Talk:Cross product

message: Currently, the section Computing the cross product expresses unit vectors using color notation. I think this should be change. First, the color is nowhere referenced in the expository paragraphs surrounding these formulae. Second, the colors conflict with the figure accompanying this section. Third, usage of color may cause accessibility issues, particularly for those who are color blind or with poor vision. Fireballs619 (talk) 00:16, 6 June 2019 (UTC)
thread_title:  Usage of color in equations 

message: 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? ABCDSocrates (talk) 07:13, 11 June 2019 (UTC)
replies: text: 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 that use these definitions. D.Lazard (talk) 09:36, 11 June 2019 (UTC)
replies: text: In fact, there was no choice for cross-product \bar{a}\times\bar{b}=\bar{c}. 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 \bar{a}\times\bar{b}= consistent with vector theory. And the remaining 3 \bar{a}\times\bar{b}=\bar{c}, \times=, \times=\bar{c} , will receive the status of uncoordinated cross product. And will be used only in geometric constructions.--Ujin-X (talk) 10:42, 19 June 2019 (UTC)
replies: text:  This talk page is WP:NOTAFORUM 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 bivectors, which however are outside the scope of this article.--Jasper Deng (talk) 10:46, 23 June 2019 (UTC)
replies: text: Read A History of Vector Analysis to see that the two "products" were taken from the quaternion 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 coquaternions, which provides an alternative mathematical model for appropriate uses. — Rgdboer (talk) 22:56, 23 June 2019 (UTC)
thread_title:  Dimensions 

message: 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. --Bill Cherowitzo  (talk) 22:35, 13 November 2019 (UTC)
replies: text: This paper 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.  — Preceding unsigned comment added by Juliacubed (talk • contribs) 20:20, 13 November 2019 (UTC)
replies: text: 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. Wikipedia does not publish "nice new ideas", only those things that have been accepted by the wider scientific community (in this case). If this preprint gets published and reviewed, then this may be looked at again. --Bill Cherowitzo  (talk) 22:35, 13 November 2019 (UTC)
thread_title:  Alternative visual notation 

message: I delete the two sentences 
 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 n is given by the left-hand rule and points in the opposite direction.
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 space.  
Hence, the end of the paragraph had to be modified.--KharanteDeux (talk) 15:56, 6 June 2021 (UTC)
thread_title:  Definition: deletion of two sentences 

message: I delete
 
 So by the above relationships, the unit basis vectors i, j and k of an orthonormal, right-handed (Cartesian) coordinate frame must all be pseudovectors (if a basis of mixed vector types is disallowed, as it normally is) since ,  and . 
The vectors i, j and k  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 2i + 3j – 5k a pseudovector?  
"Axial" and "polar" are physical qualifiers for physical vectors, that is to say vectors which represent physical quantities such as the velocity or the magnetic field. The vectors i, j and k  are mathematical vectors, neither axial nor polar. In mathematics, the cross-product of two vectors is a vector. It's that simple! 
So, we can say that the position vector is r = 2i + 3j – 5k and the magnetic field is B = 2i + 3j – 5k without any contradiction.
--KharanteDeux (talk) 15:10, 8 June 2021 (UTC)
replies: text: I agree with the deletion, although the error in the sentence is not exactly what  wrote. The truth is that the inner product 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 , if i and j are considered as vectors, then k must be considered as a pseudovector. If i and j are considered as pseudovectors, then k 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 sentence is definitively an error. D.Lazard (talk) 17:22, 8 June 2021 (UTC)

text: @KharanteDeux, do you want to delete this talk section, since the words you wanted deleted are now long-gone? MathewMunro (talk) 08:57, 15 February 2024 (UTC)
replies: text:  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 way.--KharanteDeux (talk) 15:05, 9 June 2021 (UTC)
thread_title:  Cross product and handedness: deletion of few lines 

message: Cross product appears 159 times.
Cross-product appears 5 times.
I came to wikipedia to find the most common practice.
I'm not sure google ngrams is giving me what I want. Universemaster1 (talk) 09:44, 23 May 2022 (UTC)
thread_title:  Consistent hyphenation needed within the same page 

message: 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 Multilinear algebra, 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? Jose Brox (talk) 10:22, 15 November 2022 (UTC)
thread_title:  Generalized cross product and n-Lie algebra structure 

message: 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: https://www.youtube.com/watch?v=9GOoiaQq4GY, and I suggest using a graphic like this one: https://sites.science.oregonstate.edu/~hadlekat/COURSES/ph213/gauss/images/areaVector.JPG, 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. MathewMunro (talk) 05:39, 15 February 2024 (UTC)
replies: text: No. You are conflating the cross product with bivectors, of which the cross product is the Hodge dual. The "area vector" you link is also not (directly) related; it is simply the (oriented) surface element that appears in a surface integral.--Jasper Deng (talk) 05:41, 15 February 2024 (UTC)
replies: text: @Jasper Deng, it seems to me that depicting a quantity that in fact has squared units as a simple arrow is far more conflating. MathewMunro (talk) 06:18, 15 February 2024 (UTC)
replies: text:  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 a long-settled question; you're not going to be changing the consensus here. If you still have questions, WP:RDMA is that way.--Jasper Deng (talk) 06:33, 15 February 2024 (UTC)
replies: text: @Jasper Deng 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 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. MathewMunro (talk) 08:18, 15 February 2024 (UTC)
replies: text:  Ultimately, the current contents of the article are what the community has deemed to be WP:DUEWEIGHT. Any of what you're proposing is WP:OR or simply would not be due weight. The Poynting vector's direction has a clear physical meaning and is defined using a cross product. Continue your discussion at the reference desk, please.--Jasper Deng (talk) 08:21, 15 February 2024 (UTC)
replies: text: 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. MathewMunro (talk) 08:32, 15 February 2024 (UTC)
thread_title:  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 vectors.