 Okay, let's get into something very exciting. We're going to look at the multiplication of two vectors. Now, the two types of vector multiplication. You might want to have two types of vectors here. There are two types of vector multiplication. The first one that you'll usually see in textbooks or explained to you is the dot product also known as the inner product. And I've got two vectors here in two space and I'm going to do the dot product. And we see vector v1. Here's v1 on the x-axis. It's got square root of 3 as a component on the y-axis 1. And v2 is 1 on the x-axis and square root of 3 up there. And you will notice them as these two angles that you memorize from trigonometry. Remember, if this was the right angle triangle, that's square root of pi over 3 and this is pi over 6. And you remember those. It's very easy to determine, say, the cosine of pi over 3, which is a half. The sine of pi over 3, which is square root of 3 over 2. And the same for here, the sine of pi over 6, which is just going to be a half. And that's why I chose them because we can clearly see what the angle between these two are going to be. It's going to be pi over 6 because this angle is pi over 6. That angle must be pi over 6 for this angle to be pi over 3. So very simple. And that's what the dot product, the inner product that Lisa's going to help us with is the angle between these two vectors. Anyway, let me with our qualification show you how easy it is to do vector multiplication, inner product. And we do put a little dot there between the two. And all we're going to do is we take component-wise multiplication. So this one multiplied by this one, this one multiplied by this one. And we're just going to do summation of all those products. So very simple here. It's going to be square root of 3 times 1 plus 1 times the square root of 3. And that gives us 2 times the square root of 3, a scalar. And of course, it's got to be a scalar because how else is that going to help us with the angle between these two? So the solution to a dot product is just a scalar. And again, without qualification, I'm just going to show you the other way that we do this. And that is to take the norm of one of the vectors, the norm of the other vector times the cosine of the angle between them. Times the cosine of the angle between them. Let's just check if this is correct. Both of this is 2. The norm of both of these are 2 by the Pythagorean theorem. And what is the cosine? Remember, the angle between them we decided is pi over 6. The cosine of pi over 6 is square root of 3 over 2. So the solution is this 2 square root of 3. They're both exactly the same thing, no problem. It gets us the same solution. Now you can see how we can work out the angle between them very easily. I can just take this multiplication of these two scalars, bring it over to this side. So if I do the dot product component-wise and I divide it by the product of the norm of these two, I'm going to get the cosine of the angle between them. If I take the arc cosine or the inverse cosine of that, I'm going to get the angle between them. So very easy to do. Let's go to Mathematica and I'll show you. As I said, this is done without qualification. Why exactly would we do this? What does this help us with? What does this tell us? What is really going on here? And we'll get to that in a future lecture. For now, it's very simple. Don't overcomplicate dot products or inner products. They are very easily, whether you do this or whether you do that. Same thing. Let's go to Mathematica. I'll show you how easy it is over there. And so here we are. Mathematica, let's look at this inner product also known as the dot product. I'm going to create two vectors and just to complicate your life for no good reason, I'm going to call my one vector just V. Use the computer variable V. Create a space in memory to put this list object into. And this list object is just going to be a vector. And the components of my vector, I'm going to call V1 and V2. And then we put a semicolon there. So we press the output to the screen. And my second one I'm going to call W. And the components of that, the X component will be W1. And the Y component will be W2. Semicolon, so press any output to the screen. Shift-Enter, Shift-Return, depending on whether you're a PC, a Linux or on a Mac. And there we go. I've got two vectors, V and W. And they've got X components, V1, W1. Y components, V2, W2. They live in two-dimensional space. Okay, sorry for complicating your life like this. On the board we had V1 and V2 as the two vectors. Yeah, I'm going to have V1 and V2 be the components of vector V and W1, W2 components. So don't get these two confused. I'm sure you weren't. It's really not that difficult. Let's just look at V in matrix form. There we go. We see the column vector there, V1 and V2. Let's do the same with the matrix W. And we see W1, W2. Let's get the dot product of these two, V1. And the way that we're going to do the dot product is just to put a full stop. Just a period, a full stop. W, I said V, not V1. I'm back on the blackboard in my head. V dot W. Dot product, inner product between these two. V, full stop, or V period W. As simple as that. Shift enter, shift return. And there you see the component-wise multiplication. So it's V1 and W1. And I see V2 times W2. And I just do a summation. I just add these components. If I had more components, it'll just be the addition of all of these products. Very, very simple. Now let's go and let's just recreate. V, and I'm going to create this in the form of my square root of 3 and 1. Now I can write SQRT, square root of 3. That's completely legitimate. I can also do it here on the desktop form. I can also hold down command or control and hit 2. And that gives me a beautiful square root sign. Isn't that the most fantastic thing? Comma 1. That was my one vector. And let's make the other vector. Let's make that the 1, and then we're going to have the square root of 3. So there's my two components. Let's have a look at those. I'm going to say V matrix form. Lo and behold, square root of 3 and 1. And let's just do it the proper way around. So matrix form W. That's the proper way to remember the two back slashes. That is post-fix notation. And I see the two there. What if we now have a square root of 3? What if we now have V dot W? 2 square root of 3 as promised. No problem at all. 2 times the square root of 3. Remember the other equation that we have. For that we needed the norm of V. The norm of V. That was 2. Remember that's how we do the length of a vector. It's just the norm function of W is that. Lo and behold, no problem. Remember how we calculated the angle between what the function was. The Wolfram language function. It was vector angle. Vector angle between these two. What is the vector angle between V and W? No surprise. Very beautiful there. Pi over 6. So let's see if we use the other equation to get to the inner product between two vectors. If we do get the same solution of 2 square root of 3. Remember that was the norm of the one vector times. For times I'm going to use shift and star. Norm the other vector W. Now there's another way to get a nice multiplication sign. I'm going to hold down escape. Then hit shift and 8 which is star for me and then escape again. Instead of that ugly star as a multiplication sign. Now we have a beautiful mathematical multiplication sign. A little cross. That's not what this is about. While it is I'm showing you how to use Mathematica. So it's the norm of the one vector times the norm of the other vector times the cosine of the angle between them. So that's vector angle. And the vector angle between V and W. So that's the closing square bracket for the vector angle function which is just now used as an argument to the cosine function. So I've got to close that square bracket as well. 2 square root of 3. Play around, create your own vectors as long as they live in the same vector space. Explore this wonderful world of calculating the dot product between two vectors. See also if you can figure out what the angle is between these and try and figure out what the inverse cosine of the arc cosine function is in Mathematica. I bet you can. It's very simple. See if you can work out the angle between two vectors using this information. You'll make me very proud. And you will enjoy doing it and you'll love working with Mathematica.