 Okay, in this lecture, we're going to be looking at something called the scalar triple product. So what we're dealing with here is taking three vectors and combining them in a certain way in order to yield a single one scalar quantity. So three vectors into one scalar, scalar triple product. Suppose we have A, we dot it with B which itself is crossed with C. That is the scalar triple product, that combination. Now here I've put brackets to emphasize to do the cross product first. But we can just write A dot B cross C without the brackets. Why? Because we have to do it in the correct order. If we try to do A dot B first and then cross that with C, it's a nonsense because that will be a scalar cross-producted with a vector. It doesn't make sense. Alright then. So let's do one. We'll make up some vectors. Let's have A is equal to three one minus one and B is equal to two zero four and C is equal to minus one minus two three. Okay. There are vectors and let's go ahead and work it out. So first we'll need to do the cross product B cross C. So let's write that out. So I'm bringing these down now. Remember you can work out the cross product by whatever your favorite method is. I'm just going to do it in the method I introduced before, which is we ignore the first elements and we do the falling diagonal. Here's zero and subtract the rising diagonal minus eight. That gives us the first element eight. Then we ignore the middle elements and we do the rising diagonal. Gives us minus four. Subtract the falling diagonal, which is six. So that's going to give us a minus ten entry. And then we ignore the third elements and we do the falling diagonal. Gives us a minus four and subtract zero. So that's going to be minus four. That is our candidate for our cross product, but it's always good to test. How do we test a cross product? We try dotting it with either of the input vectors and check we get zero. So here we'll get eight twos of sixteen and four minus four is minus sixteen. Add it up. That is zero. And now we try the other combination. Here we're going to have minus one on eight minus eight and then plus twenty and then minus twelve. That does indeed add up to zero. It's past our checks. Those were just checks, but it was good to do them. And so we're now very happy that that is the correct cross product. To finish the scalar triple product, we now just need to dot that with a. So let's write it out again minus ten minus four and do the dot product. That's twenty four minus ten plus four is going to be eighteen. That's the answer. That's our scalar triple product. It could have been a positive number, a negative number. It could have been zero. In this case it's eighteen. Now let's do another one. So I'll erase this, but we'll simply use the same three vectors, but we'll do them in a different order as our second example. So let's do B dotted with C cross A. So of course we have to start by doing that C cross A combination first. So let me write that down quickly. Minus one minus two three crossed with three one minus one. So we start with the falling diagonal. That's going to be two and then we subtract three. That's minus one and then we have a rising diagonal. That's going to be nine and subtract one. That's eight and then we have a falling diagonal minus one and subtract minus six. So that's going to be five in all. Okay. Did I get that cross product correct or not? Do the dot product test minus three. Minus three eight minus five. That one's passed. Let's try this dot product combination as a second check. Double check one minus sixteen plus fifteen. That's also going to come out at zero. So it's passed both of my checks. That one is zero as well. We're happy that this is indeed the cross product C cross A. We now need to complete it. So what we're doing is B which was two zero four dotted with what we found. Cross product minus one eight five. So go ahead and value this minus two zero and twenty eighteen again. All right. So our second example has also given us eighteen. Does this mean that it doesn't matter in which order we do the elements of the scalar triple product? Let me just write down the answer to that and then we'll look at it. It turns out that for any vectors A, B and C then A dot B cross C is equal to B dot C cross A. These were the two cases we looked at and it's also equal in fact to C dot A cross B. This will always be true. In this case it was equal to eighteen. But these three things will always be equal. There are three other combinations we could write down in principle. There are three other ways to combine A, B and C. We could have A dot C cross B or we could have B dot A cross C or we could have C dot B cross A. Now it turns out that those things are easy to see what they will be because let's just look at the difference from the ones above. I've just swapped the order of the cross product. We know that when we swap the order of a cross product we introduce a minus sign. If the top three cases were equal to eighteen, the bottom three cases must be equal to each other and equal to minus eighteen. In general this is the same rule for all scalar triple products. Three of them are equal and three of them are equal to one another but equal to the minus of the first three, so to speak. And how can you tell which ones are equal? It's helpful to write out this little cycle A, B and C written in a circle like this. If we are going around in a clockwise direction here B dot C cross A but that's clockwise around our wheel then and here's another one that's clockwise C dot A cross B. Those guys all belong together. So the guys that are in the clockwise direction all belong together and the anti-clockwise guys they belong together and they're the minus of one another, these two groups. Alright, so that's I think all we need to do as practice for doing the scalar triple product and knowing what we ought to get. Let's think about something else. I'm going to introduce you to something called the parallelepiped. That's why I say, I'm not sure how to pronounce it, parallelepiped. Anyway, this guy is a three dimensional shape but first I'm going to remind you of what a parallelogram looks like. So here's a rectangle and here's a parallelogram that we get if we have the pairs of the sides are parallel to each other but they are not at right angles around the vertex. Now consider this rectangular box and let's tie it up, there we are and consider what happens if we build it out of edges that are in groups of parallel edges but are not all at right angles to each other. So let's see if I can draw this reasonably realistically as a three dimensional object. So I'm going to draw this and then I'm going to stress which edges are parallel to each other. Alright, here we are. Okay, let me change colour. So consider these four edges of the object are all parallel to each other in exactly the same way that in our simple parallelogram these opposing edges were parallel and then these four edges are all parallel to one another again in our 3D shape just as these two edges are parallel and then we have another set, these four edges here in yellow are also going to be parallel to one another. That object is a particular three dimensional solid. It's clearly a generalization of the box in that we're allowing ourselves to have slanting edges if we want to. Now let's introduce three vectors A, B and C to represent these three kinds of edges. You see that all the green edges are the same vector A and so on. What happens if we do A dot B cross C? That it turns out the magnitude of that if we drop the sign then the magnitude is just the volume of this shape. So it contains of course the simple case of a rectangular box as a special case but this will work for any parallelepiped that we care to think of. Those three vectors can always be combined with the scalar triple product to give us the volume and that's the end of the video.