I've been looking at a lot of videos and forums looking for the practical meaning of a dot product. Every video and forum I found only defined what dot product is and they were very technical about it. This video gave me the answer after 5 seconds. Thank you!
You're videos are amazing! Really helped me understand super daunting subjects. You're ability to break these topics down and explain them in a straightforward way is so awesome!!!
Wow! I just started linear algebra and was already confused as to why dot products matter. The way you explain this is so good and you can really make out the effort in the videos. I can't believe I haven't seen your content before! Subbed right away!!
Outstanding explanation of the dot product! This is the first video-and I've seen plenty- that's helped me get an intuition for this concept. Thanks so much!
Wow. Finally i understand whats the point of the dot product. Should be called "similarity product" or something to point to it's meaning. Can't believe i went on all this years not understanding this basic idea.
Amazing! Our human brain is evolved such a way that we understand anything very easily if it is intuitive & your techniques are so intuitive and gripping that anyone who is very weak in maths can learn by heart the very concept behind "dot product". In that way, your techniques can be called "compatible with evolution". Richard Feynman also had this ability. Go ahead!
It's truly awesome (in every aspect) the videos you make. You don't need to feel bad about taking some time off. For me at least your videos are a gift and it is not fair with you that we demand anything. That's just my long way to say thank you!
How deeply you explained it!...OMG!...This is the only video in youtube I came across, to know more deeply dot product and it's significance...You are really doing great. 🌈
With gems like this we can partially forget or tolerate the crap that youtube is filled with. Thank you for the video and please keep making. How about PDEs or probability?
Yay i found another great math channel!, thank you very much now i can understand this not like those "explanations" that basically say the definition and the properties
finaly i understand the dot product i have been looking for this kind of an explanation for so long but only now i realy understand it thanks thanks thanks!!!
WOW. This video is better than any classes I had in school or college. I'll watch again later and do the "youtube homeworks" in paper and not half-way in my mind. Thanks
I know nothing about filming but this does look good! :) Also, it's so good to finally learn the meaning behind this formulas I've known for years. Thank you!
For Homework 2: If we take v_b as the unit vector on its axis, from the previous exercise we know that x = v_u . v_b / ||v_b|| and y = v_v . v_b / ||v_b|| Now we add v_u and v_v v_u + v_v = (x+y) * v_b + perp We apply the same theorem on x+y as on x and y (x+y) = (v_u + v_v) . v_b / [[v_b|| Now replace x and y by their expressions (the ||v_b||s cancel out) v_u . v_b + v_v . v_b = (v_u + v_v) . v_b (1) This shows the distributivity of the dot product (v_a * k) . (v_b . s) = ||v_a * k|| * ||v_b*s|| * cos(theta) = k * s * ||v_a|| * ||v_b|| * cos(theta) = k * s * v_a . v_s This shows the scalar multiplication property of the dot product (not sure if that's the right term lol) v_u = alpha * v_v1 v_v = beta * v_v2 Now we replace v_v and v_u in the formula (1) (alpha * v_v1 ) . v_b + (beta * v_v2) . v_b = (alpha * v_v1 + beta * v_v2) . v_b From last property this is the same as: alpha * v_v1 . v_b + beta * v_v2 . v_b = (alpha * v_v1 + beta * v_v2) . v_b This shows the dot product is bilinear Homework 2 finished
وَحَدَّثَنَاهُ قُتَيْبَةُ بْنُ سَعِيدٍ، عَنْ مَالِكِ بْنِ أَنَسٍ، عَنْ أَبِي بَكْرِ بْنِ نَافِعٍ، عَنْ أَبِيهِ، عَنِ ابْنِ عُمَرَ، عَنِ النَّبِيِّ صلى الله عليه وسلم أَنَّهُ أَمَرَ بِإِحْفَاءِ الشَّوَارِبِ وَإِعْفَاءِ اللِّحْيَةِ . This hadith is in Sahih Muslim. Growing beard is a wajib act and not growing it is a sin.
I realised the depth of the info you share, not just the rules of it. I love it. Just a recommendation, I think it would be much better if it was less distracted. I wanted to focus on sth, but I just kept looking here and there on the necessary graphs, forgetting the point.
Great video but I have a question. I like how using unit vectors creates a range of -1 to 1 when rotating one of the two parallel vectors wrt the other. But if the lengths of the two parallel vectors is 5, then the range is -25 to 25. Using your definition of the dot product gives "The dot product shows how much of a vector is in the direction of the other. Therefore 5i dotted with 5i...500% of one vector is in the direction of the other vector. Does that sound a little odd? Thanks!
Answer for the last question: So we have vectors a and b that can be written as a1x + a2y for vector a and b1x and b2y for b. Given the linearity formula for the dot product. a•b = (a1x + a2y)•(b) = a1x•b + a2y•b Substitute for b and we get: a1x•(b1x + b2y) + a2y•(b1x + b2y) a1x•b1x + a1x•b2y + a2y•b1x + a2y•b2y The dot products of orthogonal vectors is 0 so the equation simplifies to: a1x•b1x + a2y•b2y Since the are facing in the same directions, the is the same as multiplying their magnitudes together. Thus a•b = a1b1 + a2b2 ...
you are great....the animation is great, the moment trig enters DHAN TAN TAAAAA... i laughed very hard..😂😂😂😂 i'll never forget this for my life ,,,, I m very glad i watched it. thank you for this video, it explains very neatly and clearly the basic. I found exactly what i was looking for
Thank you! I wasn’t planning to, since 1)I don’t know the answer (though it could be fun to figure it out) 2) the cross product doesn’t generalise nicely like the dot product. Sorry!!
Is it usual to write the dot product with the dot at the bottom? Everyone I know always writes it with the dot in the center (in LateX this would be \cdot).
Yes, an identical unit vector in the opposite direction should be minus. (And this means the dot product has to be Zero). I think I get it! I really like your question, to what extent are the two vectors going the same way.
dot product between one orthonormal basis vector with itself comes out to be 1 because 1*1*cos 0 =1 and with any other orthonormal basis vector it becomes 0 because they are orthogonal i.e. cos 90. Therefore we get the neat formula to multiply corresponding coordinates to get dot product between 2 vectors. Thanks I really had that doubt why this formula works.
I have a question, isn’t the dot product also used for how much two vectors are alligned? How can the formula also use this definition and how much these are pointing the same way
I would be so so grateful if you answer my question. 2:56 how is the x ( -1) and not (1) ... I mean.. the magnitude couldn't be negative one .. it makes sense that the direction is neg 1 but then x is denotes magnitude.. I'm confused ..ease help me dear
Hi. The video started with: We want some way to describe - how much 2 vectors are pointed in same direction? And answer is dot products helps in achieving that. But why can't we use simply angle between 2 vectors for the same thing. (I am not talking about cos(theta) ) simply angle. Reason why we can't use angle in this case is, it will work in 2D but not in higher dimension. So we need some other way to describe it. Is this geometrically correct statement?
If the dot product of two vectors can be thought of as a measure of how much the two vectors are pointing in the same direction, would there be any significance to generalizing this idea to n vectors?
I’ve never seen such a thing, but maybe you could? You can at least certainly generalise the dot product for n dimensional vectors, but that’s not the same thing. So yeah, I don’t have a great answer, sorry!
I was using whiteboard cleaner and it ruined my board. Now nothing but toothpaste really works. It’s really ugly in real life, but I just put the contrast on high so you don’t see it
OK here's a weird thought. As you draw a vector in various positions around a circle, I notice that the dot product is not unique. 0 (zero) occurs twice as the vector rotates in a circle, once at 90 degrees and once at 270 degrees. But the vector is in different positions!! So are there different zeros here? Is zero not always equal to zero? It seems like there should be some way to distinguish between those different positions.
Gojira There are, but not with the dot product. We go from two vectors of two or more dimensions, to a single real number, so some information gets lost: we can’t distinguish between an angle and its full circle complement, or between acute and obtuse angles, or between negative and positive angles. The dot products tells us how much the vectors point in the same way, but not where they initially pointed.
Really nice observation/ question, and great answer. That’s exactly right, the dot product doesn’t uniquely determine where the other vector is relative to the first. Come to think of it, that’s why when you want to specify a vector wrt a orthonormal basis, you need specify the dot product of it with every basis vector
To handle with nonuniqueness, assuming all the vectors are in the same quadrant may help. Just a triggered thougt, why nonnegativity is so important for recommendation systems. Since all components have positive value, comparing two client's rankings by using dot product is a well suited approach.
the reason the dot product increases the more similar two vectors are and is smaller the less similar they are is because: 1. you can breakup each vector into the same basis components that can have opposite magnitude pieces of each other when two vectors point in different directions. 2. those negatives cancel out a lot in the calculation with the positives
9:40 hold on how does this make sense at all. we know that a•b is the part of a in the direction of b, right? so it's a bit like, where the projection of a on b is, that's the vector a, minus the "orthogonal to b" bit. so why does the length of b matter? if I make b very large, and have a stay the same, wouldn't the projection land in the same spot? and if I make a really big, wouldn't it then land on a different spot instead? and also, why is it logical at all that you have to devide by the length of b? since, how exactly does its length change the projection?
Why don't you write 1/√2 as (√2)/2 ? It's easier to understand, and more flagrant with the remarkable trigonometric value than just 1/√2. Though I appreciated this video, twas really helpful, 'cuz it's not ony about understanding the formulas and applying these, it's also about understanding what does it mean. Thanks to you, I did
I've not afraid of trig or anything (I really love math in general) but the flashy flashy of a right triangle at around 3:30 seemed kinda creepy to me.
Наука
Just for the record, I do notice the effort you put into these videos. You're doing great! Keep it up!
Thank you, that’s very very kind of you :)
Looking Glass Universe yeah, please know it doesn’t go unnoticed by us!
I really really appreciate that :)
IM SO GRATEFUL FOR THESE textbooks arent doing much for me with numbers and formulas :') so thank you mam
Honestly, I wasn't expecting those pauses in the video, but turns out, they helped a lot with just grasping the concept and understanding. Thanks!
This is incredible, taking a concept that confused me to no end and helping me understand in 13 min. Thank you so much
I've been looking at a lot of videos and forums looking for the practical meaning of a dot product. Every video and forum I found only defined what dot product is and they were very technical about it. This video gave me the answer after 5 seconds. Thank you!
Suddenly discovered this channel. Loving it!
The amount of creativity in this video , the amount of efforts you have put in editing this. Salute. 🔥🔥
You're videos are amazing! Really helped me understand super daunting subjects. You're ability to break these topics down and explain them in a straightforward way is so awesome!!!
Wow! I just started linear algebra and was already confused as to why dot products matter. The way you explain this is so good and you can really make out the effort in the videos. I can't believe I haven't seen your content before! Subbed right away!!
Outstanding explanation of the dot product! This is the first video-and I've seen plenty- that's helped me get an intuition for this concept. Thanks so much!
Finally a video on RUclips that gives a really intuitive visualisation of the dot product. Well done. Thank you.
Thanks so much!!
The first 8 seconds literally answered a question I've been trying to solve for the last 20 minutes with multiple websites and textbooks.
Wow. Finally i understand whats the point of the dot product. Should be called "similarity product" or something to point to it's meaning. Can't believe i went on all this years not understanding this basic idea.
I cant even imagine how much work you did for this video , its great
literally the thumbnail alone explained it better than everyone else
this is actually so sick, a lot of work went into this and it really did help me
Amazing! Our human brain is evolved such a way that we understand anything very easily if it is intuitive & your techniques are so intuitive and gripping that anyone who is very weak in maths can learn by heart the very concept behind "dot product". In that way, your techniques can be called "compatible with evolution". Richard Feynman also had this ability. Go ahead!
It's truly awesome (in every aspect) the videos you make. You don't need to feel bad about taking some time off. For me at least your videos are a gift and it is not fair with you that we demand anything. That's just my long way to say thank you!
Thanks so much :D!
How deeply you explained it!...OMG!...This is the only video in youtube I came across, to know more deeply dot product and it's significance...You are really doing great. 🌈
Ain't u gonna extend the playlist?
I loved ur explanation.
With gems like this we can partially forget or tolerate the crap that youtube is filled with.
Thank you for the video and please keep making. How about PDEs or probability?
Wow the production quality on this vid is crazy! Thank you for the great explanation too!
Yay i found another great math channel!, thank you very much now i can understand this not like those "explanations" that basically say the definition and the properties
finaly i understand the dot product i have been looking for this kind of an explanation for so long but only now i realy understand it thanks thanks thanks!!!
New favorite math channel! wow this was the style of explanation I've been searching for, such a helpful conceptual take.
Amazing videos. This really helped me understand the dot product.
Definitely subscribing! This is an amazing channel!!! Very well explained :)
WOW.
This video is better than any classes I had in school or college.
I'll watch again later and do the "youtube homeworks" in paper and not half-way in my mind.
Thanks
Thank you very much! Did you try the homeworks :)?
Thank you so much! You're an extremely talented teacher.
This is the best explanation to dot product. Thank u
While studying I listened to an old creepy prof. For hours and hours.
And you run that down in less that 14 min AND with your sweet voice...thanks :)
Excellent video, and I loved the toothpast tip!
We have been missing you! Every video you make is a little gem.
Thank you so so much- that means the world to me.
This channel is actually amazing
This video was awesome. A beautiful lecture that help the viewer better understand a confusing topic.
Excellent video and explanation. I applaud your efforts.
May God bless you!
This is beyond great.
I know nothing about filming but this does look good! :) Also, it's so good to finally learn the meaning behind this formulas I've known for years. Thank you!
Yay! Thanks for noticing the 1% (probably lower) increase in video quality.
I am madly in love with your voice!
Every comment section of every female youtuber ever. Very cliche way of simping.
For Homework 2:
If we take v_b as the unit vector on its axis,
from the previous exercise we know that
x = v_u . v_b / ||v_b|| and y = v_v . v_b / ||v_b||
Now we add v_u and v_v
v_u + v_v = (x+y) * v_b + perp
We apply the same theorem on x+y as on x and y
(x+y) = (v_u + v_v) . v_b / [[v_b||
Now replace x and y by their expressions (the ||v_b||s cancel out)
v_u . v_b + v_v . v_b = (v_u + v_v) . v_b (1)
This shows the distributivity of the dot product
(v_a * k) . (v_b . s) = ||v_a * k|| * ||v_b*s|| * cos(theta) = k * s * ||v_a|| * ||v_b|| * cos(theta) = k * s * v_a . v_s
This shows the scalar multiplication property of the dot product (not sure if that's the right term lol)
v_u = alpha * v_v1
v_v = beta * v_v2
Now we replace v_v and v_u in the formula (1)
(alpha * v_v1 ) . v_b + (beta * v_v2) . v_b = (alpha * v_v1 + beta * v_v2) . v_b
From last property this is the same as:
alpha * v_v1 . v_b + beta * v_v2 . v_b = (alpha * v_v1 + beta * v_v2) . v_b
This shows the dot product is bilinear
Homework 2 finished
You're a great teacher.
Excellent video, thank you for your hard work.
I cant tell exactly how gladful im after this point of view, i thought about it a lot and now thinga makes more sense!
Yay, she's back!
Amazing, you are incredible!
Amazing. Thank you very very much.
Underrated video. Thank you
Kudos! You are an amazing teacher! Well done. Keep up the good work.
وَحَدَّثَنَاهُ قُتَيْبَةُ بْنُ سَعِيدٍ، عَنْ مَالِكِ بْنِ أَنَسٍ، عَنْ أَبِي بَكْرِ بْنِ نَافِعٍ، عَنْ أَبِيهِ، عَنِ ابْنِ عُمَرَ، عَنِ النَّبِيِّ صلى الله عليه وسلم أَنَّهُ أَمَرَ بِإِحْفَاءِ الشَّوَارِبِ وَإِعْفَاءِ اللِّحْيَةِ .
This hadith is in Sahih Muslim. Growing beard is a wajib act and not growing it is a sin.
I totally forgot about this and we started using dot products for the first time since calc 3, thank you!
Great!! So happy to help :)
perfect didactic and editing, youre very talented :)
It was a really nice one :) ,it actually gave me a clear picture of dot product
I realised the depth of the info you share, not just the rules of it. I love it. Just a recommendation, I think it would be much better if it was less distracted. I wanted to focus on sth, but I just kept looking here and there on the necessary graphs, forgetting the point.
Great video but I have a question.
I like how using unit vectors creates a range of -1 to 1 when rotating one of the two parallel vectors wrt the other. But if the lengths of the two parallel vectors is 5, then the range is -25 to 25. Using your definition of the dot product gives "The dot product shows how much of a vector is in the direction of the other. Therefore 5i dotted with 5i...500% of one vector is in the direction of the other vector. Does that sound a little odd? Thanks!
Answer for the last question:
So we have vectors a and b that can be written as a1x + a2y for vector a and b1x and b2y for b. Given the linearity formula for the dot product. a•b = (a1x + a2y)•(b)
= a1x•b + a2y•b
Substitute for b and we get:
a1x•(b1x + b2y) + a2y•(b1x + b2y)
a1x•b1x + a1x•b2y + a2y•b1x + a2y•b2y
The dot products of orthogonal vectors is 0 so the equation simplifies to:
a1x•b1x + a2y•b2y
Since the are facing in the same directions, the is the same as multiplying their magnitudes together. Thus
a•b = a1b1 + a2b2 ...
Nicely done! Thanks for typing it all out :)
Holy crap I just wanted to understand dot product for my Physics class but I somehow ended up in Linear Algebra.
This is the best intuitive explanation for the dot product .Can you please do the same for cross product
So well explained. I appreciate it 🙂
Are you a theoretical physicist
you are great....the animation is great, the moment trig enters DHAN TAN TAAAAA... i laughed very hard..😂😂😂😂 i'll never forget this for my life ,,,, I m very glad i watched it. thank you for this video, it explains very neatly and clearly the basic. I found exactly what i was looking for
You deserve more views
Thanks for this video. I searched for this
Love the vid as always!!! : )
Will you do the cross product too? I actually don't know why it works...
Thank you! I wasn’t planning to, since 1)I don’t know the answer (though it could be fun to figure it out) 2) the cross product doesn’t generalise nicely like the dot product. Sorry!!
I never knew I had all these problems. But luckily I can solve (most) of them! 🧠
Yeah I take some time to get that it is cos but where did you get this concept ,
For 8:09, if x = u cos theta, then why isn't y = u sin theta? For u.v2 , wouldn't the angle be 90 - theta? Thanks.
Is it usual to write the dot product with the dot at the bottom? Everyone I know always writes it with the dot in the center (in LateX this would be \cdot).
Thror251
There’s many ways
Some write it like this
@Mi Les yup :) I write it down because I use \cdot for multiplication sometimes. But it doesn't matter too much!
@@duckymomo7935
Or .
@@LookingGlassUniverse
I use \bdot for the dot product, but I also use bold symbols for vectors. I define it as:
\usepackage{amsmath}
ewcommand{\bdot}{\boldsymbol\cdot}
and for the cross product
ewcommand{\cross}{\boldsymbol\times}
Oh, that’s a nice way to do it!
Awesome video !
Thank you!!
this type of teaching is very good and very useful for beginners .
Thanks very much!
Well done.
YaY you are back!!
Sorry for disappearing!
Can we reboot the series, starting with the cross product? 👀
This is great.
3:35 No God! No God Please no. No! NO! NOOOOO!
lol the toothpick thing is relatable. I use chalk to represent a 3d vector on a chalkboard sometimes :D
Haha! Do you teach vectors?
Yes, an identical unit vector in the opposite direction should be minus. (And this means the dot product has to be Zero). I think I get it!
I really like your question, to what extent are the two vectors going the same way.
dot product between one orthonormal basis vector with itself comes out to be 1 because 1*1*cos 0 =1 and with any other orthonormal basis vector it becomes 0 because they are orthogonal i.e. cos 90. Therefore we get the neat formula to multiply corresponding coordinates to get dot product between 2 vectors. Thanks I really had that doubt why this formula works.
Thanks for not explaining homework one now I have no clue how to do the rest
wth this video is so good
put a hat on the vectors so that you don't have to remember you have made them unit vectors ;)
6:10 Awwww, I like your toothpick vectors!
Haha! At least there’s one.
really nice video !
Thank you!
I have a question, isn’t the dot product also used for how much two vectors are alligned? How can the formula also use this definition and how much these are pointing the same way
Nice video, curl next?
Perfect
please mam make some videos about Isospin ,Hyperchage (particle physics)
I would be so so grateful if you answer my question.
2:56 how is the x ( -1) and not (1) ...
I mean.. the magnitude couldn't be negative one .. it makes sense that the direction is neg 1 but then x is denotes magnitude..
I'm confused ..ease help me dear
You are awesome...or at least your videos are!
The animations 😍
Hi. The video started with:
We want some way to describe - how much 2 vectors are pointed in same direction?
And answer is dot products helps in achieving that.
But why can't we use simply angle between 2 vectors for the same thing. (I am not talking about cos(theta) ) simply angle.
Reason why we can't use angle in this case is, it will work in 2D but not in higher dimension. So we need some other way to describe it.
Is this geometrically correct statement?
If the dot product of two vectors can be thought of as a measure of how much the two vectors are pointing in the same direction, would there be any significance to generalizing this idea to n vectors?
I’ve never seen such a thing, but maybe you could? You can at least certainly generalise the dot product for n dimensional vectors, but that’s not the same thing. So yeah, I don’t have a great answer, sorry!
I just watched is spinning angular momentum video,
When particle does larmor precession why don't its angular momentum go on increasing?
I don’t know, what do you mean?
Doesn't almost organic solvent you buy at the chemist's work for wiping off the marker from the white board?
I was using whiteboard cleaner and it ruined my board. Now nothing but toothpaste really works. It’s really ugly in real life, but I just put the contrast on high so you don’t see it
Shouldn't the answer be Cos . a? Because cos = x/a, right?
OK here's a weird thought. As you draw a vector in various positions around a circle, I notice that the dot product is not unique. 0 (zero) occurs twice as the vector rotates in a circle, once at 90 degrees and once at 270 degrees. But the vector is in different positions!! So are there different zeros here? Is zero not always equal to zero? It seems like there should be some way to distinguish between those different positions.
Gojira There are, but not with the dot product. We go from two vectors of two or more dimensions, to a single real number, so some information gets lost: we can’t distinguish between an angle and its full circle complement, or between acute and obtuse angles, or between negative and positive angles. The dot products tells us how much the vectors point in the same way, but not where they initially pointed.
Really nice observation/ question, and great answer. That’s exactly right, the dot product doesn’t uniquely determine where the other vector is relative to the first. Come to think of it, that’s why when you want to specify a vector wrt a orthonormal basis, you need specify the dot product of it with every basis vector
To handle with nonuniqueness, assuming all the vectors are in the same quadrant may help. Just a triggered thougt, why nonnegativity is so important for recommendation systems. Since all components have positive value, comparing two client's rankings by using dot product is a well suited approach.
This is really great, once in a while I look at a dot product and I'm like "why the heck sr cos() again?"
Where is the next video?
the reason the dot product increases the more similar two vectors are and is smaller the less similar they are is because:
1. you can breakup each vector into the same basis components that can have opposite magnitude pieces of each other when two vectors point in different directions.
2. those negatives cancel out a lot in the calculation with the positives
9:40 hold on how does this make sense at all. we know that a•b is the part of a in the direction of b, right? so it's a bit like, where the projection of a on b is, that's the vector a, minus the "orthogonal to b" bit. so why does the length of b matter? if I make b very large, and have a stay the same, wouldn't the projection land in the same spot? and if I make a really big, wouldn't it then land on a different spot instead? and also, why is it logical at all that you have to devide by the length of b? since, how exactly does its length change the projection?
-1
Cos
u • V1
a • b
Why don't you write 1/√2 as (√2)/2 ? It's easier to understand, and more flagrant with the remarkable trigonometric value than just 1/√2.
Though I appreciated this video, twas really helpful, 'cuz it's not ony about understanding the formulas and applying these, it's also about understanding what does it mean. Thanks to you, I did
The intro is hilarious
I've not afraid of trig or anything (I really love math in general) but the flashy flashy of a right triangle at around 3:30 seemed kinda creepy to me.