Defining a plane in R3 with a point and normal vector | Linear Algebra | Khan Academy
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- เผยแพร่เมื่อ 3 ธ.ค. 2024
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Determining the equation for a plane in R3 using a point on the plane and a normal vector
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Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
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Salim is the best . I am 62 and I am still learning from him ...the stuff i missed in college whne i was young. . . now i watch for fun and keep the brain sharp.
salman*
build something
First million dollars I gross I am donating 10% to Khan academy.
Thanks my dude
The internet will hold you to it. Best wishes, Colton!
where the million dollars at its been 6yrs
Wow he talks so clearly that the subtitles on youtube were actually accurate. Like, 100% accurate. No that is harder to achieve than any of these math questions.
You can provide translations to your youtube videos. And Khan Academy is a proper charity/non-profit and has received donations and government grants to do just that in multiple languages ;)
This is great! I only wish you had time to do more on vectors and planes in 3 space. I'm taking an exam tomorrow on this stuff and it would have been great to see you do more. Thank you so much.
This is godlike! I was so confused after watching my prof's lecture, but I understand everything after watching this :) Thank you so much for teaching others for free, you are a blessing
Thanks so much, Sal, for taking the time to do this. Lucidly clear.
6:40 this doesn't make any sense to me -- wouldn't the vector X - X0 just give you a plane that's parallel to the plane created by connecting X and X0? I tried a similar example in 2D and the whole (b-a) concept shows the same -- it creates a line parallel to the line that connecting the endpoints b and a creates, but never actually the line
what if my monitor is curved? ahá. Now seriously, you were very helpful, thank you a lot!
The moment your spaces/surfaces become curved, your brain is begging for tensors
Great explainer. In every complex thought, there is always a key note; if you miss it, you will never get the complex one. Herein the key thought is the fact that the difference between any given two position vectors having their heads as points on the hyperplane lies always on the hyperplane in question, which is always perpendicular to the normal vector to the plane. As you know already, the dot product is always equal to zero when two given vectors are normal to each other. The algebra of the dot product is a treat in linear algebra and something that can be thought of as a a special case of “linear” transformation. The hyperplane is the span of the difference of any two position vectors and they all must be perpendicular to the normal vector to that plane. In other words, the normal vector lies in the null-space of the hyperplane. This fact is of particular interest to unique solutions of linear transformations or matrix operations. Finding unique solutions of linear transformation is probably the most fascinating feature of linear algebra and could always be thought of as the coordinates of the difference of two points on the hyperplane.
In differential geometry, this incremental difference between two position vector being always perpendicular to the position vector is fascinating when the increment is always tangential to the surface created by the position vector undergoing a change. In two dimensions, this generates polar coordinates; in three dimensions, this generates cylindrical and spherical coordinates with a nice set of orthogonal basis!
Thanks for always being the best explainer!
This was the best instructional video I've ever seen. It was so well done! Everything is so clear when put this way; by definition, the dot products must equal zero for perpendicular vectors, and finding the equation of the plane is so straight-forward. Thank you so much!
You explained me in 14 mins what the Wikipedia article and my textbook couldn't. Thanks a lot.
I wish my linear algebra prof would teach this, this way
13:42 ...it's very useful for Machine Learning as well
dude i am serious here! Lucky that i watched this video about planes before my first midterm, or else i would have no idea how to do these questions. Somehow, I was able to freakin aced my first midterm, thanks to you man!! THANK YOU!!!
Hey Sal, thanks, great video. You are the man.
But you know what would be a great addition to this course? Affine spaces, affine transformation and uniform coordinates in 3D, i.e. any position (x, y, z, w) | w non zero implies (x, y, z, w) is the same vector as (x/w, y/w, z/w, 1), and (x, y, z, 0) implies a direction rather than a position.
It sounds like a weird and pointless thing to do, but it isn't, because it allows you to compose transformation, such as movement, rotation, scaling, change of coordinate systems, even ones that would be non linear (I think) in simple R3 like perspective transformations, by simply multiplying the matrices that represent the transformations together once and then applying them to each point in your dataset.
Rather than doing what might be hundreds of transformations to each individual point. And you can treat all those points in parallel as well. The notion is so useful that it forms the basis of most 3D graphics hardware and software, and understanding it is crucial for understanding 3D graphics.
I would also be super interested in a course on Tensors and Differential Geometry, because apparently it's one of the most useful and general representations in engineering and physics, to the point where relativity becomes very straightforward in it.
And while I'm airing my wishlist, how about a course in topology and manifolds? :D Or maybe one about the sort of skills you need to move from say your science engineering type applied mathematics into doing actual mathematics as a mathematician. Things like understanding and constructing proofs, etc. And other skills I'm no doubt even aware of but which can be taught...
more linear algebra please
thank you Sal
Very well explained. Thank you.
10:56 I changed the order, I use X0 minus X, then I get different equation for the plane. Why's that? does the direction of the vector also matters?
You might get different equation but it equals to zero when multiplied with n .also by changing order the resulting vector would just be In opposite direction so doesn't affect the eq Ax+By+Cz
same plane
i wish all my instructors were as good as you. u r great!
Thank you for this explanation. It was just want I needed to understand the code in a Shader I've been looking at. :)
Why am I paying over $1000 per course in university for something I can't even understand when I can just listen to this for free and actually learn something.
Thank you so much for the clear explanation! This whole concept was unclear to me for such a long time! I understand now how important the unit vector n is! Am I right in assuming that it is the unit vector n that determines the tilt of the plane?
you rock, please keep posting videos about linear algebra and vector analysis
Thank you very much, just watch it one time and I get it. God bless u. :)
Simple and understandable. Great! Thank you!
At the end, towards the example, could we also do 1 - x, 2 - y, and z - 3 instead of the other way around, or would this be incorrect?
saved my life for the finals
Thank you so much for doing what you do. You know this material so well it makes it super easy to pick up.
Let's say Ax + By + Cz = D (so the point is on the plane).
What if you have a point outside of the plane even though the equation has the correct result? It could be that the plane only has a certain size.
Salman, i am proud that you are a Bengali.
wow that was a good explanation
the example helped a lot cause i could do my own problem while you explained yours
Wo0K No, D is just a placeholder for whatever ends up on the right side.
I see planes in the sky
Could you explain the origin of the equation of the plane in the form you wrote it first (ax+by+cz=d) - without referring to vectors...avoiding circle definition? I mean - to create this equation of a plane without a knowledge in vectors. Thank you.
Awesome explanation !
You are a life savior
11:00 Correction: x and x0 don't lie on the plane. It's (x-x0) that lies on the plane.
Thank you so much! Bullet Club 4 life!
Thank you Sal!!!!
To save 10minutes and 15 seconds of your life.. the example is at 10:15
I prefer listening to everything he has to say!!!!
nice way of wasting time and forgetting everything he says after 1 day, yours is
Thanks a lot.
okay, I have a question. Instead of taking a vector that is perpendicular to the plane and another thats on the plane, why didn't we just take any arbitrary pair of vectors that would span the plane and define the plane in terms of that ?
I know you took linearly independent vectors and you can use these two to span the plane, but why only these two ? we could have taken any other pair as well right ?
The Euclidian Vector Plane this is called in Dutch. It describes our 3 d reality. As an initiate (I hold a masters in engineering, but never actually worked in that capacity, so it's been a while since I've looked at this)
I have always wondered: To calculate forces in 3d, we use matrix calculation (cfr: Willey & sons: Engineering mechanics - Statics). Now these matrices can have n (infinite) dimensions. Could it be Euclides only described what he could see and the fact that we use matrices (applicable to n dimensions) to calculate vectors (3 dimensions...as far as we know) is actually trying to tell us, there's an infinite number of dimensions and ergo more than 3 vectors that describe our world?
Anyone?
You Are The BBBBBBBBBBESTTTT
Thank you!
@IZZOmath Thanks, I appreciate the help, I'll take a look to your videos.
Best explanation!
thanks a lot, sal!
bless you!!
so what's the plane's equation represent? does it represent the plane that vector XoX lies on?
Just all the points in the plane
it tells you that all (x,y,z) that verify it are points in the plane. And yes, it represents that plane.
Thanks !!!
Thanks :)
What should i do for a equation of plane parallel to another plane. Do reply buddy!
Is vector n just normal to the the plane at the point X_c (as in to those vectors which originate from X_c) or Can I just say that it’s normal to the plane itself? (Because that’s how it’s said usually and it’s bit confusing)
Why is x-x_0 not x_0 on the tip of x rotated by 180 degrees at @6.20
It's taken me a while to get this. I had to skip to some other videos. patrickJMT's one at watch?v=ISsO9Q4UCZw was very helpful as was calctube's one at watch?v=3QLaud6SnHM. Then I came back here and it made sense. Well, mostly sense :)
great video review.
what do i do if i have 3points, or a triangle, on the same plane and want to figure out the equation of the plane?
Thank you Sal :)
Huh that's a little weird. I started a computer science course... for the math half of my 3D graphics module we just started looking at planes and normal vectors... I knew I had done the mathematics before but didn't realize it had been 3 years till you replied to my old comment.
volume of a parallelepipe needed
What is D? Is it a vector or a point?
This is supposed to be my homework, but it doesn't look like my homework at all.
I love you.
What program is he using?
how do you find a point lying in the plane given an equation??
Rachael Quirke
solve the equations to find x y nd z
use cross multiplication method to solve
so.... n dot x = n dot point?
@Tott26 Try a tablet PC and a program called NotateIt. Watch my videos to see if you like the combo.
shoudlnt it be ax + by + cz = -d???
The only thing useful here to a programmer was the fact that a normal vector is perpendicular to everything on the plain. But since i already knew that then this video was a waste of time. If you wanna see how fast Salt can write some meaningless equations that are not really related to 3d programming needs go for it.
no, it's lol, Wynaut XD
y knot
cuz thats why
Pokemon reference...
Sal your x's started looking like lambda's towards the end of the video
Where are the snakes on a plane?
Black screen ruins everything!!! Ohhh ........ thanks anyways:)
My brain hurts
thanks a lot for this