTC) 8816:06:33, 15 February 2024 (UTC) 8792:06:18, 15 February 2024 (UTC) 8774:05:41, 15 February 2024 (UTC) 8738:05:39, 15 February 2024 (UTC) 8709:10:22, 15 November 2022 (UTC) 8616: 8597: 8586:08:57, 15 February 2024 (UTC) 8514:wrote. The truth is that the 7843:22:35, 13 November 2019 (UTC) 7823:20:20, 13 November 2019 (UTC) 7795:22:35, 13 November 2019 (UTC) 5667: 5580: 5490: 4902: 4848: 4791: 4734: 4680: 4626: 4551: 4072: 3564: 458:Solution: Using the formula 231:Knowledge:WikiProject Physics 225:and see a list of open tasks. 120:and see a list of open tasks. 8966:B-Class mathematics articles 8304:I delete the two sentences 8209:{\displaystyle i^{2}\neq 1,} 7748:A History of Vector Analysis 7393:that use these definitions. 7139:{\displaystyle \mathbf {b} } 6975:09:40, 14 October 2017 (UTC) 6956:08:34, 14 October 2017 (UTC) 4487:That is, the angular vector 3540:15:21, 12 October 2017 (UTC) 3450:, that is there is a vector 343:Cross product does not exist 234:Template:WikiProject Physics 8294:20:37, 16 August 2020 (UTC) 8270:21:01, 16 August 2020 (UTC) 8240:20:37, 16 August 2020 (UTC) 8226:19:53, 16 August 2020 (UTC) 8087:19:43, 16 August 2020 (UTC) 8011:19:36, 16 August 2020 (UTC) 7993:11:12, 16 August 2020 (UTC) 7885:10:49, 16 August 2020 (UTC) 7781:Alternative visual notation 7677:{\displaystyle ={\bar {c}}} 7096:(and even given orthogonal 4100:- the basic angular vectors 1529:found by the first method, 9002: 8676:is giving me what I want. 8493:without any contradiction. 8477:and the magnetic field is 8023:Found it: On the site for 6870:that is there is a vector 2795:{\displaystyle {\vec {b}}} 257:project's importance scale 8922:20:09, 29 June 2024 (UTC) 7769:22:56, 23 June 2019 (UTC) 7729:10:46, 23 June 2019 (UTC) 7694:10:42, 19 June 2019 (UTC) 7403:09:36, 11 June 2019 (UTC) 7387:07:13, 11 June 2019 (UTC) 7359:10:39, 23 June 2019 (UTC) 7293:10:33, 23 June 2019 (UTC) 4521:{\displaystyle (100,0,0)} 1862:. It would be logical if 250: 183: 145: 78: 57: 8976:B-Class physics articles 8686:09:44, 23 May 2022 (UTC) 8645:15:05, 9 June 2021 (UTC) 8568:17:22, 8 June 2021 (UTC) 8505:15:10, 8 June 2021 (UTC) 8434:"Axial" and "polar" are 8333:15:56, 6 June 2021 (UTC) 7204:18:28, 4 July 2018 (UTC) 1610:Thus, the cross product 152:project's priority scale 8621:{\displaystyle \qquad } 8602:{\displaystyle \qquad } 7643:{\displaystyle \times } 7595:{\displaystyle \times } 5202:{\displaystyle +d_{zx}} 5167:{\displaystyle +d_{yz}} 4056:{\displaystyle +d_{zx}} 4021:{\displaystyle +d_{yz}} 3986:{\displaystyle =d_{xy}} 2773:Now we find the vector 109:WikiProject Mathematics 8936:B-Class vital articles 8622: 8603: 8210: 8174: 8069:) and a 3-dimensional 8036:William Rowan Hamilton 7977: 7945: 7916: 7678: 7644: 7618: 7596: 7573: 7511: 7464: 7341: 7277: 7178: 7140: 7118: 7090: 7035: 6914: 6861: 6659: 6454: 6249: 5758: 5476: 5228: 5203: 5168: 5133: 5005: 4963: 4908: 4869: 4812: 4755: 4701: 4647: 4593: 4557: 4522: 4479: 4417: 4337: 4318: 4293: 4268: 4243: 4194: 4145: 4078: 4077:{\displaystyle \quad } 4057: 4022: 3987: 3952: 3881: 3810: 3707: 3619: 3570: 3569:{\displaystyle \quad } 3494: 3444: 3358: 3134: 2796: 2757: 2707: 2625: 2405: 2069: 1974: 1915: 1856: 1785: 1663: 1597: 1596:{\displaystyle a'_{z}} 1566: 1523: 1494: 1493:{\displaystyle a'_{z}} 1463: 1392: 1079: 1078:{\displaystyle a'_{z}} 1049: 1022: 969:and using the formula 963: 843: 628: 549: 450: 393: 333:Lowercase sigmabot III 8623: 8604: 8252:Joseph-Louis Lagrange 8211: 8175: 7978: 7946: 7917: 7915:{\displaystyle i,j,k} 7679: 7645: 7619: 7597: 7574: 7512: 7465: 7342: 7278: 7179: 7141: 7119: 7091: 7036: 6915: 6862: 6660: 6455: 6250: 5759: 5477: 5229: 5204: 5169: 5134: 5006: 4964: 4909: 4870: 4813: 4756: 4702: 4648: 4594: 4558: 4523: 4480: 4418: 4338: 4319: 4294: 4269: 4244: 4195: 4146: 4079: 4058: 4023: 3988: 3953: 3882: 3811: 3708: 3620: 3571: 3495: 3445: 3359: 3135: 2797: 2758: 2708: 2626: 2406: 2070: 1975: 1916: 1857: 1786: 1664: 1598: 1567: 1524: 1522:{\displaystyle a_{z}} 1495: 1464: 1393: 1080: 1050: 1048:{\displaystyle a_{z}} 1023: 964: 844: 629: 550: 451: 394: 36:level-4 vital article 8613: 8594: 8258:in three dimensions. 8184: 8095: 7958: 7926: 7894: 7656: 7634: 7608: 7586: 7524: 7474: 7415: 7322: 7227: 7150: 7128: 7100: 7045: 6998: 6874: 6669: 6464: 6259: 5768: 5487: 5238: 5215: 5180: 5145: 5017: 4975: 4919: 4881: 4824: 4767: 4713: 4659: 4605: 4571: 4536: 4528:is in the plane OXY 4494: 4427: 4347: 4330: 4305: 4280: 4267:{\displaystyle =100} 4255: 4206: 4157: 4108: 4069: 4034: 3999: 3964: 3893: 3822: 3721: 3631: 3582: 3561: 3454: 3383: 3159: 2818: 2777: 2717: 2713:that is, the vector 2646: 2426: 2089: 1993: 1925: 1866: 1807: 1736: 1614: 1577: 1533: 1506: 1474: 1402: 1089: 1059: 1032: 973: 854: 641: 559: 462: 403: 358: 132:mathematics articles 8697:Multilinear algebra 8660:appears 159 times. 7223:You have a mistake 7188:of dimension 3 are 6790: 6772: 6754: 6585: 6567: 6549: 6380: 6362: 6344: 6149: 6131: 6113: 6016: 5998: 5980: 5883: 5865: 5847: 5749: 5731: 5713: 5681: 5662: 5644: 5626: 5594: 5572: 5554: 5536: 5504: 5440: 5411: 5364: 5335: 5288: 5259: 5094: 5066: 5038: 1592: 1561: 1489: 1417: 1113: 1074: 542: 524: 506: 214:WikiProject Physics 8762:that appears in a 8618: 8617: 8599: 8598: 8431:a pseudovector? 8286:TheFibonacciEffect 8232:TheFibonacciEffect 8206: 8170: 8079:TheFibonacciEffect 8034:was introduced by 8003:TheFibonacciEffect 7973: 7941: 7912: 7877:TheFibonacciEffect 7713:This talk page is 7674: 7640: 7614: 7592: 7569: 7507: 7460: 7337: 7273: 7174: 7136: 7114: 7086: 7031: 6910: 6857: 6849: 6793: 6776: 6758: 6740: 6655: 6644: 6588: 6571: 6553: 6535: 6450: 6439: 6383: 6366: 6348: 6330: 6245: 6152: 6135: 6117: 6099: 6019: 6002: 5984: 5966: 5886: 5869: 5851: 5833: 5754: 5752: 5735: 5717: 5699: 5669: 5668: 5665: 5648: 5630: 5612: 5582: 5581: 5575: 5558: 5540: 5522: 5492: 5491: 5472: 5428: 5399: 5352: 5323: 5276: 5247: 5227:{\displaystyle )=} 5224: 5199: 5164: 5129: 5082: 5054: 5026: 5001: 4959: 4954: 4904: 4903: 4865: 4849: 4808: 4792: 4751: 4735: 4697: 4681: 4643: 4627: 4589: 4553: 4552: 4518: 4475: 4413: 4333: 4317:{\displaystyle +0} 4314: 4292:{\displaystyle +0} 4289: 4264: 4239: 4190: 4141: 4074: 4073: 4053: 4018: 3983: 3948: 3877: 3806: 3703: 3615: 3566: 3565: 3490: 3440: 3354: 3130: 2792: 2753: 2703: 2621: 2401: 2065: 1970: 1911: 1852: 1781: 1659: 1593: 1580: 1562: 1549: 1519: 1490: 1477: 1459: 1405: 1388: 1092: 1075: 1062: 1045: 1018: 959: 839: 624: 545: 528: 510: 492: 446: 389: 101:Mathematics portal 45:content assessment 8666:appears 5 times. 8042:, which is a sum 7825: 7809:comment added by 7671: 7617:{\displaystyle =} 7566: 7551: 7536: 7501: 7486: 7457: 7442: 7427: 7255: 7239: 7083: 6886: 6848: 6792: 6643: 6587: 6438: 6382: 6242: 6217: 6192: 6167: 6151: 6034: 6018: 5901: 5885: 5751: 5664: 5574: 5466: 5390: 5314: 5104: 5076: 5048: 4995: 4953: 4951: 4931: 4899: 4859: 4842: 4802: 4785: 4745: 4728: 4691: 4674: 4637: 4620: 4583: 4548: 3751: 3736: 3679: 3643: 3612: 3597: 3466: 3351: 3333: 3315: 3297: 3255: 3213: 3171: 3127: 3102: 3077: 3052: 2988: 2924: 2860: 2845: 2830: 2789: 2729: 2618: 2600: 2582: 2564: 2522: 2480: 2438: 2398: 2373: 2348: 2323: 2259: 2195: 2131: 2116: 2101: 2041: 2005: 1967: 1952: 1937: 1908: 1893: 1878: 1849: 1834: 1819: 1778: 1763: 1748: 1712: 1691: 1675:comment added by 1656: 1641: 1626: 1385: 1360: 1342: 1324: 1277: 1259: 1232: 1183: 1149: 1127: 1102: 1015: 1000: 985: 937: 902: 867: 830: 808: 786: 770: 745: 720: 653: 613: 543: 479: 415: 375: 340: 339: 271: 270: 267: 266: 263: 262: 162: 161: 158: 157: 8993: 8845: 8802: 8764:surface integral 8749: 8627: 8625: 8624: 8619: 8608: 8606: 8605: 8600: 8532: 8402: 8388: 8374: 8215: 8213: 8212: 8207: 8196: 8195: 8179: 8177: 8176: 8171: 8064: 8055: 8025:Euclidean vector 7982: 7980: 7979: 7974: 7972: 7971: 7966: 7950: 7948: 7947: 7942: 7940: 7939: 7934: 7921: 7919: 7918: 7913: 7835:Bill Cherowitzo 7787:Bill Cherowitzo 7712: 7683: 7681: 7680: 7675: 7673: 7672: 7664: 7652: 7649: 7647: 7646: 7641: 7630: 7626: 7623: 7621: 7620: 7615: 7604: 7601: 7599: 7598: 7593: 7582: 7578: 7576: 7575: 7570: 7568: 7567: 7559: 7553: 7552: 7544: 7538: 7537: 7529: 7519: 7516: 7514: 7513: 7508: 7503: 7502: 7494: 7488: 7487: 7479: 7469: 7467: 7466: 7461: 7459: 7458: 7450: 7444: 7443: 7435: 7429: 7428: 7420: 7349:division algebra 7346: 7344: 7343: 7338: 7336: 7335: 7330: 7317: 7282: 7280: 7279: 7274: 7266: 7265: 7257: 7256: 7248: 7241: 7240: 7232: 7222: 7183: 7181: 7180: 7175: 7173: 7165: 7157: 7145: 7143: 7142: 7137: 7135: 7123: 7121: 7120: 7115: 7113: 7095: 7093: 7092: 7087: 7085: 7084: 7076: 7070: 7069: 7061: 7052: 7040: 7038: 7037: 7032: 7030: 7022: 7011: 6989: 6943: 6935: 6927: 6919: 6917: 6916: 6911: 6888: 6887: 6879: 6866: 6864: 6863: 6858: 6850: 6847: 6834: 6833: 6823: 6800: 6794: 6791: 6789: 6784: 6771: 6766: 6753: 6748: 6738: 6737: 6736: 6724: 6723: 6711: 6710: 6698: 6697: 6687: 6681: 6680: 6664: 6662: 6661: 6656: 6645: 6642: 6629: 6628: 6618: 6595: 6589: 6586: 6584: 6579: 6566: 6561: 6548: 6543: 6533: 6532: 6531: 6519: 6518: 6506: 6505: 6493: 6492: 6482: 6476: 6475: 6459: 6457: 6456: 6451: 6440: 6437: 6424: 6423: 6413: 6390: 6384: 6381: 6379: 6374: 6361: 6356: 6343: 6338: 6328: 6327: 6326: 6314: 6313: 6301: 6300: 6288: 6287: 6277: 6271: 6270: 6254: 6252: 6251: 6246: 6244: 6243: 6235: 6232: 6231: 6219: 6218: 6210: 6207: 6206: 6194: 6193: 6185: 6182: 6181: 6169: 6168: 6160: 6157: 6153: 6150: 6148: 6143: 6130: 6125: 6112: 6107: 6097: 6096: 6095: 6083: 6082: 6070: 6069: 6057: 6056: 6046: 6036: 6035: 6027: 6024: 6020: 6017: 6015: 6010: 5997: 5992: 5979: 5974: 5964: 5963: 5962: 5950: 5949: 5937: 5936: 5924: 5923: 5913: 5903: 5902: 5894: 5891: 5887: 5884: 5882: 5877: 5864: 5859: 5846: 5841: 5831: 5830: 5829: 5817: 5816: 5804: 5803: 5791: 5790: 5780: 5763: 5761: 5760: 5755: 5753: 5750: 5748: 5743: 5730: 5725: 5712: 5707: 5697: 5696: 5687: 5677: 5666: 5663: 5661: 5656: 5643: 5638: 5625: 5620: 5610: 5609: 5600: 5590: 5576: 5573: 5571: 5566: 5553: 5548: 5535: 5530: 5520: 5519: 5510: 5500: 5481: 5479: 5478: 5473: 5468: 5467: 5459: 5453: 5452: 5436: 5424: 5423: 5407: 5392: 5391: 5383: 5377: 5376: 5360: 5348: 5347: 5331: 5316: 5315: 5307: 5301: 5300: 5284: 5272: 5271: 5255: 5233: 5231: 5230: 5225: 5211: 5208: 5206: 5205: 5200: 5198: 5197: 5176: 5173: 5171: 5170: 5165: 5163: 5162: 5141: 5138: 5136: 5135: 5130: 5128: 5127: 5106: 5105: 5097: 5090: 5078: 5077: 5069: 5062: 5050: 5049: 5041: 5034: 5013: 5010: 5008: 5007: 5002: 4997: 4996: 4991: 4983: 4971: 4968: 4966: 4965: 4960: 4955: 4952: 4944: 4939: 4933: 4932: 4924: 4913: 4911: 4910: 4905: 4901: 4900: 4892: 4877: 4874: 4872: 4871: 4866: 4861: 4860: 4852: 4844: 4843: 4835: 4820: 4817: 4815: 4814: 4809: 4804: 4803: 4795: 4787: 4786: 4778: 4763: 4760: 4758: 4757: 4752: 4747: 4746: 4738: 4730: 4729: 4721: 4709: 4706: 4704: 4703: 4698: 4693: 4692: 4684: 4676: 4675: 4667: 4655: 4652: 4650: 4649: 4644: 4639: 4638: 4630: 4622: 4621: 4613: 4601: 4598: 4596: 4595: 4590: 4585: 4584: 4576: 4562: 4560: 4559: 4554: 4550: 4549: 4541: 4527: 4525: 4524: 4519: 4490: 4484: 4482: 4481: 4476: 4474: 4473: 4458: 4457: 4442: 4441: 4422: 4420: 4419: 4414: 4406: 4405: 4384: 4383: 4362: 4361: 4342: 4340: 4339: 4336:{\displaystyle } 4334: 4326: 4323: 4321: 4320: 4315: 4301: 4298: 4296: 4295: 4290: 4276: 4273: 4271: 4270: 4265: 4251: 4248: 4246: 4245: 4240: 4202: 4199: 4197: 4196: 4191: 4153: 4150: 4148: 4147: 4142: 4104: 4097: 4093: 4089: 4083: 4081: 4080: 4075: 4065: 4062: 4060: 4059: 4054: 4052: 4051: 4030: 4027: 4025: 4024: 4019: 4017: 4016: 3995: 3992: 3990: 3989: 3984: 3982: 3981: 3960: 3957: 3955: 3954: 3949: 3944: 3943: 3934: 3933: 3921: 3920: 3911: 3910: 3889: 3886: 3884: 3883: 3878: 3873: 3872: 3863: 3862: 3850: 3849: 3840: 3839: 3818: 3815: 3813: 3812: 3807: 3802: 3801: 3792: 3791: 3779: 3778: 3769: 3768: 3753: 3752: 3744: 3738: 3737: 3729: 3717: 3712: 3710: 3709: 3704: 3681: 3680: 3672: 3645: 3644: 3636: 3624: 3622: 3621: 3616: 3614: 3613: 3605: 3599: 3598: 3590: 3578: 3575: 3573: 3572: 3567: 3499: 3497: 3496: 3491: 3468: 3467: 3459: 3449: 3447: 3446: 3441: 3433: 3432: 3414: 3413: 3395: 3394: 3363: 3361: 3360: 3355: 3353: 3352: 3344: 3335: 3334: 3326: 3317: 3316: 3308: 3299: 3298: 3290: 3257: 3256: 3248: 3215: 3214: 3206: 3173: 3172: 3164: 3139: 3137: 3136: 3131: 3129: 3128: 3120: 3117: 3116: 3104: 3103: 3095: 3092: 3091: 3079: 3078: 3070: 3067: 3066: 3054: 3053: 3045: 3039: 3038: 3029: 3028: 3016: 3015: 3006: 3005: 2990: 2989: 2981: 2975: 2974: 2965: 2964: 2952: 2951: 2942: 2941: 2926: 2925: 2917: 2911: 2910: 2901: 2900: 2888: 2887: 2878: 2877: 2862: 2861: 2853: 2847: 2846: 2838: 2832: 2831: 2823: 2801: 2799: 2798: 2793: 2791: 2790: 2782: 2762: 2760: 2759: 2754: 2731: 2730: 2722: 2712: 2710: 2709: 2704: 2696: 2695: 2677: 2676: 2658: 2657: 2630: 2628: 2627: 2622: 2620: 2619: 2611: 2602: 2601: 2593: 2584: 2583: 2575: 2566: 2565: 2557: 2524: 2523: 2515: 2482: 2481: 2473: 2440: 2439: 2431: 2410: 2408: 2407: 2402: 2400: 2399: 2391: 2388: 2387: 2375: 2374: 2366: 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1362: 1361: 1353: 1344: 1343: 1335: 1326: 1325: 1317: 1305: 1304: 1295: 1294: 1279: 1278: 1270: 1261: 1260: 1252: 1234: 1233: 1225: 1222: 1221: 1185: 1184: 1176: 1173: 1172: 1157: 1156: 1151: 1150: 1142: 1135: 1134: 1129: 1128: 1120: 1109: 1104: 1103: 1095: 1084: 1082: 1081: 1076: 1070: 1054: 1052: 1051: 1046: 1044: 1043: 1027: 1025: 1024: 1019: 1017: 1016: 1008: 1002: 1001: 993: 987: 986: 978: 968: 966: 965: 960: 958: 957: 945: 944: 939: 938: 930: 923: 922: 910: 909: 904: 903: 895: 888: 887: 875: 874: 869: 868: 860: 848: 846: 845: 840: 838: 837: 832: 831: 823: 816: 815: 810: 809: 801: 794: 793: 788: 787: 779: 772: 771: 763: 760: 759: 747: 746: 738: 735: 734: 722: 721: 713: 710: 709: 694: 693: 681: 680: 668: 667: 655: 654: 646: 633: 631: 630: 625: 614: 612: 611: 599: 598: 586: 585: 576: 571: 570: 554: 552: 551: 546: 544: 541: 536: 523: 518: 505: 500: 491: 486: 481: 480: 472: 469: 455: 453: 452: 447: 442: 441: 417: 416: 408: 398: 396: 395: 390: 382: 377: 376: 368: 365: 335: 319: 283: 275: 239: 238: 237:physics articles 235: 232: 229: 208: 203: 202: 192: 185: 184: 179: 171: 164: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 9001: 9000: 8996: 8995: 8994: 8992: 8991: 8990: 8926: 8925: 8899: 8856:Poynting vector 8839: 8796: 8760:surface element 8743: 8717: 8693: 8678:Universemaster1 8655: 8629: 8611: 8610: 8592: 8591: 8520: 8438:qualifiers for 8404: 8390: 8376: 8362: 8340: 8313: 8302: 8187: 8182: 8181: 8093: 8092: 8060: 8043: 7961: 7956: 7955: 7929: 7924: 7923: 7892: 7891: 7853: 7783: 7706: 7654: 7653: 7650: 7632: 7631: 7628: 7624: 7606: 7605: 7602: 7584: 7583: 7580: 7522: 7521: 7517: 7472: 7471: 7413: 7412: 7375: 7325: 7320: 7319: 7311: 7245: 7225: 7224: 7216: 7148: 7147: 7126: 7125: 7098: 7097: 7056: 7043: 7042: 6996: 6995: 6983: 6967:Boris Tsirelson 6941: 6933: 6925: 6872: 6871: 6825: 6824: 6801: 6739: 6725: 6715: 6699: 6689: 6688: 6672: 6667: 6666: 6620: 6619: 6596: 6534: 6520: 6510: 6494: 6484: 6483: 6467: 6462: 6461: 6415: 6414: 6391: 6329: 6315: 6305: 6289: 6279: 6278: 6262: 6257: 6256: 6223: 6198: 6173: 6098: 6084: 6074: 6058: 6048: 6047: 6040: 5965: 5951: 5941: 5925: 5915: 5914: 5907: 5832: 5818: 5808: 5792: 5782: 5781: 5774: 5766: 5765: 5698: 5688: 5611: 5601: 5521: 5511: 5485: 5484: 5441: 5412: 5365: 5336: 5289: 5260: 5236: 5235: 5213: 5212: 5209: 5186: 5178: 5177: 5174: 5151: 5143: 5142: 5139: 5116: 5015: 5014: 5011: 4984: 4973: 4972: 4969: 4917: 4916: 4879: 4878: 4875: 4822: 4821: 4818: 4765: 4764: 4761: 4711: 4710: 4707: 4657: 4656: 4653: 4603: 4602: 4599: 4569: 4568: 4534: 4533: 4492: 4491: 4488: 4462: 4446: 4430: 4425: 4424: 4394: 4372: 4350: 4345: 4344: 4328: 4327: 4324: 4303: 4302: 4299: 4278: 4277: 4274: 4253: 4252: 4249: 4204: 4203: 4200: 4155: 4154: 4151: 4106: 4105: 4102: 4095: 4091: 4087: 4067: 4066: 4063: 4040: 4032: 4031: 4028: 4005: 3997: 3996: 3993: 3970: 3962: 3961: 3958: 3935: 3925: 3912: 3902: 3891: 3890: 3887: 3864: 3854: 3841: 3831: 3820: 3819: 3816: 3793: 3783: 3770: 3760: 3719: 3718: 3715: 3629: 3628: 3580: 3579: 3576: 3559: 3558: 3532:Boris Tsirelson 3452: 3451: 3424: 3405: 3386: 3381: 3380: 3157: 3156: 3108: 3083: 3058: 3030: 3020: 3007: 2997: 2966: 2956: 2943: 2933: 2902: 2892: 2879: 2869: 2816: 2815: 2775: 2774: 2715: 2714: 2687: 2668: 2649: 2644: 2643: 2424: 2423: 2379: 2354: 2329: 2301: 2291: 2278: 2268: 2237: 2227: 2214: 2204: 2173: 2163: 2150: 2140: 2087: 2086: 1991: 1990: 1923: 1922: 1864: 1863: 1805: 1804: 1734: 1733: 1717: 1612: 1611: 1607: 1575: 1574: 1536: 1531: 1530: 1509: 1504: 1503: 1472: 1471: 1431: 1421: 1400: 1399: 1366: 1296: 1286: 1213: 1164: 1139: 1117: 1087: 1086: 1057: 1056: 1035: 1030: 1029: 971: 970: 949: 927: 914: 892: 879: 857: 852: 851: 820: 798: 776: 751: 726: 701: 685: 672: 659: 639: 638: 603: 590: 577: 562: 557: 556: 460: 459: 433: 401: 400: 356: 355: 345: 331: 320: 314: 288: 236: 233: 230: 227: 226: 204: 197: 177: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 8999: 8997: 8989: 8988: 8983: 8978: 8973: 8968: 8963: 8958: 8953: 8948: 8943: 8938: 8928: 8927: 8898: 8895: 8894: 8893: 8892: 8891: 8890: 8889: 8888: 8887: 8886: 8885: 8884: 8883: 8808:is that way.-- 8716: 8712: 8692: 8689: 8654: 8651: 8650: 8649: 8648: 8647: 8590: 8589: 8588: 8570: 8494: 8344: 8339: 8336: 8306: 8301: 8298: 8297: 8296: 8282: 8281: 8280: 8279: 8278: 8277: 8276: 8275: 8274: 8273: 8272: 8205: 8202: 8199: 8194: 8190: 8169: 8166: 8163: 8160: 8157: 8154: 8151: 8148: 8145: 8142: 8139: 8136: 8133: 8130: 8127: 8124: 8121: 8118: 8115: 8112: 8109: 8106: 8103: 8100: 8076: 8075: 8074: 8027:it says that: 8016: 8015: 8014: 8013: 7996: 7995: 7970: 7965: 7952: 7938: 7933: 7911: 7908: 7905: 7902: 7899: 7874: 7872: 7871: 7862: 7861: 7852: 7849: 7848: 7847: 7846: 7845: 7827: 7826: 7782: 7779: 7778: 7777: 7776: 7775: 7774: 7773: 7772: 7771: 7736: 7735: 7734: 7733: 7732: 7731: 7699: 7698: 7697: 7696: 7670: 7667: 7661: 7639: 7613: 7591: 7565: 7562: 7556: 7550: 7547: 7541: 7535: 7532: 7506: 7500: 7497: 7491: 7485: 7482: 7456: 7453: 7447: 7441: 7438: 7432: 7426: 7423: 7406: 7405: 7374: 7371: 7370: 7369: 7368: 7367: 7366: 7365: 7364: 7363: 7362: 7361: 7334: 7329: 7300: 7299: 7298: 7297: 7296: 7295: 7272: 7269: 7264: 7261: 7254: 7251: 7244: 7238: 7235: 7209: 7208: 7207: 7206: 7172: 7168: 7164: 7160: 7156: 7134: 7112: 7109: 7106: 7082: 7079: 7073: 7068: 7065: 7060: 7055: 7051: 7029: 7025: 7021: 7017: 7014: 7010: 7006: 7003: 6978: 6977: 6921: 6909: 6906: 6903: 6900: 6897: 6894: 6891: 6885: 6882: 6867: 6856: 6853: 6846: 6843: 6840: 6837: 6832: 6828: 6822: 6819: 6816: 6813: 6810: 6807: 6804: 6797: 6788: 6783: 6779: 6775: 6770: 6765: 6761: 6757: 6752: 6747: 6743: 6735: 6732: 6728: 6722: 6718: 6714: 6709: 6706: 6702: 6696: 6692: 6684: 6679: 6675: 6665: 6654: 6651: 6648: 6641: 6638: 6635: 6632: 6627: 6623: 6617: 6614: 6611: 6608: 6605: 6602: 6599: 6592: 6583: 6578: 6574: 6570: 6565: 6560: 6556: 6552: 6547: 6542: 6538: 6530: 6527: 6523: 6517: 6513: 6509: 6504: 6501: 6497: 6491: 6487: 6479: 6474: 6470: 6460: 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2121: 2115: 2112: 2106: 2100: 2097: 2079: 2078: 2077: 2076: 2064: 2061: 2058: 2055: 2052: 2049: 2046: 2040: 2037: 2031: 2028: 2025: 2022: 2019: 2016: 2013: 2010: 2004: 2001: 1984: 1983: 1982: 1981: 1966: 1963: 1957: 1951: 1948: 1942: 1936: 1933: 1907: 1904: 1898: 1892: 1889: 1883: 1877: 1874: 1848: 1845: 1839: 1833: 1830: 1824: 1818: 1815: 1801: 1795: 1794: 1793: 1792: 1777: 1774: 1768: 1762: 1759: 1753: 1747: 1744: 1730: 1729: 1722: 1721: 1713: 1710:JohnBlackburne 1696: 1655: 1652: 1646: 1640: 1637: 1631: 1625: 1622: 1609: 1605: 1603: 1591: 1587: 1583: 1572: 1560: 1556: 1552: 1548: 1543: 1539: 1516: 1512: 1501: 1488: 1484: 1480: 1469: 1458: 1455: 1452: 1449: 1446: 1443: 1438: 1434: 1428: 1424: 1420: 1416: 1412: 1408: 1398: 1384: 1381: 1373: 1369: 1365: 1359: 1356: 1350: 1347: 1341: 1338: 1332: 1329: 1323: 1320: 1314: 1311: 1308: 1303: 1299: 1293: 1289: 1285: 1282: 1276: 1273: 1267: 1264: 1258: 1255: 1249: 1246: 1243: 1240: 1237: 1231: 1228: 1220: 1216: 1212: 1209: 1206: 1203: 1200: 1197: 1194: 1191: 1188: 1182: 1179: 1171: 1167: 1163: 1160: 1155: 1148: 1145: 1138: 1133: 1126: 1123: 1116: 1112: 1108: 1101: 1098: 1085: 1073: 1069: 1065: 1042: 1038: 1014: 1011: 1005: 999: 996: 990: 984: 981: 956: 952: 948: 943: 936: 933: 926: 921: 917: 913: 908: 901: 898: 891: 886: 882: 878: 873: 866: 863: 850: 836: 829: 826: 819: 814: 807: 804: 797: 792: 785: 782: 775: 769: 766: 758: 754: 750: 744: 741: 733: 729: 725: 719: 716: 708: 704: 700: 697: 692: 688: 684: 679: 675: 671: 666: 662: 658: 652: 649: 637: 623: 620: 617: 610: 606: 602: 597: 593: 589: 584: 580: 574: 569: 565: 540: 535: 531: 527: 522: 517: 513: 509: 504: 499: 495: 489: 485: 478: 475: 468: 457: 445: 440: 436: 432: 429: 426: 423: 420: 414: 411: 388: 385: 381: 374: 371: 364: 344: 341: 338: 337: 325: 322: 321: 316: 312: 310: 307: 306: 290: 289: 284: 278: 269: 268: 265: 264: 261: 260: 253:Mid-importance 249: 243: 242: 240: 223:the discussion 210: 209: 206:Physics portal 193: 181: 180: 178:Mid‑importance 172: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 8998: 8987: 8984: 8982: 8979: 8977: 8974: 8972: 8969: 8967: 8964: 8962: 8959: 8957: 8954: 8952: 8949: 8947: 8944: 8942: 8939: 8937: 8934: 8933: 8931: 8924: 8923: 8919: 8915: 8910: 8908: 8903: 8896: 8882: 8878: 8874: 8869: 8868: 8867: 8864: 8861: 8857: 8853: 8849: 8843: 8838: 8837: 8836: 8832: 8828: 8823: 8819: 8818: 8817: 8814: 8811: 8807: 8800: 8795: 8794: 8793: 8789: 8785: 8781: 8777: 8776: 8775: 8772: 8769: 8765: 8761: 8757: 8753: 8747: 8742: 8741: 8740: 8739: 8735: 8731: 8727: 8723: 8711: 8710: 8706: 8702: 8698: 8690: 8688: 8687: 8683: 8679: 8675: 8674:google ngrams 8672:I'm not sure 8670: 8667: 8665: 8664:Cross-product 8661: 8659: 8658:Cross product 8652: 8646: 8642: 8638: 8633: 8632: 8631: 8630: 8587: 8583: 8579: 8575: 8571: 8569: 8565: 8561: 8556: 8552: 8548: 8544: 8540: 8536: 8531: 8527: 8523: 8517: 8516:inner product 8513: 8509: 8508: 8507: 8506: 8502: 8498: 8492: 8488: 8484: 8480: 8476: 8472: 8468: 8464: 8459: 8457: 8453: 8449: 8445: 8441: 8437: 8432: 8430: 8426: 8422: 8417: 8413: 8409: 8401: 8397: 8393: 8387: 8383: 8379: 8373: 8369: 8365: 8360: 8356: 8352: 8348: 8343: 8337: 8335: 8334: 8330: 8326: 8321: 8319: 8311: 8305: 8299: 8295: 8291: 8287: 8283: 8271: 8267: 8263: 8259: 8257: 8253: 8247: 8246:Cross product 8243: 8242: 8241: 8237: 8233: 8229: 8228: 8227: 8223: 8219: 8203: 8200: 8197: 8192: 8188: 8167: 8161: 8158: 8155: 8152: 8149: 8143: 8137: 8134: 8131: 8128: 8125: 8119: 8113: 8110: 8107: 8104: 8101: 8090: 8089: 8088: 8084: 8080: 8077: 8072: 8068: 8065:(also called 8063: 8059: 8054: 8050: 8046: 8041: 8038:as part of a 8037: 8033: 8029: 8028: 8026: 8022: 8021: 8020: 8019: 8018: 8017: 8012: 8008: 8004: 8000: 7999: 7998: 7997: 7994: 7990: 7986: 7968: 7953: 7936: 7909: 7906: 7903: 7900: 7897: 7889: 7888: 7887: 7886: 7882: 7878: 7875:Kind regards 7868: 7867: 7866: 7858: 7857: 7856: 7850: 7844: 7840: 7836: 7831: 7830: 7829: 7828: 7824: 7820: 7816: 7812: 7808: 7802: 7799: 7798: 7797: 7796: 7792: 7788: 7780: 7770: 7766: 7762: 7758: 7757:coquaternions 7754: 7750: 7749: 7744: 7743: 7742: 7741: 7740: 7739: 7738: 7737: 7730: 7727: 7724: 7720: 7716: 7710: 7705: 7704: 7703: 7702: 7701: 7700: 7695: 7691: 7687: 7665: 7659: 7637: 7611: 7589: 7560: 7554: 7545: 7539: 7530: 7504: 7495: 7489: 7480: 7451: 7445: 7436: 7430: 7421: 7410: 7409: 7408: 7407: 7404: 7400: 7396: 7391: 7390: 7389: 7388: 7384: 7380: 7372: 7360: 7357: 7354: 7350: 7332: 7315: 7310: 7309: 7308: 7307: 7306: 7305: 7304: 7303: 7302: 7301: 7294: 7290: 7286: 7270: 7267: 7262: 7259: 7249: 7242: 7233: 7220: 7215: 7214: 7213: 7212: 7211: 7210: 7205: 7202: 7199: 7195: 7191: 7187: 7186:vector spaces 7166: 7158: 7071: 7066: 7063: 7053: 7023: 7015: 7004: 6993: 6992:zero divisors 6987: 6982: 6981: 6980: 6979: 6976: 6972: 6968: 6964: 6960: 6959: 6958: 6957: 6953: 6949: 6945: 6944: 6937: 6936: 6929: 6928: 6904: 6901: 6898: 6895: 6892: 6880: 6868: 6854: 6851: 6844: 6841: 6838: 6835: 6830: 6826: 6820: 6817: 6814: 6811: 6808: 6805: 6802: 6795: 6786: 6781: 6777: 6773: 6768: 6763: 6759: 6755: 6750: 6745: 6741: 6733: 6730: 6726: 6720: 6716: 6712: 6707: 6704: 6700: 6694: 6690: 6682: 6677: 6673: 6652: 6649: 6646: 6639: 6636: 6633: 6630: 6625: 6621: 6615: 6612: 6609: 6606: 6603: 6600: 6597: 6590: 6581: 6576: 6572: 6568: 6563: 6558: 6554: 6550: 6545: 6540: 6536: 6528: 6525: 6521: 6515: 6511: 6507: 6502: 6499: 6495: 6489: 6485: 6477: 6472: 6468: 6447: 6444: 6441: 6434: 6431: 6428: 6425: 6420: 6416: 6410: 6407: 6404: 6401: 6398: 6395: 6392: 6385: 6376: 6371: 6367: 6363: 6358: 6353: 6349: 6345: 6340: 6335: 6331: 6323: 6320: 6316: 6310: 6306: 6302: 6297: 6294: 6290: 6284: 6280: 6272: 6267: 6263: 6236: 6228: 6224: 6220: 6211: 6203: 6199: 6195: 6186: 6178: 6174: 6170: 6161: 6154: 6145: 6140: 6136: 6132: 6127: 6122: 6118: 6114: 6109: 6104: 6100: 6092: 6089: 6085: 6079: 6075: 6071: 6066: 6063: 6059: 6053: 6049: 6041: 6037: 6028: 6021: 6012: 6007: 6003: 5999: 5994: 5989: 5985: 5981: 5976: 5971: 5967: 5959: 5956: 5952: 5946: 5942: 5938: 5933: 5930: 5926: 5920: 5916: 5908: 5904: 5895: 5888: 5879: 5874: 5870: 5866: 5861: 5856: 5852: 5848: 5843: 5838: 5834: 5826: 5823: 5819: 5813: 5809: 5805: 5800: 5797: 5793: 5787: 5783: 5775: 5771: 5745: 5740: 5736: 5732: 5727: 5722: 5718: 5714: 5709: 5704: 5700: 5693: 5689: 5682: 5678: 5674: 5670: 5658: 5653: 5649: 5645: 5640: 5635: 5631: 5627: 5622: 5617: 5613: 5606: 5602: 5595: 5591: 5587: 5583: 5577: 5568: 5563: 5559: 5555: 5550: 5545: 5541: 5537: 5532: 5527: 5523: 5516: 5512: 5505: 5501: 5497: 5493: 5469: 5460: 5449: 5446: 5442: 5437: 5433: 5429: 5425: 5420: 5417: 5413: 5408: 5404: 5400: 5393: 5384: 5373: 5370: 5366: 5361: 5357: 5353: 5349: 5344: 5341: 5337: 5332: 5328: 5324: 5317: 5308: 5297: 5294: 5290: 5285: 5281: 5277: 5273: 5268: 5265: 5261: 5256: 5252: 5248: 5241: 5221: 5194: 5191: 5187: 5183: 5159: 5156: 5152: 5148: 5124: 5121: 5117: 5110: 5098: 5091: 5087: 5083: 5079: 5070: 5063: 5059: 5055: 5051: 5042: 5035: 5031: 5027: 5020: 4998: 4988: 4985: 4978: 4956: 4945: 4940: 4934: 4925: 4893: 4887: 4884: 4862: 4853: 4845: 4836: 4830: 4827: 4805: 4796: 4788: 4779: 4773: 4770: 4748: 4739: 4731: 4722: 4716: 4694: 4685: 4677: 4668: 4662: 4640: 4631: 4623: 4614: 4608: 4586: 4577: 4564:individually. 4542: 4530: 4512: 4509: 4506: 4503: 4500: 4470: 4467: 4463: 4459: 4454: 4451: 4447: 4443: 4438: 4435: 4431: 4410: 4407: 4402: 4399: 4395: 4391: 4388: 4385: 4380: 4377: 4373: 4369: 4366: 4363: 4358: 4355: 4351: 4311: 4308: 4286: 4283: 4261: 4258: 4233: 4230: 4227: 4224: 4221: 4218: 4215: 4209: 4184: 4181: 4178: 4175: 4172: 4169: 4166: 4160: 4135: 4132: 4129: 4126: 4123: 4120: 4117: 4111: 4099: 4098: 4048: 4045: 4041: 4037: 4013: 4010: 4006: 4002: 3978: 3975: 3971: 3967: 3940: 3936: 3930: 3926: 3922: 3917: 3913: 3907: 3903: 3896: 3869: 3865: 3859: 3855: 3851: 3846: 3842: 3836: 3832: 3825: 3798: 3794: 3788: 3784: 3780: 3775: 3771: 3765: 3761: 3754: 3745: 3739: 3730: 3724: 3697: 3694: 3691: 3688: 3685: 3673: 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1726: 1725: 1724: 1723: 1720: 1716: 1711: 1706: 1702: 1697: 1694: 1693: 1692: 1690: 1686: 1682: 1678: 1674: 1650: 1644: 1635: 1629: 1620: 1589: 1585: 1581: 1558: 1554: 1550: 1546: 1541: 1537: 1514: 1510: 1486: 1482: 1478: 1456: 1453: 1450: 1447: 1444: 1441: 1436: 1432: 1426: 1422: 1418: 1414: 1410: 1406: 1379: 1371: 1367: 1363: 1354: 1348: 1345: 1336: 1330: 1327: 1318: 1309: 1306: 1301: 1297: 1291: 1287: 1280: 1271: 1265: 1262: 1253: 1247: 1244: 1238: 1235: 1226: 1218: 1214: 1210: 1207: 1201: 1195: 1192: 1189: 1186: 1177: 1169: 1165: 1158: 1153: 1143: 1136: 1131: 1121: 1114: 1110: 1106: 1096: 1071: 1067: 1063: 1040: 1036: 1009: 1003: 994: 988: 979: 954: 950: 946: 941: 931: 924: 919: 915: 911: 906: 896: 889: 884: 880: 876: 871: 861: 834: 824: 817: 812: 802: 795: 790: 780: 773: 764: 756: 752: 748: 739: 731: 727: 723: 714: 706: 702: 698: 690: 686: 682: 677: 673: 669: 664: 660: 647: 634: 621: 618: 615: 608: 604: 600: 595: 591: 587: 582: 578: 572: 567: 563: 538: 533: 529: 525: 520: 515: 511: 507: 502: 497: 493: 487: 473: 438: 434: 430: 427: 424: 421: 409: 386: 383: 369: 352: 350: 342: 334: 329: 324: 323: 309: 308: 305: 304: 300: 296: 292: 291: 287: 282: 277: 276: 273: 258: 254: 248: 245: 244: 241: 224: 220: 216: 215: 207: 201: 196: 194: 191: 187: 186: 182: 176: 173: 170: 166: 153: 149: 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 8911: 8906: 8904: 8900: 8848:WP:DUEWEIGHT 8842:MatthewMunro 8718: 8696: 8694: 8671: 8668: 8663: 8662: 8657: 8656: 8637:KharanteDeux 8574:KharanteDeux 8554: 8550: 8546: 8542: 8538: 8534: 8529: 8525: 8521: 8512:KharanteDeux 8497:KharanteDeux 8490: 8486: 8482: 8478: 8474: 8470: 8466: 8462: 8460: 8455: 8451: 8447: 8443: 8439: 8435: 8433: 8428: 8424: 8420: 8415: 8411: 8407: 8406:The vectors 8405: 8399: 8395: 8391: 8385: 8381: 8377: 8371: 8367: 8363: 8358: 8354: 8350: 8346: 8341: 8325:KharanteDeux 8322: 8317: 8314: 8309: 8303: 8070: 8066: 8061: 8052: 8048: 8044: 8031: 7873: 7863: 7854: 7805:— Preceding 7784: 7746: 7715:WP:NOTAFORUM 7379:ABCDSocrates 7376: 6961:Irrelevant; 6940: 6939: 6932: 6931: 6924: 6923: 6869: 4531: 4086: 4085: 3556: 3552: 1701:pseudovector 1671:— Preceding 1055:, naming it 635: 353: 346: 327: 293: 285: 272: 252: 212: 148:Mid-priority 147: 107: 73:Mid‑priority 51:WikiProjects 34: 8873:MathewMunro 8860:Jasper Deng 8827:MathewMunro 8822:Jasper Deng 8810:Jasper Deng 8799:MathewMunro 8784:MathewMunro 8780:Jasper Deng 8768:Jasper Deng 8746:MathewMunro 8730:MathewMunro 8578:MathewMunro 8256:tetrahedron 8248:begin with 8058:Real number 7723:Jasper Deng 7353:Jasper Deng 7219:Jasper Deng 7198:Jasper Deng 6963:WP:NOTFORUM 123:Mathematics 114:mathematics 70:Mathematics 8930:Categories 8756:Hodge dual 8040:quaternion 7811:Juliacubed 7801:This paper 7753:quaternion 7373:Dimensions 7190:isomorphic 7146:such that 8752:bivectors 8701:Jose Brox 8342:I delete 8250:In 1773, 8030:The term 7719:bivectors 7196:manner.-- 1705:bivectors 39:is rated 8715:vectors. 8560:D.Lazard 8444:physical 8440:physical 8436:physical 8262:D.Lazard 8218:D.Lazard 7985:D.Lazard 7870:vectors. 7819:contribs 7807:unsigned 7395:D.Lazard 7194:rigorous 4423:, where 4084:, where 1803:vectors 1728:product. 1685:contribs 1673:unsigned 328:365 days 286:Archives 8806:WP:RDMA 7761:Rgdboer 849:, then 255:on the 228:Physics 219:Physics 175:Physics 150:on the 41:B-class 8863:(talk) 8813:(talk) 8771:(talk) 8635:way.-- 8071:vector 8067:scalar 8032:vector 7726:(talk) 7709:Ujin-X 7686:Ujin-X 7356:(talk) 7314:Ujin-X 7285:Ujin-X 7201:(talk) 6994:(e.g. 6986:Ujin-X 6948:Ujin-X 3713:, then 3502:Ujin-X 2075:, then 1677:Ujin-X 47:scale. 8852:WP:OR 8533:, if 8318:space 8056:of a 7745:Read 4914:,then 1715:deeds 1608:axis. 28:This 8918:talk 8877:talk 8831:talk 8788:talk 8734:talk 8705:talk 8682:talk 8641:talk 8582:talk 8564:talk 8549:and 8537:and 8501:talk 8454:and 8414:and 8389:and 8359:must 8353:and 8329:talk 8320:. 8290:talk 8266:talk 8236:talk 8222:talk 8083:talk 8007:talk 7989:talk 7881:talk 7839:talk 7815:talk 7791:talk 7765:talk 7690:talk 7399:talk 7383:talk 7289:talk 6971:talk 6952:talk 3536:talk 3506:talk 3473:1000 3400:1000 3304:1000 1921:and 1681:talk 8766:.-- 8489:– 5 8485:+ 3 8481:= 2 8473:– 5 8469:+ 3 8465:= 2 8427:– 5 8423:+ 3 8403:. 6604:100 6411:100 4501:100 4367:100 4262:100 3241:100 3187:100 2748:100 2701:100 2607:100 1457:100 247:Mid 142:Mid 8932:: 8920:) 8907:If 8879:) 8833:) 8790:) 8736:) 8707:) 8684:) 8643:) 8584:) 8566:) 8528:= 8524:× 8503:) 8495:-- 8450:, 8410:, 8398:= 8394:× 8384:= 8380:× 8375:, 8370:= 8366:× 8349:, 8331:) 8292:) 8268:) 8238:) 8224:) 8198:≠ 8085:) 8051:+ 8047:= 8009:) 7991:) 7883:) 7841:) 7821:) 7817:• 7793:) 7767:) 7692:) 7669:¯ 7651:b̑ 7638:× 7629:ȃ 7627:, 7625:c̑ 7603:b̑ 7590:× 7581:ȃ 7579:, 7564:¯ 7549:¯ 7540:× 7534:¯ 7518:c̑ 7499:¯ 7490:× 7484:¯ 7455:¯ 7440:¯ 7431:× 7425:¯ 7401:) 7385:) 7291:) 7283:-- 7260:− 7253:¯ 7243:× 7237:¯ 7159:× 7081:^ 7064:− 7054:× 7016:× 6973:) 6965:. 6954:) 6942:b̄ 6934:c̄ 6926:d̑ 6899:10 6884:→ 6827:10 6818:⋅ 6815:10 6812:− 6806:⋅ 6713:− 6650:10 6622:10 6613:⋅ 6607:− 6601:⋅ 6598:10 6508:− 6417:10 6408:⋅ 6402:− 6396:⋅ 6303:− 6240:¯ 6215:¯ 6190:¯ 6165:¯ 6072:− 6032:¯ 5939:− 5899:¯ 5806:− 5464:¯ 5426:− 5388:¯ 5350:− 5312:¯ 5274:− 5210:n̑ 5175:m̑ 5140:l̑ 5111:× 5102:¯ 5074:¯ 5046:¯ 5012:d̑ 4999:× 4993:¯ 4970:d̑ 4957:× 4949:¯ 4929:¯ 4897:¯ 4888:− 4876:n̑ 4863:× 4857:¯ 4840:¯ 4831:− 4819:m̑ 4806:× 4800:¯ 4783:¯ 4774:− 4762:l̑ 4749:× 4743:¯ 4726:¯ 4708:n̑ 4695:× 4689:¯ 4672:¯ 4654:m̑ 4641:× 4635:¯ 4618:¯ 4600:l̑ 4587:× 4581:¯ 4546:¯ 4489:d̑ 4325:n̑ 4300:m̑ 4275:l̑ 4250:n̑ 4231:⋅ 4228:10 4225:− 4219:⋅ 4201:m̑ 4185:10 4182:⋅ 4176:− 4170:⋅ 4152:l̑ 4133:⋅ 4127:− 4124:10 4121:⋅ 4118:10 4103:d̑ 4096:n̑ 4092:m̑ 4088:l̑ 4064:n̑ 4029:m̑ 3994:l̑ 3959:n̑ 3923:− 3888:m̑ 3852:− 3817:l̑ 3781:− 3749:¯ 3740:× 3734:¯ 3716:d̑ 3692:10 3677:¯ 3650:10 3641:¯ 3610:¯ 3601:× 3595:¯ 3577:d̑ 3538:) 3508:) 3464:→ 3349:→ 3331:→ 3313:→ 3295:→ 3280:⋅ 3277:10 3274:− 3268:⋅ 3253:→ 3238:⋅ 3232:− 3226:⋅ 3211:→ 3196:⋅ 3190:− 3184:⋅ 3181:10 3169:→ 3125:→ 3100:→ 3075:→ 3050:→ 3018:− 2986:→ 2954:− 2922:→ 2890:− 2858:→ 2849:× 2843:→ 2828:→ 2787:→ 2727:→ 2616:→ 2598:→ 2580:→ 2562:→ 2547:⋅ 2541:− 2538:10 2535:⋅ 2532:10 2520:→ 2505:⋅ 2502:10 2499:− 2493:⋅ 2478:→ 2466:10 2463:⋅ 2457:− 2451:⋅ 2436:→ 2396:→ 2371:→ 2346:→ 2321:→ 2289:− 2257:→ 2225:− 2193:→ 2161:− 2129:→ 2120:× 2114:→ 2099:→ 2054:10 2039:→ 2012:10 2003:→ 1965:→ 1956:× 1950:→ 1935:→ 1906:→ 1897:× 1891:→ 1876:→ 1847:→ 1838:× 1832:→ 1817:→ 1776:→ 1761:→ 1752:× 1746:→ 1687:) 1683:• 1654:→ 1639:→ 1630:× 1624:→ 1547:≠ 1451:10 1448:⋅ 1445:10 1383:→ 1358:→ 1340:→ 1322:→ 1307:− 1275:→ 1257:→ 1230:→ 1202:× 1181:→ 1147:→ 1137:× 1125:→ 1100:→ 1013:→ 998:→ 989:× 983:→ 935:→ 900:→ 865:→ 828:→ 806:→ 784:→ 768:→ 743:→ 718:→ 651:→ 619:± 605:10 601:− 592:10 588:− 579:15 477:→ 428:10 422:10 413:→ 387:15 373:→ 301:, 297:, 8916:( 8875:( 8844:: 8840:@ 8829:( 8820:@ 8801:: 8797:@ 8786:( 8778:@ 8748:: 8744:@ 8732:( 8703:( 8680:( 8639:( 8580:( 8572:@ 8562:( 8555:k 8551:j 8547:i 8543:k 8539:j 8535:i 8530:k 8526:j 8522:i 8499:( 8491:k 8487:j 8483:i 8479:B 8475:k 8471:j 8467:i 8463:r 8456:k 8452:j 8448:i 8429:k 8425:j 8421:i 8419:2 8416:k 8412:j 8408:i 8400:j 8396:i 8392:k 8386:i 8382:k 8378:j 8372:k 8368:j 8364:i 8355:k 8351:j 8347:i 8327:( 8310:n 8288:( 8264:( 8234:( 8220:( 8204:, 8201:1 8193:2 8189:i 8168:. 8165:) 8162:1 8159:, 8156:0 8153:, 8150:0 8147:( 8144:, 8141:) 8138:0 8135:, 8132:1 8129:, 8126:0 8123:( 8120:, 8117:) 8114:0 8111:, 8108:0 8105:, 8102:1 8099:( 8081:( 8073:. 8062:s 8053:v 8049:s 8045:q 8005:( 7987:( 7969:3 7964:R 7937:3 7932:R 7910:k 7907:, 7904:j 7901:, 7898:i 7879:( 7837:( 7813:( 7789:( 7763:( 7711:: 7707:@ 7688:( 7666:c 7660:= 7612:= 7561:c 7555:= 7546:b 7531:a 7505:= 7496:b 7481:a 7452:c 7446:= 7437:b 7422:a 7397:( 7381:( 7333:3 7328:R 7316:: 7312:@ 7287:( 7271:0 7268:= 7263:1 7250:a 7234:a 7221:: 7217:@ 7171:c 7167:= 7163:b 7155:a 7133:b 7111:c 7108:, 7105:a 7078:a 7072:= 7067:1 7059:a 7050:a 7028:0 7024:= 7020:a 7013:) 7009:a 7005:k 7002:( 6988:: 6984:@ 6969:( 6950:( 6908:) 6905:0 6902:, 6896:, 6893:0 6890:( 6881:c 6855:0 6852:= 6845:0 6842:+ 6839:0 6836:+ 6831:2 6821:0 6809:0 6803:0 6796:= 6787:2 6782:z 6778:b 6774:+ 6769:2 6764:y 6760:b 6756:+ 6751:2 6746:x 6742:b 6734:x 6731:z 6727:d 6721:x 6717:b 6708:z 6705:y 6701:d 6695:y 6691:b 6683:= 6678:z 6674:c 6653:, 6647:= 6640:0 6637:+ 6634:0 6631:+ 6626:2 6616:0 6610:0 6591:= 6582:2 6577:z 6573:b 6569:+ 6564:2 6559:y 6555:b 6551:+ 6546:2 6541:x 6537:b 6529:z 6526:y 6522:d 6516:z 6512:b 6503:y 6500:x 6496:d 6490:x 6486:b 6478:= 6473:y 6469:c 6448:, 6445:0 6442:= 6435:0 6432:+ 6429:0 6426:+ 6421:2 6405:0 6399:0 6393:0 6386:= 6377:2 6372:z 6368:b 6364:+ 6359:2 6354:y 6350:b 6346:+ 6341:2 6336:x 6332:b 6324:y 6321:x 6317:d 6311:y 6307:b 6298:x 6295:z 6291:d 6285:z 6281:b 6273:= 6268:x 6264:c 6237:k 6229:z 6225:c 6221:+ 6212:j 6204:y 6200:c 6196:+ 6187:i 6179:x 6175:c 6171:= 6162:k 6155:) 6146:2 6141:z 6137:b 6133:+ 6128:2 6123:y 6119:b 6115:+ 6110:2 6105:x 6101:b 6093:x 6090:z 6086:d 6080:x 6076:b 6067:z 6064:y 6060:d 6054:y 6050:b 6042:( 6038:+ 6029:j 6022:) 6013:2 6008:z 6004:b 6000:+ 5995:2 5990:y 5986:b 5982:+ 5977:2 5972:x 5968:b 5960:z 5957:y 5953:d 5947:z 5943:b 5934:y 5931:x 5927:d 5921:x 5917:b 5909:( 5905:+ 5896:i 5889:) 5880:2 5875:z 5871:b 5867:+ 5862:2 5857:y 5853:b 5849:+ 5844:2 5839:x 5835:b 5827:y 5824:x 5820:d 5814:y 5810:b 5801:x 5798:z 5794:d 5788:z 5784:b 5776:( 5772:= 5746:2 5741:z 5737:b 5733:+ 5728:2 5723:y 5719:b 5715:+ 5710:2 5705:x 5701:b 5694:z 5690:b 5683:= 5679:′ 5675:z 5671:b 5659:2 5654:z 5650:b 5646:+ 5641:2 5636:y 5632:b 5628:+ 5623:2 5618:x 5614:b 5607:y 5603:b 5596:= 5592:′ 5588:y 5584:b 5578:; 5569:2 5564:z 5560:b 5556:+ 5551:2 5546:y 5542:b 5538:+ 5533:2 5528:x 5524:b 5517:x 5513:b 5506:= 5502:′ 5498:x 5494:b 5470:= 5461:k 5455:) 5450:x 5447:z 5443:d 5438:′ 5434:x 5430:b 5421:z 5418:y 5414:d 5409:′ 5405:y 5401:b 5397:( 5394:+ 5385:j 5379:) 5374:z 5371:y 5367:d 5362:′ 5358:z 5354:b 5345:y 5342:x 5338:d 5333:′ 5329:x 5325:b 5321:( 5318:+ 5309:i 5303:) 5298:y 5295:x 5291:d 5286:′ 5282:y 5278:b 5269:x 5266:z 5262:d 5257:′ 5253:z 5249:b 5245:( 5242:= 5222:= 5219:) 5195:x 5192:z 5188:d 5184:+ 5160:z 5157:y 5153:d 5149:+ 5125:y 5122:x 5118:d 5114:( 5108:) 5099:k 5092:′ 5088:z 5084:b 5080:+ 5071:j 5064:′ 5060:y 5056:b 5052:+ 5043:i 5036:′ 5032:x 5028:b 5024:( 5021:= 4989:′ 4986:b 4979:= 4946:b 4941:1 4935:= 4926:c 4894:k 4885:= 4854:i 4846:, 4837:j 4828:= 4797:k 4789:, 4780:i 4771:= 4740:j 4732:, 4723:i 4717:= 4686:k 4678:, 4669:k 4663:= 4632:j 4624:, 4615:j 4609:= 4578:i 4543:c 4516:) 4513:0 4510:, 4507:0 4504:, 4498:( 4471:x 4468:z 4464:d 4460:, 4455:z 4452:y 4448:d 4444:, 4439:y 4436:x 4432:d 4411:0 4408:= 4403:x 4400:z 4396:d 4392:, 4389:0 4386:= 4381:z 4378:y 4374:d 4370:, 4364:= 4359:y 4356:x 4352:d 4312:0 4309:+ 4287:0 4284:+ 4259:= 4237:) 4234:0 4222:0 4216:0 4213:( 4210:+ 4188:) 4179:0 4173:0 4167:0 4164:( 4161:+ 4139:) 4136:0 4130:0 4115:( 4112:= 4094:, 4090:, 4049:x 4046:z 4042:d 4038:+ 4014:z 4011:y 4007:d 4003:+ 3979:y 3976:x 3972:d 3968:= 3946:) 3941:z 3937:c 3931:x 3927:b 3918:x 3914:c 3908:z 3904:b 3900:( 3897:+ 3875:) 3870:y 3866:c 3860:z 3856:b 3847:z 3843:c 3837:y 3833:b 3829:( 3826:+ 3804:) 3799:x 3795:c 3789:y 3785:b 3776:y 3772:c 3766:x 3762:b 3758:( 3755:= 3746:c 3731:b 3725:= 3701:) 3698:0 3695:, 3689:, 3686:0 3683:( 3674:c 3668:, 3665:) 3662:0 3659:, 3656:0 3653:, 3647:( 3638:b 3607:c 3592:b 3586:= 3534:( 3504:( 3488:) 3485:0 3482:, 3479:0 3476:, 3470:( 3461:b 3438:0 3435:= 3430:z 3426:b 3422:, 3419:0 3416:= 3411:y 3407:b 3403:, 3397:= 3392:x 3388:b 3346:k 3340:0 3337:+ 3328:j 3322:0 3319:+ 3310:i 3301:= 3292:k 3286:) 3283:0 3271:0 3265:0 3262:( 3259:+ 3250:j 3244:) 3235:0 3229:0 3223:0 3220:( 3217:+ 3208:i 3202:) 3199:0 3193:0 3178:( 3175:= 3166:b 3122:k 3114:z 3110:b 3106:+ 3097:j 3089:y 3085:b 3081:+ 3072:i 3064:x 3060:b 3056:= 3047:k 3041:) 3036:x 3032:d 3026:y 3022:c 3013:y 3009:d 3003:x 2999:c 2995:( 2992:+ 2983:j 2977:) 2972:z 2968:d 2962:x 2958:c 2949:x 2945:d 2939:z 2935:c 2931:( 2928:+ 2919:i 2913:) 2908:y 2904:d 2898:z 2894:c 2885:z 2881:d 2875:y 2871:c 2867:( 2864:= 2855:d 2840:c 2834:= 2825:b 2784:b 2751:) 2745:, 2742:0 2739:, 2736:0 2733:( 2724:d 2698:= 2693:z 2689:d 2685:, 2682:0 2679:= 2674:y 2670:d 2666:, 2663:0 2660:= 2655:x 2651:d 2613:k 2604:+ 2595:j 2589:0 2586:+ 2577:i 2571:0 2568:= 2559:k 2553:) 2550:0 2544:0 2529:( 2526:+ 2517:j 2511:) 2508:0 2496:0 2490:0 2487:( 2484:+ 2475:i 2469:) 2460:0 2454:0 2448:0 2445:( 2442:= 2433:d 2393:k 2385:z 2381:d 2377:+ 2368:j 2360:y 2356:d 2352:+ 2343:i 2335:x 2331:d 2327:= 2318:k 2312:) 2307:x 2303:c 2297:y 2293:b 2284:y 2280:c 2274:x 2270:b 2266:( 2263:+ 2254:j 2248:) 2243:z 2239:c 2233:x 2229:b 2220:x 2216:c 2210:z 2206:b 2202:( 2199:+ 2190:i 2184:) 2179:y 2175:c 2169:z 2165:b 2156:z 2152:c 2146:y 2142:b 2138:( 2135:= 2126:c 2111:b 2105:= 2096:d 2063:) 2060:0 2057:, 2051:, 2048:0 2045:( 2036:c 2030:, 2027:) 2024:0 2021:, 2018:0 2015:, 2009:( 2000:b 1962:b 1947:d 1941:= 1932:c 1903:d 1888:c 1882:= 1873:b 1844:c 1829:b 1823:= 1814:d 1791:. 1773:k 1767:= 1758:j 1743:i 1679:( 1651:k 1645:= 1636:j 1621:i 1606:z 1590:′ 1586:z 1582:a 1559:′ 1555:z 1551:a 1542:z 1538:a 1515:z 1511:a 1487:′ 1483:z 1479:a 1454:= 1442:= 1437:y 1433:a 1427:x 1423:a 1419:= 1415:′ 1411:z 1407:a 1380:k 1372:z 1368:a 1364:+ 1355:j 1349:0 1346:+ 1337:i 1331:0 1328:= 1319:k 1313:) 1310:0 1302:y 1298:a 1292:x 1288:a 1284:( 1281:+ 1272:j 1266:0 1263:+ 1254:i 1248:0 1245:= 1242:) 1239:0 1236:+ 1227:j 1219:y 1215:a 1211:+ 1208:0 1205:( 1199:) 1196:0 1193:+ 1190:0 1187:+ 1178:i 1170:x 1166:a 1162:( 1159:= 1154:y 1144:a 1132:x 1122:a 1115:= 1111:′ 1107:z 1097:a 1072:′ 1068:z 1064:a 1041:z 1037:a 1010:k 1004:= 995:j 980:i 955:z 951:a 947:= 942:z 932:a 925:, 920:y 916:a 912:= 907:y 897:a 890:, 885:x 881:a 877:= 872:x 862:a 835:z 825:a 818:+ 813:y 803:a 796:+ 791:x 781:a 774:= 765:k 757:z 753:a 749:+ 740:j 732:y 728:a 724:+ 715:i 707:x 703:a 699:= 696:) 691:z 687:a 683:, 678:y 674:a 670:, 665:x 661:a 657:( 648:a 622:5 616:= 609:2 596:2 583:2 573:= 568:z 564:a 539:2 534:z 530:a 526:+ 521:2 516:y 512:a 508:+ 503:2 498:x 494:a 488:= 484:| 474:a 467:| 444:) 439:z 435:a 431:, 425:, 419:( 410:a 384:= 380:| 370:a 363:| 303:3 299:2 295:1 259:. 154:. 53::

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Index