I've been thinking of dot product as simply the product of one vector's magnitude with the component of another vector that is in the first vector's direction. Functionally, this is what the |a||b|cos(theta) expression does. I really like this thought process, because it shows why if we tried to use the dot product with 1 dimensional vectors, it would still work just like multiplication that we're used to. In 1 dimension, all vectors are parallel, so the cosine of the angle between them will always be 1 or -1. Unfortunately, the same reasoning doesn't quite seem to apply to cross product, so maybe I'm wrong about the relationship between vector multiplication and scalar multiplication. It did not occur to me to think of the cosine of theta as representing how much the two vectors are "working together", as if they were both force vectors. That's a nice way to think about it.
Beautiful explanation of the dot product - thank you so much for your visualization of the concept and overall passion for teaching!! I finally understand this.
I understood really not much. can you explain, what is the geometric interpretation of the dot product? is it a line? a magnitude of a line? or is it just a characteristic magnitude for two vectors? what is it?
Once you understand the dot product, you realize it just shows how similar in direction two vectors are. In other words, it is only a measure of similarity. It doesn't have direction, so it's not a vector. The video that helped me most understand what the dot product really is comes from Zach Star at ua-cam.com/video/TBpDMLCC2uY/v-deo.html I highly recommend it. 🙂
We live in a time where the internet is full of videos anout mathematics to find at our discretion, and for some reason this is the one that best helped me understand the dot product of vectors. I am grateful for this video and the fact that I can search through the internet until I find the teacher that helps it make sense to me 😊
Might be a bit late, but dot product is used to calculate Work in physics. We have two vectors, force and displacement, and we relate them by their angle. Work is used in the work energy theorem, which allows physics students to ditch kinematics when acceleration isnt constant. (I think, i honestly cant remember much of physics 1) Thats the only application i can think of for the dot product. ALSO GO WATCH 3B1B SERIES ON LINEAR ALGEBRA, HES GOOD AT TEACHING AND SHOWING THE CONCEPTS BEHIND VECTORS AND MATRICES.
The dot product shows how similar in direction two vectors are. This has many practical applications, and one of the videos that best talks about those applications comes from Zach Star ua-cam.com/video/TBpDMLCC2uY/v-deo.html
Are you with me Doctor Woo ? Are you really just a shadow of the man that I once knew ? Are you crazy..are you high ? Or just an ordinary guy. Have they really got to you ? Are you with me Doctor ?
I see nonstandard notation here that makes me wonder whether or not this is pedagogically wise. If these students arrived next year in a physics class using the Geometric Algebra notation developed by David Hestenes et al., would they pick up the new multivector notation quickly or feel confused by the polyglot? I've never seen tilde used to denote a vector before. Why no mention of the projection definition of the dot product?
I was always taught that the dot product a•b is the component of a in the direction of b. Basically, the magnitude of the vector projection of a onto b.
Teaching with passion. Love it
I've been thinking of dot product as simply the product of one vector's magnitude with the component of another vector that is in the first vector's direction. Functionally, this is what the |a||b|cos(theta) expression does. I really like this thought process, because it shows why if we tried to use the dot product with 1 dimensional vectors, it would still work just like multiplication that we're used to. In 1 dimension, all vectors are parallel, so the cosine of the angle between them will always be 1 or -1.
Unfortunately, the same reasoning doesn't quite seem to apply to cross product, so maybe I'm wrong about the relationship between vector multiplication and scalar multiplication.
It did not occur to me to think of the cosine of theta as representing how much the two vectors are "working together", as if they were both force vectors. That's a nice way to think about it.
This summation just helped me understand the answer so much better! Thanks!
@@stevedoetsch You're welcome! I'm glad you found it insightful.
Beautiful explanation of the dot product - thank you so much for your visualization of the concept and overall passion for teaching!! I finally understand this.
I understood really not much.
can you explain, what is the geometric interpretation of the dot product?
is it a line? a magnitude of a line? or is it just a characteristic magnitude for two vectors? what is it?
Great video but I still don't get why they are only scalers.
Once you understand the dot product, you realize it just shows how similar in direction two vectors are. In other words, it is only a measure of similarity. It doesn't have direction, so it's not a vector.
The video that helped me most understand what the dot product really is comes from Zach Star at ua-cam.com/video/TBpDMLCC2uY/v-deo.html I highly recommend it. 🙂
Please do cross product I am begging you please
At 14:50, Eddie casually makes the quadratic formula make sense in about 10 seconds, and suddenly the thing becomes intuitive...
nice try ;-)
We live in a time where the internet is full of videos anout mathematics to find at our discretion, and for some reason this is the one that best helped me understand the dot product of vectors. I am grateful for this video and the fact that I can search through the internet until I find the teacher that helps it make sense to me 😊
Great as always, Mr Woo!
I love the pushing the car analogy
Dot Product is fun to prove
so beautifully setup Eddie. high five
But what does Dot Product represent? And what problem does it solve?
Might be a bit late, but dot product is used to calculate Work in physics. We have two vectors, force and displacement, and we relate them by their angle. Work is used in the work energy theorem, which allows physics students to ditch kinematics when acceleration isnt constant. (I think, i honestly cant remember much of physics 1)
Thats the only application i can think of for the dot product.
ALSO GO WATCH 3B1B SERIES ON LINEAR ALGEBRA, HES GOOD AT TEACHING AND SHOWING THE CONCEPTS BEHIND VECTORS AND MATRICES.
The dot product shows how similar in direction two vectors are. This has many practical applications, and one of the videos that best talks about those applications comes from Zach Star ua-cam.com/video/TBpDMLCC2uY/v-deo.html
@@patricktanoeyjaya4430what about cross product? since the vectors involved are perpendicular to each other
Thank you for this! You are amazing.
Are you with me Doctor Woo ?
Are you really just a shadow of the man that I once knew ?
Are you crazy..are you high ?
Or just an ordinary guy.
Have they really got to you ?
Are you with me Doctor ?
That “independent” thing is kinda sketchy, some might confuse it with linear independence.
Now I actually know what a dot product means, this is great
Hello , I ❤ your teaching by the way , do you have a phd in math?
❤❤❤
اسمي كمان علي ☺
I see nonstandard notation here that makes me wonder whether or not this is pedagogically wise. If these students arrived next year in a physics class using the Geometric Algebra notation developed by David Hestenes et al., would they pick up the new multivector notation quickly or feel confused by the polyglot? I've never seen tilde used to denote a vector before.
Why no mention of the projection definition of the dot product?
No offense but having the student speak adds zero value to the lesson
I was always taught that the dot product a•b is the component of a in the direction of b. Basically, the magnitude of the vector projection of a onto b.