Cross product introduction | Vectors and spaces | Linear Algebra | Khan Academy
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Introduction to the cross product
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that is one beautiful hand
paurush chaudhery exactly I saying while watching. That boy good!
fap
forreal
And for all this time I was thinking math teachers couldn't draw for shit. This guy comes along and draws a near perfect hand.
dries v He was a art editor for his school magazine I guess.
Just here to show some love for that sweet hand.
"assuming that you're anatomically similar to me" that line cracked me up XD
Sal, you are HILARIOUS and it's interesting to see how you teach as you literally explain EVERY step of the way not only to the audience, but to yourself too as you do it to confirm your own knowledge. Just brilliant dude.
That hand drawing really woke me up. I was paying attention and noting things down but I wasn't really into it until that beautiful drawing. Thanks, Sal! Who says math teachers can't draw!
I gasped when I saw the drawing, the structure of the bones is pretty good too considering he spent like 10s to draw it.
@@ian.ambrose thanks for bringing me back to this video haha, it’s indeed a great drawing to be done in such little time and drawing isn’t even the focus of the video
@@hathbeenslain Pardon me, how are you doing now? It's been 5 years since you learnt all this stuff. Are you putting these math concepts into use?
@@ian.ambrose I am not haha, I actually switched majors after my first year from Math to Philosophy, so no mathematical concepts needed (although my Logic courses were math-like and the use of logic in other aspects of my philosophical education was not wholly dissimilar to studying mathematical proofs)
@@hathbeenslain Thanks for sharing. It's great that you didn't mind leaving everything that you put effort and time into behind, to follow what you love and enjoy more. Most people don't have guts to do that.
Hope you will achieve what you're working for. Have a good day!
Hand drawing pretty good
And surprised to find comments r all about the hand haha
Holy shit that thumb thing was a game-changer
+Reid Mcgowan Lol
right?!
Hands are almost universally the hardest body part to draw for artists but you just pulled that off effortlessly, can we have some art tutorials as well as maths ones?
As a college engineering freshman, this stuff is gold!!!! Thanks!
i like the hand you draw
I think the crossproduct is easier to remember if you draw the a vectors underneath again. for the second element, you do a3 times the b1 you drew underneath the original b3, - a1 that you drew under a3, times b3. that makes it easier to remember without 'doing opposites and such', you just expand your box. for the third, basically you do the cross of the elements underneath again, which is again a1b2 - a2b1.
Today I came to know that SAL is not only a good math teacher & mathematician, he is also good at drawing.
definitely he is a good drawer
The satisfaction of cancelling out till 0 is just something i cant explain.
this guys is a fucking boss. not only can he do math, but he can draw
almost everyone here is commenting just for the hand. i'm just over here really glad you proved the cross product dot a or b cancel out back to 0. i never thought to prove that in the definition. it makes much more sense knowing that.
but btw, nice bloody hand drawing.
I just came here from your site to say that the hand you drew is AMAZING!
This is all great stuff! I'm going to have a nice head start on vectors!
Omg that hand WOKE ME UP!!
this is literally a great video
grade 8-10 math is simple, and you can solve the equations doing just that
but he's explaining everything to the fundamental level, which is quite important for college or university since
we dont get easy questions where we only have to put variables on both sides.
Thanks for making this!
I got my answer which I was searching for so many days thanks a lot.
thank you, your videos are great
Where would i be without Sal! Great hand btw.
Going to uni in a month and I didn't learn this in high school, thank the stars for this video.
The cross product is not limited to R^3 only. There is a generalization for any sample of (n-1) vectors in R^n...
Your an artist as well. That is awesome.
that is one good drawing of a right hand. I had to pause the video and type this in.
wow this is very understandably
I Love your way of teaching Sal..........
Thank you so much.........
that was a sick hand you drew
ur picture final helped me understand what the heck that right hand rule meant
09:40 "hopefully you don't have a thumb hanging down here" 😂
I must say, that hand is quite awesome.
Thank you alot Sal
thank you Khan Sir, your videos will always be relevant
There's an easier way to do it: instead of taking the terms all confusing together x=(x1, x2, x3) y=(y1, y2, y3) you can make a matrix with z=(i, j, k) of the form
| i , j , k |
det |x1, x2, x3| = i(x2*y3 - x3*y2)+j(x3*y1 -
|y1, y2, y3| x1*y3)+k(x1*y2 - x2*y1)
and you have your terms
+Imortal ZoD isn't that exactly the same just written in a different format
+Imortal ZoD thats how i learned it as well. and it is the same, HisSolitude..
I do want to point out, Imortal; that you forgot to make your second term negative (you have a plus sign ahead of the j, where it should have been negative).
Joshua Mulleady that's because it should be positive. Can't understand why you think it should be negative
That hand, tho... smh... so good
You can do cross product in higher dimensions, you just need more than two vectors. In R4, you take three vectors and compute a vector orthogonal to all three. So in n-space a cross product requires n-1 vectors.
That is actually incorrect. By definition the cross product can only be done in the 3rd and 7th dimension
8:47 for hand drawing
@eileenBrain Anyway on a computer the above does not work the way things cancel out on the board.
Thank you
hi sir, u said when proving that a.b are orthogonal ,then a.b=0 but in the example u did the dot product of the result of de axb.a.may you please explain that
Your drawing is AWESOME =))))
I'm curious, what do you do your math on? What is the software? Great video btw! :)
Anime drawing style right there. I know you are a fan of anime. >.> Khan. Let us exchange some words and drink tea.
weeb
This is good stuff. Thanks.
I liked this video just for the hand drawing.
Khan Academy, Could you translate cross product and dot product to Turkish please?🙏
another mindhint for memorizing think of
[a1,a2,a3] then exclude then from the equation so first row don't have a1 [a2b3-a3b2]vertically the order of first terms 231
I owe you my decent grade in linear algebra.
Its been 10 years
awesome thank you
@karevkarev Anatomical is position, not size.
trivia: the cross-product also applies to vectors in R7
10/10 would watch that hand drawn again
my brain wasnt fully functioning at the time i watched this.
reading these hand comments, had to skip ahead to see this. and it was worth it.. haha, btw thanks heaps for the vid !
Could you also define the cross product of n-1 n-th dimensional vectors as the determinant of the matrix made from each vector. i.e:
For 2 3-dimensional vectors, the cross product is defined as:
a i x x' y' z'
b X j = y , such that; Det[ a b c ] = x'x - y'y + z'z
c k z i j k
For 3 4-th dimensional vectors, the cross product is defined as:
a i s x x' y' z' w'
b j t y a b c d
c X k X u = z, such that; Det[ i j k l ] = x'x - y'y + z'z - w'w
d l v w s t u v
Note: n vectors are needed to calculate the cross product in dimension n+1
Therefore, the cross product of a single 2-dimensional vector is:
a x x' y'
Cross[ b ] = y, where Det[ a b ] = bx' - ay'
Therefore x = b, y = -a,
a b
Therefore: Cross[ b ] = -a
This just extends the defined dimensional-domain from domain=3 to domain>=2
so instead of memorizing a formula of an easy determinant whose columns are ijk, a and b, you would have me remember some sort of half determinant sign switched in the middle?
i just wanted to ask , if i watch all these videos in this playlist of "linear algebra", will i be ready to take on Quantum mechanics and understand matrix mechanics and Dirac notations and relativistic equations?
Jagadeshrao Thalur how*
what kind of platform this video was made? using some kind of table with pen? or on microsoft surface?
The inventor of dot product and cross product why it is defined one as a scalar and the other as a vector
Both are useful and make sense in Physics. :)
are there any drawing tutorials?
I like all khan tutorials.... but as almost 99% of the written and video tutorials over the internet is the disassociation between pure math, physics or any science AND the real life.
Almost none of the tutorials talks about the application of whatever topic they discuss with the real life.
I think any such tutorials should start with what could be the practical application i.e. where and how it can be used in the real life...
Nothing wrong with learning new concepts, but w/o real app.its devoid
good video although did not quiet get it
4:38 You mean cross product, right?
Yeah seriously, that was a good hand.
Can you cross two matrices that are not vector matrices (e.g. 2x2 matrix by 2x2 matrix)?
came here expecting everyone to be praising that hand drawing lol
is this for calc 3?
are you a professional hand artist or something
Almost broke my hand trying to make that gesture
i jumped over here from khan academy to say.. NICE HAND
nice hand bro
7:00
There's something wrong with your calculations. The sole reason the cross product was created was to create a vector orthogonal to the two other vectors that were crossed in R3, and any vectors that are orthogonal, by definition, have a dot product of zero. So it just makes sense that (a × b) · a = 0, and (a × b) · b = 0, because (a × b) is, by definition, orthogonal to a and b.
Wow dude, you totally didn't understand. Watch that part again. He was introducing the definition of orthogonal.
Ok i get that the formula is nice. By actually i dont understand how to derive this formula to get the perpendicular vector to both vectors. For example i see that he uses the determinant but i don’t understand why is x component of a normal vector is a2b3 - a3b2. I need to understand how they derived it.Can anyone give me a link to an article on this?
Just here for the hand drawing.
8:49 is what I'm here for.
Hey sal isn't determinant only defined for square matrix
There was NOTHING overly complicated in this.
Minor mistake: At 4:38 you say "dot product" when I think you mean "cross product".
I was on Khan academy but I came here to talk about the hand
I wanted you to complete the proof by deriving the original method that's used to calculate the cross product but you didn't .. I did it on my own , but i left with 3 equations one of them is non-linear so the equation gets complicated. i guess, this definition basically comes from solving 3 linear equations making the orthogonal vector the determinant of these 3 equations.
I can visualise the dot product in the meaning of having a component acting in the direction of the other vector but I never understood the cross product or can never visualise it whatsoever, It's a definition but isn't real?. The cross product has to be the most difficult concept I have ever come accross and Ikeep asking people "what is it " and they all say the same thing more or less "it's the product of two vectors and acts orthogonal to these two vectors ", but I keep asking the back "Why does it become orthogonal? " and they look at me like I am a moron from the planet moronay.
The day someone actually explains the cross product so I can visualise it then I will kiss their feet.
3 and 7 dimensions actually
where does the 'n' in the formula a x b = absinx n ..... come from ?
n is a vector of length 1 that is orthogonal to vectors a and b.
It's there to force the result of the equation into a vector that's orthogonal to a and b, so the equation conforms to the definition of the cross product.
If you remove n from the equation, then it will give you a scalar result.
'now that i have you excited with anticipation' lol
came to the comments for the hand drawing
Why cross product is written in matrix form....
what is r3?
Bharath Nagarajan 3 dimensions. X, y, z planes.
Sal draws hand better than lines.
❤️❤️❤️❤️❤️❤️❤️❤️❤️...... ♾️
Those handss
After learning absolutely nothing from this 15+(47/60) minute video, I was able to master linear algebra by simply reading your comment. You, little 8th-grader, are clearly more knowledgeable than this dottering old MIT graduate. Look at how many more math videos you've made than him! Look at your accomplishments! 10th grade math! 10th grade social studies! Clearly, your innate aptitude for doing exactly what your teachers tell you cannot be the manifestation of anything short of pure genius.
hhh everyone talked about the hand
Cuz it's beautiful lol
jesus, what a draw, you could be the next picasso
Sorry @yoster3423 he's the mathematician of Bob Ross XD
sup nerds