 In our previous video, we introduced the notion of orthogonal vectors. This was exactly the idea that when we take two vectors, u and v, if they're inner products equal to zero, we say the vectors are orthogonal. I try to make a geometric connection between this algebraic notion of orthogonality with the geometric notion of perpendicular erity, that is the right angles and such. I want to explain how this can be useful in constructing hyperplanes in our vector space. For example, let's imagine we're in R3 for a moment. This is a vector space we know and love very well, and perhaps we're trying to describe some plane in R3. Now, the way we usually been doing it is the following. We pick our favorite point on the plane, maybe call it x naught here, which it's a point in the plane. If you want to think of this like an arrow, right? It's an arrow pointing to this point in the plane, but there's really not much benefit of distinguishing between these arrows and the points. We have a point which is in the plane, and it's a vector, we'll call it x naught, and then we define the plane using these spanning vectors, u and v, so that a generic vector x is just equal to x naught plus something times the first vector plus something times the second vector. So we get these linear combinations of the spanners, and this gives us a plane in R2. Now, a plane in R2 is also a hyperplane, and it turns out we can describe it, instead of using two planars, we can actually describe it using a so-called normal vector, a vector which is orthogonal to every point in the plane. And so geometrically speaking, our normal vector is gonna look something like this, call it n right here. The normal vector, if orthogonality does represent right angles, the normal vector is gonna be a vector which is perpendicular to every vector in this plane. So if you think of this plane as this infinite sheet, we're looking for a vector which will form a right angle with that sheet, with that plane. And so if we have some normal vector, say n, and we won't be specific about its coordinates here, we'll just call them a, b, c here. It's normal vector, meaning it's orthogonal to everything in the plane right there. If we take a generic vector on the plane, let's just say it's x or x here, x, y, z, then we want that the dot product of n and x, the normal vector with the vector x, which in expanded form would look like ax plus by plus cz. We want that to equal zero. And so this produces an equation for that plane. Now I should note, mention that this equation right here gives us a plane through the origin, right? Because notice if I plug in zero, zero, zero for x, y, and z, that's a solution. So this is a plane through the origin. If we don't necessarily want the plane to go through the origin, but we want it to go through this point x naught, we can replace the vector x, the vector x right here with instead x minus x naught. And so then we come and look at this equation right here. This is the so-called point normal form of the plane, the point normal form. And that's because it has built into it a point which is on the plane and the normal vector. If we expand this point normal form out, well by properties of the inner product, you can distribute the n across the division there. And given that this thing right here is just, it's gonna be x minus a particular value. It'll expand to be something like the following. You get a times x minus x naught plus b times y minus y naught plus c times z minus z naught where x naught, y naught, and z naught are the coordinates of a specific point on the plane. x, y, and z are the variables and a, b, c are specific numbers, specific coefficients. And so when you then distribute the a's here, you're gonna get a x minus a x naught. You're gonna get a b y minus b y naught. You're gonna get a c z minus c z naught for which you're gonna get an a x plus a b y plus a c z. Now notice that when you distribute the a onto these constants as a, b, c are constants and x naught, y naught, z naught are constants. You can combine all those terms together and move it to the side and you're gonna get some constant as we're gonna call it d right now. And therefore this right here, this equation right here is what we typically see as a generic equation of a plane inside of R3 where you just choose x, or a, b, c, d. Though those parameters will determine sort of like this standard form for the equation of a plane inside of R3. And this is where it comes from. It comes from this point slope, this point normal form right here that if we pick the normal vector when we pick a specific point on the plane, we get this equation right here. And notice of course, if you actually insert x naught in for x here, you're gonna get x naught minus x naught which is zero and zero dot anything will give you zero. So in fact, x naught is a vector on there and then the normal vector determines essentially the slope. The one normal vector does the work of the two spanning vectors. And so let's look at an example of such a thing right here that if we wanna construct a plane in R3 which contains the point four negative five zero and is perpendicular to three five negative two, perpendicular here suggesting this is the normal vector in play here. What we then have is the following. X naught is given by the coordinates four negative five zero. Our normal vector n is given by a three five and negative two. And so using the point normal form, we're gonna get n times x minus x naught. This should equal zero. And then expanding it out. We should get three times x minus four plus five times y plus five. Notice the negative minus negative five is a plus five. And then lastly, we're going to get a negative two times z minus zero. This should equal zero. And so then we just have to work to sort of simplify this thing, distribute things through. We get three x minus 12. We're going to get five y plus 25. And then we're gonna get negative two z equals zero. 25 take away 12 is 13. And when you move that to the side, you get three x plus five y minus two z equals negative 13. And this would then give us an equation for that plane in R3. Now this idea here of constructing planes using normal vectors, this can be generalized to any dimension whatsoever. If we just want to talk about a hyperplane, a hyperplane say in FN, then we can describe a hyperplane using a single normal vector and a single point on the plane. We just need a point and we have a normal vector. And that describes the entire hyperplane. And so this is essentially just the top down approach, right? So we've talked a lot about the bottom up approach in this course. And we can alluded to the top down approach. I mean, we mentioned it as like, oh, you have a system of equations, you're intersecting hyperplanes, but we never really gave a construction in how you come up with the equation of the hyperplanes. You just start intersecting them. Well, the top down approach is gonna describe, it's gonna describe hyperplanes using normal vectors and then you start intersecting, as you start intersecting those hyperplanes, you're basically joining together the normal vectors. And so the top down approach is about describing affine sets using normal vectors as opposed to the bottom up approach which describes the affine sets using spanning vectors.