nice teaching with very basic concept unlike many just simply dump out formula without explaining :D thumbs up for that

This lecture is really better than my university professor. I really appreciate it.

Amazing video. Every 13 years later this has explained projections better than my linear algebra book (admittedly skipped all calculation instruction in favor of waiting until inner products were defined, but then it just went straight to subspaces!). Thank you for your amazing information, my ancestors praise you

Love, love, love these videos!!! Can't express how grateful I am for them. Can't wait to finish these and pursue more and more. So generous of Sal and the team! One thing that I hope people find interesting is that there's another way of arriving at the formula for the scalar multiplier of v for the projection of x on L. I think of it as first determine the length of the projection, then ensure it's directed appropriately. The length to me is simply x cos(theta) where theta is the angle between x and v. But we know that x cos(theta) is then just v dot x divided by the length of v, which in the numerical example yields 7/sqrt(5) as the required length of the projection using his example vectors for v and x. Then to find the right direction of the projection vector, it must be some scalar c times the starting vector of L, v, to yield the projection. And c times v must have that length v dot x divided by the length of v which we calculated as being 7/sqrt(5) using his example vectors as stated above. Solving for c gives you v dot x divided by (the length of v times the length of v) which is just (v dot x) divided by (v dot v) which is exactly what Sal arrived at as well with the resulting vector being [14/5, 7/5]. He just got there using the difference of vectors and leveraging the resulting normal vector to the line and this version outlined above doesn't, felt a bit more intuitive to me, just focusing on the projection itself using cosines.

Thank you for the series of these amazing lectures. This helped me a lot to brush up and understand more about linear algebra, so a big thank you to you Sal!!

I don't think I have ever been so surprised how easy a concept could be explained. I've struggled with understanding formulas and the books explanation of projections and with info from this video noted down. I'll never forget it :) Thanks for guiding us all toward a more rich understanding of math Sal & khan team

Thank for the upload. My instructor's explanation was a little tough to follow, and I am usually able to understand him pretty well, but your video helped out a lot.

thanks sal it was killing me not knowing where the equation for a projection came form

I succeeded in my uni only thanks to Khan Academy!!! Ty so much! Love, Hannabeth

@christoferkalda1396

Жыл бұрын

tuupur Christofer

I needed a refresher (like this one) on projections for me to write a hit detection/test on an arbitrarily-oriented rectangle.

Linear algebra is my favorite topic. As I am engineer , this topic is the fundamental of infinite element method or called an application of it. Thinking of the car we drive daily, we actually benefit from this topic .

Love it ! Its worth mentioning that V-hat is a normalized vector and that the scalar c when multiplied by a normalized vector (v hat) gives us the vector projection of x onto v. cheers, b

You're like one million times better at explaining this than my teachers!! Thank you alot, I'd be so lost if it wasn't for Khan Academy :D

Great video!!! Helps a lot!!

thank you for your awesome explanations.

awesome, I went through three other videos and they threw me a magic scalar formula without explaining why :) sal never disappoints

Thank you Khan Academy for your free online courses!

thanks man that was really easy to understand

Thnx a lot SIR....I really foumd it utile...wish the best from you

thank you!!!! Really helpful!

Love the graphs

PERFECTLY EXPLAINED.

as usual- life saver!

Thanks for helping me in my EE class :D

Thank you so much.

Thanks a lot!

Could you also find the angle between the two vectors and then just multiply the vectors magnitude by Cos(theta)?

its very helpful thx

I wish my professor explained it this way.. 4 wasted lectures lmao, instead of a simple 14 mins explanation

Thank you Soo much.....

Thank you for the video, it is great. My only suggestion would be to add the tag: orthogonal.

Thank you

mate ur a legend

Old but gold ✨✨

I watch explanations of you only.....as you explain what I want exactly,but even one video is there like this....which explains what I want

Impressed

Never buy books of projection matrix, just listen what the man are talking about!!Thanks again.

thanks

Linear Algebra Comeback season

Sal, QR decomposition uses this process. What is QR used for? Thanks,

Even me did understand :D Thanks

You make calc so much easier :)

@jahitrst2863

4 жыл бұрын

This isn't calc

@oni8337

2 жыл бұрын

@@jahitrst2863 he still does in his other vids tho

Gracisas.

Dont forget to mention about youtube n khan academy for your graduation speech!

I'm going through Eric Lengyel's book " Mathematics for 3D game programming & computer graphics". I've been staring at his explanation of vector projection and it's not very well explained. This video has been a great additional resource. Thanks Sal!

@ssshj

3 жыл бұрын

hey 10 yrs ago i was 10 , lol,

@newbarker523

3 жыл бұрын

@@ssshj Blimey - and I was 40. How time flies! (I am LAnonHubbard by the way but use that account rarely now).

I've done this out on graph paper and x - proj(x) does not = that orthogonal line. Just by looking at your drawing you can see that. I've watched all the videos and I have no idea how you came up with that. And what is v? Is that just arbitrary? Can someone please help??

@meeharbin4205

5 жыл бұрын

The projection of x going in the same direction as v with length ||x||*cos(theta), where theta is the angle between the vectors. That would cause x, proj(x) and proj(x) to make a right angled triangle. v is a given vector.

I always thought Sal sounded quite different in most of his videos older than 3 years, but in this video he sounds almost as if it were made in 2012 or 2013.

Thank you Sal :)

@ssshj

3 жыл бұрын

really 11 yrs ago , 11 yrs ago i was 9 , lol , now i have my exam in 3 days

Thank you! my book gives me the formula for finding projections but doesn't show how the formula is derived. This helped a lot!

@ssshj

3 жыл бұрын

8 yrs ago woahhhhhhhh i have exam in 3 days lol

@56jmoney

3 жыл бұрын

@@ssshj Good luck. I don't remember any of this.

Just wondering, does anyone know why he multiplied the projection equation by the original vector v at the end?

@MyNguyen-xu4oj

5 жыл бұрын

Well, you should carefully watch the video again. That original vector v comes from the original formula: L=c*V

Please upload Linear Block Codes tutorial

doing gods work sal doing gods work.....

thank you thank you thank u thank u

Orthogonal, I'd been calling that function "rightangle" the way I had implemented it.

Salman may God rewards you Jenna

I got very confused for a while thinking that a vector dot itself is 1. But that's only true with unit vectors, in other cases the dot is the square of the magnitude. So, the area of the magnitude, like a square box coming off the vector. Which now has me thinking: what does the determinant look like geometrically? Is it the area between the vectors?

dam man i REALY REALY LOVE YOU best ever , finally i understand it . thanks thanks very gratefully

Thank you so much!

Thanks a lot. Even studying in 2023.

Just for clearity it's not (x-cv)*v. Because you start in cv and go up to x, so it is (cv-x)*v however you will get the same result because they are both orthogonal to v :)

Cheers, understand it now.... But why would I ever use this? (Other than being required to for a test). What is the application of this? Thanks very much!!

@ardensetiawan353

7 жыл бұрын

If you're doing IT in graphic study, you'll find this calculation used a lot. Displaying 3-dimensional objects are actually projection to 2-dimensional surface (the screen), so each time we "render" 3D objects, we project them into 2D. While people just use library or plugin these days, it's useful to know the basic behind how it works so you can tweak it to your own desire

What do I get? It. I get it!

Watching the video before my maths exam

Say, who is this guy? He does all of the math videos.

the quality of your previous video is much better then this one and i am wondering why that is (in this one your voice is echoed and sound like you are in a 'tin can' and the video is pixelly )

How do I find the projection of a point onto a line on the form x=[x1, x2] + t[x3, x4] using the equation?

@noellundstrom7447

7 жыл бұрын

you cant project a point onto a line, you can project vectors onto vectors, but by the time you read this you will probably have learnt that, oh well.

@theghost3133

7 жыл бұрын

hahahah love it when i see this answers after a while haha

@ScilexGuitar

7 жыл бұрын

Haha ja det har jag. Thanks Noel :)

@ScilexGuitar

7 жыл бұрын

Haha happens all the time xD

Can anyone say why on the earth we should project a vector onto another orthogonally, but not like an arch

I'm learning about projections in calculus right now, but it's not taught like the linear algebra way.

@ssshj

3 жыл бұрын

wow 9 yrs ago, you okay now?

How is the pink vector x - projection of x?

Is it by accident that his formula in 9:08 c = c*v/v*v reminds me of the ols formula b = x*y/x*x???

@mlaumusic

7 жыл бұрын

The two are clearly related, I think that in OLS, the estimated Y is the projection of Y in the subspace generated by X

ECE 485 UofA

how do you get x - proj(x) ??

@That_One_Guy...

5 жыл бұрын

remember pythagorean theorem ? it's similar to that , x is the hypotenuse while proj(x) is the adjacent, than you would need the opposite to create a triangle

First comment! First view! Must be my lucky day! :-)

@jahitrst2863

4 жыл бұрын

Are u still alive?

@G_Koch

4 жыл бұрын

@@jahitrst2863 hahaha good question, is he still alive???

awesomeeeeeeeeeeeeeeeeeeee

@ssshj

3 жыл бұрын

god really 10 yrs ago , guys what were u ecen thinking

gREET

Instead of black background, white background would be more effective. Problem in viewing.

whata fuck is this? xD

dis sounds so wrong

@ssshj

3 жыл бұрын

whyyyyyyyyyyyyyyyyyyyyyy tell me i have exam in 3 days

I hate the sound of this voice.

isn't the scalar x dot v /(magnitude of v squared)?

@gulgaffel

7 жыл бұрын

magnitude of v squared and v dot v is the same thing.