The Physical Meaning of the Cross Product and Dot Product

00:00 Introduction
00:30 Work and the Dot Product
3:01 Proof of the Dot Product Formula
4:52 Torque and the Cross Product
7:35 Tutoring Advertisement
7:54 The Right Hand Rule
9:36 The Physical Meaning of the Torque Vector
11:09 Physical Proof of the Cross-Product Formula
15:36 Summary
16:12 Outro

[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!

[ryanjbuchanan]

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

[Serghey_]

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

[allenanderson]

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.

[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.

[bpark]

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?

[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

[jeanlucas]

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

[anirbanmukhopadhyay]

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

[AlbertTheGamer-gksn]

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> = [[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.

[scienceclick]

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

[nitinjain]

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 ...

[stanbleszynski]

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

[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.

[sinfinite]

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

[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.

[kisho]

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

[perseushire]

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?

[timothyjohnson]

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

[bjornfeuerbacher]

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?