The Physical Meaning of the Cross Product and Dot Product

[@FractalWoman]

I really like this video. Not only is it extremely informative, but it made me laugh out loud. A good way to start my day. Thanks.

[@Inductica]

Thanks!

[@dialectphilosophy]

Teaching the cross product through torque is a pretty smart way to go about it! Torque is (for the most part) fairly intuitive -- you have to push orthogonally to some lever or bar to rotate it about a pivot point, so that explains why you have y components multiplied by x components and vice-versa. The "minus" seems to come from the fact a rotation can be split into an "up and over" (counter-clockwise) or an "over and down" (clockwise) motion, which requires the moving components be oppositely signed. Still always some frustrating sense of abstraction that seems to linger when we use vectors, but that's hard to avoid. Thanks for another great video!

[@Inductica]

I really appreciate it! A comment I would make is that my goal in my explanation videos is not to reduce abstraction nor to make things intuitive. Abstractions are not bad or hard to understand if all the observations and reasoning steps required to see that they are true are explained. Similarly, intuitions are not something we should try to appeal to, because they aren't necessarily correct. I think "making things intuitive," or, "less abstract," are approximations for what we really need in an explanation: a complete connection to observational evidence.

[@Serghey_83]

У тебя 100% - ная эталонная дикция. Отчётливо слышно каждое слово. Для изучающих американский английский это идеальный вариант.

[@MrBumbo90]

great visual and instinctive explanation. Thanks a lot for your effort.

[@Inductica]

Thank you, though I try not to make the explanation instinctive, rather, I try to connect the ideas to observations.

[@ryanjbuchanan]

The best explanation of a dot product that I've heard is that it's basically like a Mario Kart turbo boost

[@Inductica]

Let's a-go!

[@user-lu6yg3vk9z]

@@Inducticareal question is why people who are going into healthcare have to learn this?

[@Inductica]

@@user-lu6yg3vk9z I'm not sure, but it probably relates to government involvement in education. For a healthcare professional getting an advanced degree, like his MD, there is a decent argument that he should learn advanced mathematics to train his mind to think in a systematic, disciplined way.

[@romado59]

@@user-lu6yg3vk9z If your a tech in MRIs or Pets scan you might need give you some understanding?

[@allenanderson5529]

The cross product is the determinant of a 3x3 matrix, where row 1 is x-hat, y-hat, z-hat. Row 2 is Ax, Ay, Az. Row 3 is Bx, By, Bz.

[@tomctutor]

Yes that's a very good way to remember it. The dot product is simply A⁺ B where A⁺ means transpose matrix. (A vector is simply a column nX1 matrix remember). This pictorially looks like..⍈ ⍗. The matrix representation allows easy change of basis (we do not need to stick to *i,j,k* thing) and can compute the result. We can further determine vector triple products and scalar triple products with ease using matrices.

[@anirbanmukhopadhyay6902]

A beautiful explanation of the dot product is here. Thank you.

[@BuckPowers]

Loving the humor bits. Just the right amount. And nice editing for those bits, as well. Humor can easily die in a bad edit. But you nailed it. This content also dovetails well with the angular momentum lecture. Including some portion of this as a sidebar might even make that lecture more effective.

[@Inductica]

Thank you very much!

[@bpark10001]

What you (& everyone else explaining this) are missing is WHERE the name "cross product" comes from. I ran into this when helping guy write CAD circuit board layout program. There is requirement to calculate the distance between a point (X,Y) & a line segment (X1,Y1), (X2, Y2). The calculation boils down to translating everything until one end of the line is at (0,0), then taking cross product of vectors (0,0) (X2',Y2') & (0,0) (X',Y'). These form angle which when extended form a parallelogram. The distance is the height of the parallelogram, which is the area divided by the base (the base is SQRT of dot product of line vector with itself). The area is the cross product of the 2 vectors (Result is scalar because we are working in the plane.) (So you can also describe the cross product as the area of a parallelogram formed by the 2 vectors in plane containing the 2 vectors.) When I worked out this formula, the terms have X1Y2 & X2Y1 in them (as your formula also shows). THE PRODUCT TERMS ARE CROSSED! This is my theory where the name came from. What do you think?

[@AlbertTheGamer-gk7sn]

The dot product is also used in matrix multiplication. Vector dot products is equal to multiplying a row matrix by a column matrix. For example, ∙ = [[1, 2]] * [[5], [7]] = 5+14=19. Dot products are derived from projections, where proj(a, b) = [(a ∙ b)/(b ∙ b)]*|b|. Cross products, however, comes from the cross-operation sequence. The cross operation involves taking a vector or a group of vectors and outputting a vector that is orthogonal to all vectors being used. For example, a vector in 2D can be crossed to find its perpendicular vector, which proves the perpendicular slope formula, and vectors in 3D can have cross products with 2 vectors, vectors in 4D with 3 vectors, and so on. Area can be interpreted by a cross product of 2 length vectors, as A = bh, with b being a length vector and h being the perpendicular component of the second length vector. Volume can be interpreted by using 3 vectors and using the 4D cross product, as V = Bh, where B is the area of the base, a cross product itself, and h being a perpendicular component of the third vector, so V = Bh = (r ⨉ r)h r ⨉ r ⨉ r (as h = r⊥), but in our 3D world, volume can also mean the DOT product of length and area, due to the box product. Finally, comes the interpretation of cross products in Flatland. We all know that in Flatland, angles exist, so rotations exist. 2D shapes and planar laminae have rotational inertia, so angular momentum and torque exists in Flatland, but since Flatlanders cannot really see the objects rotating due to a 1D vision, they usually don't think about torque, as the torque will be bending into the 3rd dimension. We 3D beings can see objects rotate about an axis, but we cannot interpret solid angular motion. This is because solid angular momentum is changed by 3-torque, which is equal to r ⨉ A ⨉ F, which goes into the 4th dimension. However, 4D beings can comprehend solid angular velocity and objects rotating about a plane rather than an axis. Finally, comes the 2nd moment of area, which is equal to A ⨉ A, or (r ⨉ r) ⨉ (r ⨉ r). This requires 6 dimensions, as the first cross product gives 3 dimensions, and the second gives 3 more dimensions.

[@EnriqueAnt.Raudales]

thank you. The lack of linear algebra in this video was annoying, and i still do t unferstand the connection between the angles shown in the generalized form and the way the operacions are done between column vectors or taking the discriminant of a group of vectors put together

[@tomctutor]

@@EnriqueAnt.Raudales That is why Linear Algebra is such a rich source of information. We think of spaces made up of groups of vectors, the algebra defines how we can manipulate these spaces to get answers to otherwise intractible problems,e.g. as mentioned in ATG's post. Can use Linear Algebra methods to work in other number-fields like complex spaces and so on. Not so academic as used a lot in Electrical Engineering, Quantum Mechanics, Relativity to name a few.

[@spelunkerd]

Outstanding review!

[@Inductica]

Thanks!

[@hrishikeshaggrawal]

cross product makes sense intuitively, but when I ask someone for the n'th time what the cross product is and they start explaining the formula i really do go to sleep. 10/10 direction

[@Inductica]

You are exactly the kind of person I was trying to reach with this video. Thanks!

[@jeanlucas2834]

I've never seen it like this before, even in books. Thank you! Make more videos like this.

[@Inductica]

Thanks! My inductive videos, which will begin in two weeks, will explain every concept of math and physics in this kind of way!

[@xninja2369]

Tell me you haven't read any books without telling me you haven't read any books 😂

[@Inductica]

@@xninja2369 i’d be interested to know which textbook actually explains the cross product in the way I do. Not trying to nail you, just actually curious. I checked pretty thoroughly to make sure there wasn’t another video that explained it this way before making this.

[@jeanlucas2834]

@@xninja2369 Haha good one 😂. I mean, learning physics by operational definition is easy but it's not the physics you want, is it? In my opinion we constantly need to innovated and find new connections between things we already know. A lot of mathematical tools that were created without any practical applications have found a place in physics because someone tried to explain something differently and it worked, that is awesome. Oh boy, you just don't know me. But read isn't enough. We read novels but books like these we have to study, practice and apply or you're wasting your time. If you are here I believe you think books sometimes isn't enough, because by the end of the day the books were made by professors like him.

[@scienceclick9092]

Nowadays, learning mathematical physics depends a lot on books. In some books the way a law/formula is derived that it seems really tough to understand. When I first learned about vectors from book I was fully confused. But when I changed book it was not so difficult for me to understand. The proof of theories are written in such a way that you dont have to be a very high IQ person to understand it on that book. While in the first book it was really really tough to understand. So books are my first priority to learn mathematics for physics

[@stanbleszynski8825]

Dot products and cross products are two components of general vector product representing superposition of operators in Clifford algebra.

[@nitinjain1605]

Thanks a lot ... Very intuitive ... I always had issue with cross produc, why someone came up with such a weird type of product but it makes sense now ... while watching your video I was able to imagine and understand the crux behind Cross product as well as Dot products ...

[@Inductica]

Thank you! I should mention though that my purpose was not to appeal to intuition, but rather to show you the actual evidence behind the concept. Once you see the evidence behind an idea, you might find it intuitive and simple to understand, but the real goal is to connect these ideas to reality, not just to make it feel natural.

[@roberttawiah3693]

I love this video its really good

[@dinsefateshome8412]

welcome back boss

[@Inductica]

Thanks chief!

[@ricshaw4358]

excellent explanation

[@Inductica]

Thanks!

[@profhalimboutayeb]

Excellent!

[@kottybeats]

Good explanation, well done

[@Inductica]

Thank you very much!

[@timothyjohnson1511]

Vortrix algebra used to describe Etheral Mechanics by Robert Distinti overcomes weaknesses of the Cross and Dot Product.

[@munchingdragon6]

this was fun to watch!

[@Inductica]

Thanks!

[@gary.richardson]

In track and field, runners time based on distance also accounts for headwinds and tailwinds. I imagine these dot products come into play here.

[@Inductica]

Yes, I think it would.

[@sinfinite7516]

Great video!!!!

[@antomarioni]

muy buen video, muchisimas gracias

[@sinfinite7516]

Hmm I’m looking at the rest of your channel and I wish you did more content like this.

[@Inductica]

@@sinfinite7516 noted!

[@bjornfeuerbacher5514]

The explanation starting at 4:00 is a bit similar to the one I'm using when I'm teaching this, but simpler - I'll try if I can incorporate this into my own teaching, thanks! :) (My own way of doing it goes like this: first I argue, using the angle formula, that for two parallel vectors, the dot product just gives the product of their magnitudes, and for two orthogonal vectors, the dot product is zero. Then I decompose the vectors A and B into their components along the axes, similar to what you are doing, and then simply multiply out the two sums and use the facts I showed before in order to calculate the dot products of the coordinate vectors with each other.) However, a crucial step is missing here: For that argument to work, you first have to show (or at least give an argument in words) why the dot product is distributive, i. e. why the dot product of a sum of vectors with another vector is the same as the sum of all the dot products of the partial vectors with the other vector. I tried to gave an argument for that in my own lectures, but unfortunately, it's in German. Would you like to have a link to that argument anyway?

[@Inductica]

That's a good point. I did think about that while writing the script and decided I didn't need to explain that, but revisiting this, I think I do. Yeah, let's see that video!

[@bjornfeuerbacher5514]

@@Inductica I don't have a video, only a text document. I try to provide the link, but probably KZbin will delete it. :/ www.feuerbachers-matheseite.de/Eigenschaften_des_Skalarprodukts.pdf

[@markszlazak]

Yours is a description or definition as opposed to a real derivation which you won’t find in math of physics books unless you know where to look. Cross products and the rest come from quaternions which were then simplified to vectors and their operators. Quaternions are complicated (see Eater and 3blue2brown Visualizing Quaternions) but recently there was a derivation of the dot and more complicated cross product from a linear combination of vectors that was published in 2018. "The linear combination of vectors implies the existence of the cross and dot products" by Jose Pujol. International Journal of Mathematical Education in Science and Technology Volume 49, 2018 - Issue 5.

[@poet.in.flight]

Such a fun video 🎉

[@kisho2679]

When/where were first the cross/dot products "invented"?

[@Inductica]

Good question, I actually don't know!

[@andrewseary]

@@Inductica Quaternions. See "A History of Vector Analysis", Crowe

[@APaleDot]

They arise naturally in the study of Quaternions, which are 4D complex numbers. Quaternions were invented by Hamilton before our modern concept of a vector, but then Gibbs and Heaviside took the 3D component of the Quaternions alone and broke apart the multiplication into a scalar part and a vector part (the dot-product and the cross-product)

[@perseus4hire210]

If rxFy and ryFx are both producing torque in the negative z direction, why is one subtracted from the other? Also, what is the significance of the negative sign for the 'j' vector?

[@APaleDot]

They're not. One is producing clockwise torque while the other produces counterclockwise torque. The negative sign in the 'j' component comes from the particular way the coefficients were ordered in that formula. They chose to order them alphabetically (so rxFz - rzFx instead of rzFx - rxFz), but since a - b = -(b - a), this produces a stray negative sign.

[@zyansheep]

Dot Product + Cross Product = Geometric Product

[@tomctutor]

Thinking outa the box, scalar + vector =? You might try ( AB= A.B + iAxB, imaginary 'i' here) this is actually used in relativistic mechanics as it can be shown that |AB| = √{(A.B)² - |AxB|²}

[@ukaszbaranowski1006]

I like very much when someone gives those intuitions for mathematical concepts and this video is of great help, but actually it produced even more questions eg. as of why is a torque actually a cross product - after doing some resarch I am still not sure. Some people say that it is for a part a useful convention but on the other hand it seems to actually be a product of strict mathematics.

[@Inductica]

If you watch carefully, you will see that I explain why a cross product is required for torque, the full physical meaning is explained.

[@ukaszbaranowski1006]

​@@Inductica As I understand so far: it is helpful to present torque as a vector perpendicular to radius and linear force as this way it represents the axis along which rotation takes place. The rotation also have a direction and so this is another commonality with a vector. Whether the vector points up or down depends on a convention. Applied force and radius are also vectors as they have direction and magnitude and so when you multiply them you get another vector and this is consistent with describing torque as a vector positioned along the axis of rotation. Is there more to it? I will rewatch the video in some time, but now I can not. There is still some doubt in me. I do understand that if you multiply the vectors it follows from equations that you get another vector perpendicular to them. But I am still not sure if it is actually a convention to describe torque as this vector pointing up or not. If you consider something like gyroscopic precession it seems like there is some deeper meaning to it. I believe that you think that it represents physical reality. I am not sure if I know what my problem is, maybe it just seems that it was just a convinient way to represent such a motion this way and it happened to work but in fact it makes perfect sens and is not just a convention, maybe I have just to sleep on it. Naming some other reason for which I might still doubt it: I also saw a lot of people on the internet say that representing it this way is a convention, say that cross product does not work in higher dimensions, on wikipedia it says that torque is actually a Pseudovector and I dont know what it means. I just started my first year at a university, also english in not my first language, maybe this will become more clear later on.

[@Inductica]

@@ukaszbaranowski1006 Your summary is great. I would add the part of this video where I talk about why a cross product adds up the torques for each direction 11:10. In my view, this justifies why torque is a cross product. It is true that there is more to the story that relates to gyroscopic precession as you say. Good luck with your continued studies and let me know your additional thoughts!

[@ukaszbaranowski1006]

​@@Inductica Thanks for response! Gyroscopic precession was what sparked my interest in the subject - I came across an explanation regarding representation of torque as a vector pointing up that said that you just want to point to the area that is spinning using that vector and show am axis of rotation, and it sounded to me like it is just a convention useful in just naming what is happening, but when I read about gyroscopic precession it really seemd as if the vector of torque is REALLY following a vector of angular momentum, like it was hard to not think about those arrows as something representing a thing that exsist in physical reality. So your video was very helpful for me when I was trying to think about all of it. I think your youtube channel is great, I defenitly think that learing about physics should involve building some intuitions about what is happening in math and not just learing to write proofs etc, so I believe content like this will be very helpful in my studies. Sorry for responding after a month, but I was busy and was thinking that maybe I will come up with something more to say and thus was procrastinating with writing it.

[@hansfrancsco71]

Is there like a book that would help learn k-12 mathematics conceptually instead of rote memorization from government schooling?

[@ryantellez2871]

Mathnasium is pretty good if you want to homeschool a kid.

[@Inductica]

The Singapore method and the Japanese method are really good for the early years. The Japanese method is more inductive, but that might only work when you have an actual teacher trained in their school system; the Singapore method might be better if you are just teaching yourself out of a book (less inductive though) I would sample both if I were you. Neither are perfectly inductive.

[@pixelapse9613]

5:05 How my math teacher teaches in class using power point

[@rainerzufall42]

Oh, I've almost expected Quaternions, but that would be longer than 16:35...

[@Serghey_83]

(·) product - скалярное произведение (×) product - векторное произведение

[@heronimousbrapson863]

Mathematics: how to make simple concepts horrifically complicated.

[@bradleygaddis5155]

Right on, as I like to say they really enjoy keeping the club small

[@alphalunamare]

7:35 This is briliant :-) I'd make it easier to understand though and label i^ as x^, j^ as y^ and k^ as z^.

[@РайанКупер-э4о]

Instead of the cross product you should really use the geometric product and the bivectors from the geometric algebra. Also bivector lies in the plain of the rotation, not in some random axis.

[@Inductica]

It would be interesting to know the physical meaning of those concepts of in geometric algebra, but the cross-product has a straightforward physical meaning that we can hold in mind to understand it, and it works for many applications. We don't need something fancy in cases where something simple will suffice.

[@Inductica]

And part of my point is that the cross-product does not lie in a random direction, it lies along the axis of rotation!

[@РайанКупер-э4о]

@@Inductica, geometric algebra isn't fancy. It's straight forward and intuitive. The multivectors are so good you can do calculus directly with them. The torque bivector that we will get by multiplying the r with the F will be numerically same as the vector that we will get with the cross product but it will have so much better and useful algebraic properties that after using it once you will never use cross product again. The geometric algebra is just so good you need to try it and you will love it. There are a swift introduction to it on KZbin, it's short and it presents you with applications, in the end you'll see how Maxwell equations become just one simple differential equation with one differential operator and two multivectors and it is computable in this form it will blow your mind how easy all the math becomes.

[@РайанКупер-э4о]

@@Inductica, I know that cross product lies on the axis of rotation, I know how it works, I've learned it in school and uni. You don't have axis of rotation in 2 or 4 dimensions. But you'll always have the plain of rotation in any number of dimensions where rotation is possible. That's why that the object describing rotation should line in the plane of rotation and not on the axis of rotation and that's what bivectors give you. Bivector is a part of a plane that has area and sign same way as a vector is a part of a line with length and sign. This works so well you'll love it, please look into it.

[@Inductica]

@@РайанКупер-э4о That's very interesting! I've watched the swift introduction and found it interesting. Perhaps one day I'll do my own video on it if I find it to be essential to my project. Thanks for telling me about it!

[@gabe-d8u]

question: where does the minus in the equation at 13:36 come from?

[@Inductica]

Good question, it comes from the fact that if Fx is positive, and rx is positive, then the torque produced by that force across that arm will be negative (into the screen.)

[@jnhrtmn]

Cross product works analogously for torque AFTER a right-hand rule, so everything after a right hand rule is void of any mechanism. Maxwell's equations use a cross product between electricity and magnetism, which means no one understands anything mechanically between those two. Modern science is happy merely describing what they see it do. "The wheels on the bus go round and round." This song describes everything you see a bus do exactly like math does, but you certainly can't claim to understand the bus with that "description," and an accurate analogy can be completely fake making fools out of lots of people. The variables in gravity math are not causal, and that's clue #1 to the Universe. Look at my gyro explanation to see what causality looks like.

[@aek03030731]

Why is it called the dot product? Also, the only cross product I remember is used in comparing fractions.

[@Inductica]

@@aek03030731 These are concepts which are covered in a 3rd semester calculus class.

[@cleison-em7bk]

Its only a physics example . Main idea i guess is when a thing(vector of any physical variable) gets maximum or 0 , with cross or dot product

[@jimmonroe5801]

The wedge product is superior to the cross product in every possible way. Why the wedge product is not taught boggles my mind.

[@bernardofitzpatrick5403]

😂 nice one ! Subd

[@Inductica]

@@bernardofitzpatrick5403 thanks!

[@oversquare6625]

Stop - you are hurting my head. At 1:54 you show F*Cos*D, but you say F*D*Cos, then at 2:24 you show F*D*Cos and correlate it to A*B*Cos which is algrabracially parallel to D*F*Cos. While sure this is a commutative property of multiplication, it turns your example into a conceptual train wreck. Did you see what I did there? its a train joke. But seriously, it needlessly confuses an otherwise simple idea.

[@kptib1988]

sticking the middle finger.. in the F direction..

[@solapowsj25]

Collide dot products and charge cross ❌products.

[@Li.Siyuan]

Spoiled by not mentioning the significance of the i hat, j hat and k hat components in the solution to the cross product formulae. Shame.

[@MechLearn-360]

That's nothing, it just a notation for writing vectors. If is typed then you bold it rather.

[@krelly90277]

There's a type of person who has to complain about everything. The video is well-done, but this commenter has to complain. Instead of complaining, make the video you want.