 In our previous video, we saw how we can use an inner product to find a length or a norm function for vectors. From the norm function, we can actually create the notion of distance for vectors, because the idea is the following. If we have two vectors, say U and two vectors V, and we think of these as arrows in space, we can position the arrows so that they have a common point right here. Well, then we're gonna say that the distance between them, these vectors are pointing at points, right? The distance between them is really just gonna be the vector that points from one head to the other. And so this would be U minus V. And it doesn't matter which direction you go, because the length of those vectors shouldn't change either way. So the distance between them is gonna be the length of their difference vector. And that's how we define the distance function inside of an arbitrary vector space. So the distance function, which we'll call distance of U and V, it's the length, the norm of U minus V. So if we wanted to compute the length of a vector, let's say an R2, V1, or U, which will be seven and one, and then V, which is three and two, the distance between these two vectors, we would compute this to be, well, this would just be the length of U minus V, that part is what we just said defined a moment ago. And so as you compute the difference function, you're gonna get three, well, seven minus three, which is four, and then one minus two, which is a negative one. Now the norm doesn't really care about negatives or positive here, because when you take the sum of squares, you're gonna get the square root of four squared plus one squared, which gives you the square root of 17, like so. And I wanna mention that with this example here, I deliberately took vectors in R2, which we can identify with points in the plane, because you're probably used to the distance formula being something like the following. The distance is equal to the square root of, let's say X2 minus X1 squared plus Y2 minus Y1 squared. You know, I'm like a traditional analytic geometry type course. Now notice what we did here, that for each of the coordinates, we subtracted the X coordinates, we subtracted the Y coordinates, which X is just the first coordinate, Y is just the second coordinate in our list there. So we took the difference of the coordinates, then we squared them out together inside of a square root. Oh wait, that's exactly what this formula is doing right here. If you subtract the coordinates, when you subtract the vectors, you subtract component-wise, that's what we see right here. Then when you take, because this right here is gonna be U minus V, dot U minus V. And so when you do this, you're gonna be taking the sum of squares and all inside the square root. Don't forget that part. So actually the usual distance formula, sometimes called the Euclidean distance formula, is really just the manifestation in two dimensions of this more general formula of distance, the length of the difference of two vectors.