 And so in the original setting, we were defining cross products for any scalar field F. For the next discussion, I want to focus on reels here, because that's where we're going to use these the most. Like if we're in a calculus setting or for example, multivariable calculus, we're probably talking about real vectors right here. But even still, if we have three vectors, UVW that live inside of R3, the real vector space there, we can define the scalar triple product of UV and W as the product of U dot V cross W. Now pay attention to two different multiplications. We have the cross product on the right and we have the dot product on the left. And that's actually why we restrict to the real numbers right here. Because for general vector spaces, we didn't define an inner product. We have those for the real numbers. We also have those with the complex numbers. I'm not going to introduce any complex numbers right here because when it comes to inner product there, we want the Hermitian product. And the Hermitian product requires we conjugate, but determinants don't. And so there's a little bit of an incompatibility going on there. So we're just going to look at real numbers for the rest of this section right here. So we get this scalar triple product. We take the product of the cross, we take the cross product and then the dot product. Well, it turns out that this scalar triple product is not that alien of a subject right here, U dot V cross W. It turns out this is none other than just the determinant of these various vectors right here. And I kind of want to show you quickly how that is. V cross W is how it should end. So if we look at first at the cross product U dot, well, we're going to get a vector, which by the rules we saw earlier, this vector would look something like the following. We're going to get, sorry, back up there. V two, W two, V three, W three. That's the first entry. Then you're going to subtract from that the second entry should be V one, W one, V three, W three, like so. And then finally, you're going to get your third determinant there, V one, W one, V two, W two. So if we use that determinant formula we saw on the previous slide, the cross product of V and W looks like this. Well, then if we take the dot product of that, we take the entries of U and dot them with all these determinants and add those together, you're going to get U one times this determinant, V two, W two, V three, W three. Then minus U two times V one, W one, V three, W three. And then lastly, you're going to get U three times this determinant, V one, W one, V two, W two. And you can see that this final expression right here, this is just the cofactor expansion along the first column of this determinant, which then gives us the equality we're looking for. So this triple product that this triple product we have right here coincides with the determinant calculation in the context of three by threes. And so why do we even need it if it just gives us what we already have? Well, one advantage here is that you can then get the determinant without actually talking about determinants. That is, you can get the determinant with just using the cross product and the dot product right here. And so this is really useful in like those physics and calculus settings I keep on talking about where students don't necessarily have a strong notion of linear algebra yet. They're just barely learning about vectors and things like dot products and cross products. So the thing is you can very quickly teach students about dot products. You can very quickly teach them about cross products as formulas. And then from that, you can get this triple product which would then recapture the determinant without all of this theory of determinants. It's sort of like a shortcut to the determinant in the special case of three by threes. That is an R3. And why we would care about that is the following theorem right here. If you take three vectors, UVW and R3, then the area of the parallelogram spanned by U and V, its area is gonna be the norm of U cross V. And then the volume of the parallel pipe head spanned by UV and W. So remember the parallel pipe head, this is like a three dimensional parallelogram. The volume of that thing is gonna be the absolute value of this scalar triple product. Now I wanna mention that this second result right here, the volume of the parallel pipe head, this is immediately true because we saw this earlier as this triple product is the determinant of a matrix. And we saw previously that the determinant of the matrix spanned by the determinant of the matrix whose columns are the spanners of the parallel pipe head gives you the volume. That's an immediate result. But that way calculus students can calculate volumes without determinants. They can sort of use this back door using this triple product. And similar things kind of over here, we can use the cross product which is itself kind of a determinant. Not exactly, but it's kind of like a determinant. We can then get capture things like area and volume which determine its measure without actually introducing areas and volumes. And so one can calculate this triple product right here. So the triple product of u dot v cross w. Admittedly if I were to calculate the triple product, let me switch the color there. If I was to calculate the triple product u dot v cross w, frankly speaking, I would just do the determinant. We get three, negative two, negative five, one, four, negative five and then zero, three, two. I'd calculate this determinant. And honestly to maximize the amount of zeros, I would cofactor expand across the first row right there. So you end up with three times the minor four, two, three, negative four and then minus one times the minor negative two, three, negative five and two. And when you do those two by two minors, so three times, you end up with eight plus 12 and then minus, we're gonna get negative four plus 15. Eight and 12 of course is 20 times that by three, we get 60. And then we end up with 15 take away four, which is a nine, so we get a negative nine right there. I'm sorry, not nine, that should be 11. 15 take away four is 11. And then 60 take away 11 gives us 49. So that's the triple product, which would measure the volume of a parallel pipe head and things like that. So if you know determinants, you really don't need cross products whatsoever. But if you were to calculate this thing using the formulas we used for the cross product at the beginning and then with the usual dot product right there, it'll give you the exact same thing as this determinant. Of course, the advantage of you thinking as determinant is that this formula follows from co-factory expanded along the first column right here, which as the determinant calculation, that's not the best column to use. I would actually use the first row. So determinants simplify the calculations of cross products. So use those whenever you need to.