 All right. One problem with vectors is that if we think about them as being the directed distance from the origin to some particular point, we can have two vectors that run in the same direction, but because they have different lengths, they're going to look like different vectors. And so we can recover from this by looking at what are called unit vectors. And so in order to do that, we have to introduce the idea of the magnitude of a vector. So let's take some vector in RK, and the magnitude is going to usually be defined as follows. Magnitude, that's double bars, almost looking like an absolute value sign on steroids, and it's going to be the square root of the sum of the squares of the vector components. Now, this magnitude might also be designated with a single set of bars, although this is somewhat discouraged because that single set of bars has multiple meanings in mathematics. The other thing is that while we talk about this as the magnitude, there's a couple of different ways that we could express. So one of those ways that we might talk about this, we might call it the vector norm. Sometimes we call it the L2 norm for a variety of reasons. We also call it the Euclidean norm, and it might even be called the modulus of the vector by analogy with what we do with complex numbers. So any one of these terms are equivalent, and they all refer to the same thing, which we can most easily think about as this Euclidean distance between the points that the vector would correspond to and the origin. Now, we have a few important properties of vector magnitudes. It should be fairly obvious, and you should be able to prove them fairly easily. If I take the scalar multiple of a vector, then the norm of that scalar multiplied vector is going to be the absolute value of that real number times the norm of the vector. Likewise, somewhat less obviously, we have the following. If I have two vectors and I consider the magnitude of the sum of the two vectors, that should be less than or equal to the individual magnitude sum together. This is something times known as Schwarz's inequality or the triangle inequality. So, for example, let's take a look at the magnitude of the vector v, where v is 3, 5, 1, negative 8, and 2. Note that this is a vector in five-dimensional space. Yeah, no problem. We're not trying to draw this, and so we can calculate our norm. It's just going to be the square root of the sum of the squares of the individual components. And after all the dust settles, we find that that magnitude is square root of 103. Now, having defined what the magnitude of a vector is, we have certain special vectors. If I have a vector whose magnitude is equal to 1 exactly, then that vector is referred to as a unit vector. In some sense, what we have is a vector because its length is a very specific value. This is something that is, in some sense, our purest example of the direction represented by the vector. And because of that property that the scalar multiple of a vector will have magnitude equal to the absolute value of the scalar times the vector, if I want to create a unit vector, I can take my original vector and scalar multiply it by the reciprocal one over the magnitude. And the useful term here is that we say that we have normalized v. So we have our original vector v, which might or might not be a unit vector, but by normalizing it, I have converted it into a vector which has a length of 1. So, for example, let's take a look at our vector 3, 5, 1, negative 8, and 2, and we've determined that the magnitude of that vector is square root of 103. So I can normalize that vector by multiplying it by 1 over root 103. So that's going to look like this. And if I do that multiplication, I'll distribute that scalar multiple into the various components.