 Now we talked about scalar multiplication of two vectors but to do proper arithmetic and proper algebra with any sort of mathematical construct what we need to be able to do is to define how the objects operate with themselves. So if I have two vectors I can find what's called the dot product of the two vectors and that emerges as follows. So what I'm going to do is I'm going to take my two vectors v being the set of components and w being the set of components. I'm going to define the dot product to be this product v1 times w1 plus v2 times w2 and so on. I'm going to add together the component wise product of the two vectors. Now this also is known as the scalar product and the reason it's known as the scalar product is that this thing that we end up with here is not actually a vector. It's a real number if our vector components are real numbers. So we end up with a scalar not a vector. We also sometimes call the dot product the inner product. And so for example let's take two vectors in R4 and find the dot product and I'm just going to multiply the corresponding components. So v dot w is going to be the dot product of the two vectors and I'm going to multiply them component wise. So that's 3 times 2, 5 times negative 1, 1 times negative 5, 8 times negative 1. I'm going to find all those products and I'm going to then add them and there's my dot product. Now why would you want to do this? Well there's some useful properties of the dot product that make it very important. So one of the useful properties is that it's commutative v dot w is the same thing as w dot v. And this emerges directly from the definition of the dot product. It's something you should be able to prove without any difficulty. Another useful connection to make is that if I take the dot product of a vector with itself, what I get is the square of the magnitude of the vector. And this follows just from the definition of how we find the magnitude of the vector. Probably the most important property of the dot product is the following. If I find the dot product of two vectors, what I get is a product of the magnitude of the two vectors times cosine of theta, the angle between those two vectors. And this gives me quite a bit of insight into how those two vectors are related to each other. Let's see how we can make use of that. For example, let's consider two vectors this time in three dimensional space. We'll take a look at the angle between the vectors 3, 2, 5, and 4, negative 1, negative 6. So what I have is I can find the dot product first. And again, that's going to be the component-wise product added together. So that's 3 times 4, 2 times negative 1, 5 times negative 6. I multiply those components together and add them. I get negative 20 as my dot product. And what I have is that that dot product is the product of the magnitude of the two vectors times the cosine of the angle between them. And so I can find the magnitudes of each vector. So again, that's going to be the square root of the sum of the squares of the components. So this vector v has magnitude square root 3 squared plus 2 squared plus 5 squared. And this vector w has magnitude 4 squared plus negative 1 squared plus negative 6 squared. And I'll take the square root. And after all the dust settles, I end up with the cosine of the angle between them is negative 20 over square root 38 times square root 53, which tells me that the angle between them is the arc cosine of some expression. And I can find this numerically. This is about 2.037 radians. Remember, after calculus, degrees don't exist. You should be speaking entirely in terms of radians. But since it's a little harder for us to get some insight into that, we're more familiar with degrees that work out to be about 116 degrees. Well, here's a very useful idea. If I have two vectors, I call them orthogonal, if the angle between them is plus or minus pi over 2. And in this case, if I have two orthogonal vectors, well, let's consider what happens to the dot product. If that angle is plus or minus pi over 2, then the cosine of that angle is going to be 0, and that's going to tell us that the dot product is 0. Now, we also have one other consequence of this. This also works backwards. If the dot product of two vectors is 0, then the two vectors are also orthogonal. One minor qualifier here, we have to make sure that neither vector is actually the zero vector. So we can do this as a quick check to see if two vectors are in fact orthogonal. So for example, I have three vectors, u, 1, 2, 5, v, 5, negative 4, 1, w, 3, 1, negative 1, and let's see if these vectors are orthogonal. So I'll find the dot product of each pair, u, dot, v. I'll multiply the components and add, and that gets me negative 2. It's not zero, so these two vectors are not orthogonal. The next pair, u, dot, w, I'm going to multiply the components together and add, and I do get zero, which tells me that u and w are orthogonal. And then finally, that last pair of vectors, v and w, I find the dot product, and after all the dust settles, I get something that's equal to 10. And the only one whose dot product is equal to zero are the two vectors, u and w. So only the vectors u and w are orthogonal.