 Hi, I'm Zor. Welcome to a new Zor education. Let me address another aspect of vector product. Actually, in this particular case, it would be a combination of scalar and vector product. I'm talking about this particular expression. So first, we have vector product of vectors B and C, and then we have a scalar product of vector A by the result of B times C. So, the first thing which I would like to prove is the following. Let me start from the second part of this lecture. The second part of this lecture is that if you have three vectors like this, let's say this is A, this is B, and this is C. Well, you can always build a parallelogram based on B and C, right, something like this. Now, let's build a parallelipipid based on the A as the third dimension. Okay. Do you see this parallelipipid? These are vertical edges, and this is a horizontal plane B and C. So, the result of this particular lecture is that this particular expression actually is a volume of this parallelipid, which is not obvious, obviously. But now, what I would like actually to say is that we will have exactly the same result here as if we will change the order. So, instead of A times BC, we can have BA and the C as a third dimension, or we can have from A to C and B as the third dimension. So, this purely algebraic expression can be proven using the co-ordinate representation of vector and scalar product. But geometry is basically the base for this thing, because no matter how you calculate it, you will still get the volume of this particular figure. So, that's the reason. I don't want to give you, okay, let me prove this and just say, nothing can start putting some formulas, etc., etc. What's important is that this is a representation of the physical aspect of the vector and scalar product. So, but let me start with this one. Let's say the first equality. Now, B has a certain coordinates. Let's say A is A1, A2, A3. B is B1, B2, B3, and C is C1, C2, C3. So, B times C, I was just deriving in the previous lecture the coordinate representation of this thing and let me recall. So, it's, I think it's B2, C3 minus B3, C2, comma. Then the second is B3, C1 minus B1, C3 is a second coordinate. And the third coordinate would be B1, C2 minus B2, C1. I think that's how the three coordinates of the B times C would look like. Now, if you multiply it by A, which is A1, A2, A3, you have to multiply each one correspondingly on A1, A2, and A3 and add them together. So, that's what it is. Now, if you do exactly the same thing with this or this, just changing the B and C and A places, you will have exactly the same result. And I don't want actually to spend much time on this. Maybe in one particular case. Well, let's say this one. So, B and A would be B2, A3 minus B3, A2, comma. B3, A1 minus B1, A3, comma, and B1, A2 minus B2, A1. That's the vector product. And multiply it by C1, C2, and C3, and add them together. So, let's think. A3, B2, C1, A3, B2, C1. The only difference is, I have a different sign here. But maybe I'm wrong with this one. Let me check. From A to B, from B to A, you get C. How about this? A2, B3, C1, A2, B3, C1. Yes, it's all different signs. So I have to do A times B here. That looks better. In which case, I will have different signs. Because vector product is anti-commutative. Now I have exactly the same, right? A3, B2, C1 with a minus. A3, B2, C1 with a minus, right? A2, B3, C1 with a plus. A2, B3, C1 with a plus. Now A1, B3, C2, minus. A1, B3, C2, minus, right? A3, B1, C2, A3, B1, C2 with a plus here and here. Fine. And the third one is A2, B1, C3 with a minus. A2, B1, C3 with a minus, right? And A1, B2, C3 with a plus. A1, B2, C3 with a plus. Correct. So everything is fine. Now, and the same will be with another. So this part is easy. The only thing is make sure you don't mix the signs. All right, so we have proven this. So the only thing we should have to prove right now is that this represents the volume of this parallel epipod. Now, I would like to tell you, although it looks like an orthogonal basis, like x, y, and z, they are not orthogonal to each other. There is no 90 degree angles here. It's general, which means here you might be less than 90 degrees here, which means that this is not a rectangle. And these are not rectangles. These are all parallelograms. And the whole figure is called parallel epipod. Now, what's the volume of the parallel epipod? Well, as in many cases like this, it's the area of the base times the altitude. Now, what is the base? Base is parallelogram. And the area of the base, we know it from the properties of the vector product, is the vector product of these two vectors. It was an absolute value, obviously. So the area of the base is equal to absolute value of b times c, vector product y, again. It's multiplication of lengths of this, times lengths of that, times the sign of the end. But if you have a parallelogram, now, this is the flat on this surface. So it's b and c. And this is perpendicular. This is phi. Well, obviously, the altitude of the parallelogram is lengths of this, which is, let's say, c or b. It doesn't matter, times the sign of this angle. And that's exactly what vector product is. Vector product is the lengths of the b times the lengths of the c times the sign of the phi of the angle. So we are considering a simple case that angle is measured in the correct direction. So it's positive, actually. If you want, you can put it this way. That it would be always correct. So this is the area of the parallelogram at the base. So you see? That's what it is. That's good, right? So we have an area. Now, the area as a vector is multiplied by a vector a as a scalar product. Now, what is a scalar product? Well, let's remember that if you have a scalar product of two vectors, then it's the u times v, lengths of the v, times the cosine of the angle between them. An angle, I mean the angle the shortest distance, less than 100 and 80 degrees. So that's why it's always like this. Now, what else? Well, you can put absolute area as well, just in case so you don't really have any problem with the direction of the angle you're measuring, OK? All right, now, let's talk about vector a and this particular plane, which is defined by b and c. Let's draw a perpendicular from a down. This is the altitude of the parallelogram. Now, what is b times c as a vector product? Well, this is the absolute value of this, but as a vector, what is this? Well, you remember that this is the vector, which is perpendicular to both of them, right? So it's somewhere here. This is b times c. This is vector product of b and c. What is the angle between a and b times c? Let's call this angle phi. So this color product of a and the result of the vector product of b and c is equal to lengths of the a, lengths of the a, lengths of the b times c, and the cosine of this angle phi, angle phi. Now, obviously, it's exactly the same as this angle. Why? Because from this point, we have a perpendicular to the plane and from this point, we drop the perpendicular to the plane. So these two perpendiculars are parallel to each other, obviously, because they're both parallel to the plane. So what is, in this case, a times cosine. So instead of u and v, I will use now b and a and b and c. So it's a and this is b times c. So this is a and this is b times c, right? So a times cosine of phi, a lengths of the a times cosine of this angle, that's exactly the altitude. Because this is the lengths from the point to the plane. We drop the perpendicular. So here, when we were building our vector product, we put the perpendicular up to the plane and here, we just drop the perpendicular down. That's why these angles are the same and that's why the cosine of angle between a and b times c is exactly the same as the cosine of the angle between a and the altitude, which means that this and this together is an altitude. Now, this b times c is an area of the parallelogram at the base and that's why we have this particular formula. So we have the area of the base, which is this, and the combination of these is an altitude of our parallel epithet. That's it. That's the end of the story. And obviously, if you change the orientation of this, if you start with, let's say, a and b as a base and c as a third dimension or if you do, let's say, a and c as a base and b as a third dimension, that's why you have all these a times scalar product of vector product of b and c is exactly the same versus you regardless of whether you multiply a scholarly by vector product of b times c or b scholarly by a times c, et cetera. So the only thing is you have to really watch the sign because of the orientation. But if you are talking about volumes and areas, you always have these absolute value of the lengths and these do not have actually any negative values anyway. Well, that's it. That's an interesting, I would say, story about vector and scalar product from geometrical standpoint. So the vector product is area of the parallelogram defined by these two vectors. And if you multiply by another vector in the third dimension as a scalar product, you will get the volume of the parallel people formed by these three vectors. I hope it was interesting. So anyway, that's it for today. Thank you very much and good luck.