 One important use of the dot product is to find what are known as orthogonal bases. So remember that a basis for a vector space describes every vector in our vector space as a unique linear combination of the vectors in our bases. However, not all bases are created equal. For example, consider the two vectors in R2 and notice that these two vectors have very similar x1 components and very similar x2 components. So without worrying too much about what we mean by similarity, we might suspect that R2 vectors are similar. Now since these vectors live in R2, I need two vectors in a basis, so maybe I'll have the basis 1, 0, 0, 1. And in this basis, these vectors have coordinates for 3 and 3, 4. And the thing to notice is that for these basis vectors, our coordinates are also very close to each other. Our first coordinates are very close, and our second coordinates are very close. But what if I take a different basis? So maybe I'll take the basis 5, 9s, 4, 9s, and 4, 9s, 5, 9s. Now again, if I use these vectors, I can write our vectors 3, 4, and 4, 3 as linear combinations, but this time the coordinates will be 8, negative 1, and negative 1, 8. And this time our first components are very different and our second components are very different. And what this means is that in some sense this is a bad basis because it takes vectors that are actually very similar and makes it appear that they are very different. And the question that you might want to ask is, what makes one basis better than another? And one of the things that will make a basis a good basis is if it is an orthogonal basis. And we'll define that as follows. Let v be a set of basis vectors. We'll say that v is an orthogonal basis if vi.vj is 0 whenever i is not equal to j. If it also turns out that the norm of vi is equal to 1 for all of our vectors, then we also say that v is an orthonormal basis. So how shall we find these orthogonal bases? One way we might proceed is suppose I have a couple of vectors that are already orthogonal. Can I include a third vector that is orthogonal to the others? So to find an orthogonal basis we should solve a system of equations. So if we want our set v to form an orthogonal basis, we need all of the dot products to be 0. i, not being a particularly trusting sort, would verify that the dot product of these two given vectors are in fact equal to 0. Next, we'll want to make sure that the dot product of the new vector with each of the others is also 0. So I want to find that the dot product of x1, x2, x3 with 1, 1, 3 is 0. Well, that gives us the equation 1x1 plus 1x2 plus 3x3 equals 0. I also want to make sure that the dot product of x1, x2, x3 with the other vector 2, 1, negative 1 is also 0. And so that gives me 2x1 plus 1x2 plus negative 1x3 equals 0. And here's our system of equations. And we can row-reduce the coefficient matrix and get our parameterized solutions. To get our actual vector, we'll pick a value of s. We could use any value of s. So let's take s equal to 1 to get our simplest solution for negative 7. 1 as our third vector in our orthogonal basis.