Defining a plane in R3 with a point and normal vector | Linear Algebra | Khan Academy

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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[sajidullah]

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.

[KoltronZer]

First million dollars I gross I am donating 10% to Khan academy.
Thanks my dude

[MASTERable]

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.

[shivamkataria]

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 <3

[voidzilla]

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.

[manekineko]

Thanks so much, Sal, for taking the time to do this. Lucidly clear.

[Tofugrass]

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!

[dalisabe]

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!

[flvyu]

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?

[Screen]

I wish my linear algebra prof would teach this, this way

[MrGoatflakes]

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...

[DiegoMathemagician]

what if my monitor is curved? ahá. Now seriously, you were very helpful, thank you a lot!

[Aznproz]

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!!!

[tidaimon]

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?

[NARESHSINGH-olez]

You explained me in 14 mins what the Wikipedia article and my textbook couldn't. Thanks a lot.

[AlteraLin]

Thank you for this explanation. It was just want I needed to understand the code in a Shader I've been looking at. :)

[BS-quwy]

i wish all my instructors were as good as you. u r great!

[ozzyfromspace]

13:42 ...it's very useful for Machine Learning as well

[norwayte]

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.

[VicfredSharikver]

you rock, please keep posting videos about linear algebra and vector analysis