Review of the Dot Product

[@MathTheBeautiful]

Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.

[@Stefabro]

Your lectures are criminally underrated, you know exactly what to say and how to phrase to elicit inherent curiosity,into the subject! You are doing so much good for the world having this content free on youtube! Wish i had more professors with your deep understanding of the subject, as well as the know-how into how to express a mathematical concept/idea so as to develop motivation to keep learning! Thank you!

[@MathTheBeautiful]

Thank you for your kind words. It's deeply appreciated!

[@TheBuilder]

probably the best video on the dot product on the internet

[@MathTheBeautiful]

"Probably"?

[@ycombinator765]

@@MathTheBeautiful probability = 1.0 tho ;)

[@krishnadasbhagwat5785]

Kudos to "MathTheBeautiful", the first time i saw this lecture on why inner products, i was floored! The lectures are testimony to reveal the beauty and power of Math !! Kudos !!

[@romitabiswas5490]

i am blessed. thank you ... you saved me. i was so helpless and could not find any channel or person that would help me like yours did

[@MathTheBeautiful]

Thank you for letting me know - it means a lot!

[@darrenpeck156]

Beautiful introduction of the inner product. Thank you!

[@priovag2632]

Very dedicated work on this channel. Thank you very much.

[@mominanoor3882]

i dont know why people dislike such a great content

[@kailasnathastro]

Superb Professor. I have not met with such a great Professor 😊

[@MathTheBeautiful]

Thank you and nice to meet you too!

[@louis9116]

The fact that the instructor confused the terms 'inner product' and 'dot product' for the whole duration of this lecture and didn't notice that during the talk proves that these can be used interchangeably and nothing will happen. But great lecture as always :)

[@MathTheBeautiful]

Yes, indeed!

[@gguevaramu]

Professor you said that vector(a) dot vector (b) doesn´t have geometric interpretation, and I agree, but in physical way it has meaning. Just to show an example, if vector (a) = Force and vector (b) = diplacement, then their dot product mean Work = Energy. I think dot product concept has it origin in physics

[@mikhailveselov7733]

energy is an abstract notion

[@zukofire6424]

at last I find an explanation of the scalar product that answers my questions and more! Thanks for this!

[@MathTheBeautiful]

Glad it was helpful!

[@mittenbear]

This is a smart guy.

[@scitwi9164]

03:18 Not true at all. There _is_ geometric meaning behind it: The `len(a)·cos(γ)` part is the projection of the vector `b` onto the direction of vector `a` to see how much of it lies in that direction (that is, its _component_ in that direction). If we denote this component of `a` in the direction of `b` as `a_b`, then the dot product is simply `len(b)·len(a_b)`, and we can now easily multiply these lengths, since they already lie in the same direction (so it's equivalent to multiplying numbers). So it's not that this particular combination was chosen at random just because this was the only that worked :q But I liked what you said before: that in geometry, lengths and angles were "givens", and the dot product has been derived from them, while when we then try to generalize this concept, we define the analogous inner product for something else, and kind of "climb down the tree" to derive the corresponding notions of length and angle in the new area of application. That's how I see it. It doesn't mean that inner products are "more important" than lengths and angles, or the other way around - they're equally important. If we've been given the corresponding notions of lengths and angles for polynomials, we wouldn't have to do that gimmick with inner products at all. But since these notions were not given, we had to first "climb up" the generalization tree, constructing the notion of inner product, and then "climb down" to the new branch to recreate the analogous lengths and angles from the inner product.

[@AlexandreG]

haha lol this guy wasting his time writing this thinking someone ever will read that little book about how the professor is wrong and hes right...

[@camilomartinez5452]

chill

[@biesman5]

@@AlexandreG He is right, tho.

[@AlexandreG]

@@biesman5 You really spent 5 minutes reading all that, props to you, and him too I guess 😅

[@electric_sand]

Thing is, to get the projection in the direction of a, in the general case if a is not a unit vector, you have to divide by the length of a...and the length of a is not included in the formula for the dot product. That's why he said - "the dimension is wrong".

[@paulobezulle]

ExcellentTeacher!!!!!ThankYouSoMuch!!!!

[@frankdearr2772]

Hello, without cos it more fast and easy like: VecA*unitVecB*unitVecB ,that is all, and we get a vector projection of VecA in direction of VecB . but there so many ways to get projection. Main is it sticks to your way you want :) Thanks and please have a great day Laurent

[@None-ss1zi]

Oh my, first time I saw the dot product my very first thought was that it doesn't make any sense

[@mauriciovinco6143]

Isn't the invention of scalar product due to Hamilton's Quartenions? So it has an author, I guess.

[@MathTheBeautiful]

Do you have a specific reference in mind? I'd love to see it!

[@mauriciovinco6143]

So, the oldest reference is the treatise "On Quaternions; or on a new System of Imaginaries in Algebra" by Hamilton. Vector multiplication is featured in the 21 paragraph. It may be no the oldest overall but this is the version I stick to.

[@danialdunson]

you've earned my sub. great videos

[@MathTheBeautiful]

Thank you! I worked hard for that sub.

[@soulmansaul]

Very good videos. Well explained, thanks! 👍

[@Hythloday71]

No, is the answer, it is deeply uncomfortable for some reason thinking in terms of geometric vectors. Strange, you would think it would be most intuitive. I guess trigonometrical conversion to Cartesian co-ordinates is so ingrained that expressing 'the way to get to' say X is by Y - Z is deeply unintuitive. I've always had a problem with it.

[@nipunsachinda]

super teaching...

[@paracyber.x6797]

mind pothondi raa rei..........su sir