 Hi, so what are we doing here? We are studying vectors in three-dimensional space, right? So last time, we talked about what vectors are and what we can do with them. So what can we do with vectors? We can take the sum of two vectors. We get another vector. It is on. Let me see if I can adjust it. Is that better? No? What is it about microphones in this class? By the way, I did find batteries after I finished my last lecture. They were hidden in the cabinet. So don't make me do this, because you don't want that. Let's see. Like this? Better? Yes. OK. But tell me if you lose me, if I lose batteries. All right. Yes, we can take the sum of two vectors. And that's a vector. Or we can multiply a vector by a number. That's a vector. But in addition, there are also more sophisticated operations we can do on vectors, for example, dot product. Now, a dot product is a whole different deal, because the input of a dot product is two different vectors, but the output is a number. The output is a number. And I explained last time that there are two ways to define the dot product of two vectors. The first way is geometric. We talk about the magnitudes of the two vectors and the cosine of the angle between them, which we call theta. So it's just a product of these three numbers. But there is also a second way, which is very convenient in calculations, which is defined in terms of the components of the vectors. And those components will be called x, y, and z. So I put index one for the first three components and index two for the components of the second vector. So the formula looks like this. Now, we can play with these formulas. And we can derive useful information about vectors by using the fact that both formulas give us the same answer. For example, we can find the angle theta, the angle between the two vectors, by combining these two formulas. From the first formula, we learn that the cosine of this angle is equal to the dot product divided by the magnitudes. And now we use the second formula. So this is the first formula. And now we use the second one to write this in the second form, x, y, x1, x2, plus y1, y2, plus z1, z2, divided by the lengths. But the lengths can also be found from the components. For example, the lengths of the first one is given by this expression. And likewise for the lengths of the second expression. So we can write this also very explicitly using the components or coordinates, if you will. So you see, this is an unexpected formula in the following sense. You have two vectors which are given by this coordinates, x1, y1, z1, and x2, y2, z2. When we just look at these coordinates, we don't know where exactly the vectors are. We'd have to plot them in space to really understand their relative position. So it looks like it would be very difficult to find out what the angle is between them. But in fact, there is a very explicit formula which uses nothing but the expression for A and B in terms of components. So this formula only uses x1, y1, z1, which are the components of A, and x2, y2, and z2, which are the components of B. And a priori, there is no reason to believe that there would be such a formula. But we were able to derive it by combining two different definitions of the dot product. It's not like it's very simple, but it's not too complicated either. I mean, it involves some square roots and so on. But it is very explicit. So for example, you can program it on your computer. And then every time your computer would be like that, black box, the dot product, for example. You can program your computer to calculate the dot product. You get a number. But you can also program your computer to calculate the cosine of the angle by simply calculating this. And that's very simple, which can be done very quickly. So that's the power of this method that we are talking about. That using this method, we can learn a lot about the geometry of vectors, say, or other geometric objects, by using as input algebraic information, such as the components of these vectors with respect to the x, y, z coordinate system. One application of this is the following. There is a very special angle for which, special angle theta for which the cosine is equal to zero. So cosine theta is equal to zero if theta is equal to pi over two. We're actually, not just one, but two different values, pi over two and negative pi over two. If you have an angle like this or like this, the projection onto the x plane, which is the cosine, is equal to zero. So in this case, the dot product formula tells us that the dot product is zero in this case. So we can use it, you can use this formula to determine whether the vectors a and b are perpendicular to each other or not. When I say perpendicular, I mean that the angle between them is 90 degrees. But to say that the angle between them is 90 degrees, well, it means that they are either like this or like this, it doesn't matter. Both cases, they are perpendicular. And in this case, a dot b is equal to zero. So if you are given two vectors, it's very easy to find out if they are perpendicular or not. You simply take the dot product given by this very simple formula, number two, and you see whether if it's zero, they are perpendicular. If it's non-zero, they are not. So that's a useful application of all this formulas. There is one more operation that we will need for vectors and that's called a cross product. Now, the terminology here is really not very, doesn't really make much, how should I say? It's not very imaginative in some ways. It's just that in the first case, we denote it by dot, so we call it dot product. In the second case, we denote it by cross, so it's called cross product. There is no underlying reason for calling them this way. It's just a tradition. So don't try to read too much into this terminology. It's just the way it is. Because of the kind of notation that historically, people have gotten used to over the years. So the second operation is a cross product. And again, I would like to think of it as a black box which has certain input and certain output. But now the input will be the same as in the case of a dot product. The input will consist of two vectors, vector number one and vector number two. But the output will be instead of a number in the case of a dot product, a vector itself, a vector as well. So the notation for this is a cross b. And again, there are two different definitions just as in the case of a dot product. The first definition being geometric and the second definition being algebraic. So what's the first definition? The first definition, to explain the first definition, I have to draw a picture. So here I have two vectors a and b. And we should think that they don't necessarily lie on this plane, but in fact, there's some plane which might be sticking out. So I want to draw, I wouldn't like to imagine this piece of this plane which is kind of floating somewhere. It's not inside the blackboard, but it's sticking out. In fact, I can use this blackboard to illustrate. So that's a kind of plane. I'm talking about. So in fact, I might use it. I never thought of that before. It's a very nice three dimensional visual aid here. So here is my, so it gives me a lot of flexibility. Not, in fact, it gives me, I can do all possible planes by turning it a little bit and then rotate. So, no, really I'm amused by this. It's nice to have good surprises. Okay, so here are the two vectors. So now I don't have to explain to you, okay, it's sticking out and so on, because now you can see it. So here's a and here's b. Now, and that's the plane which contains them. So the result of the cross product of these two vectors is going to be vector, which will be perpendicular to this plane. Here I need a big stick, which I don't have, but I have a small stick. I have this. I have my pen. Okay, so it's going to be the eraser. Okay, thank you. So I'm doing really well in visual aids today. I'm very proud. So with your help, thank you. So to people watching, this is how it's done. This is a cross product of A and B. Now, so it's a vector which is perpendicular to this plane, which is good because I give you the direction of the vector. Actually, not exactly, because if I just say that it is perpendicular, there are two possibilities for it, it could go like this or it could go like this. But that's where I use the corkscrew rule, right? And the corkscrew rule tells me that it should, I use the corkscrew rule in the following way, that this vector, this vector, and this vector should form a oriented system, coordinate system. In other words, if I turn my corkscrew from A to B, it should move in the direction of this vector, so which means this is a correct position, not like this. So I explain to you what the direction of this vector is, but it's not enough. To define a vector, I have to give it, tell you what the direction is and I have to tell you what the magnitude is. So the magnitude of this vector is going to be the area of this parallel graph. So in words, the cross product is the vector perpendicular, and I will say the word oriented, where oriented means that A, B, and this vector form the satisfied corkscrew rule or the right hand rule, whatever you like. And this is a vector perpendicular, I'm gonna raise this now because I have a much better illustration perpendicular to the plane, I will say it's plane spanned by A and B. Spanned by A and B means a plane which contains both of them. It is in fact a unique plane, unless of course these two vectors are proportional to each other. If the two vectors are proportional to each other, the answer is going to be zero vector. So let me assume that they are not proportional to each other like this, so they are not pointing in the same direction. Then they do span a plane. There is a unique plane which contains them. Let me qualify this. There is a unique plane which contains them as soon as I specify a particular point. In other words, as soon as I say to which point I'm applying these two vectors. As I explained to you last time, representation of a vector is not unique. In fact, I could parallel transform it. So I could apply it to a different point which would simply mean moving this blackboard. I'm not going to do this. Now, it's perpendicular to the plane spanned by A and B. And by a plane spanned by A and B, I mean what I just explained. Of magnitude equal to the area which is spanned, which is drawn here. I'm not going to write it in words, just to save time. So what is this area? We can find it very easily because we know we know that the area of parallelogram can be computed by dropping a perpendicular line like this. So this is 90 degrees. So let's call this H. And so this area is the length of A times H. I have to be careful not to shift it. A times H, but H in turn is the length of B times the sign of this angle right here. Which I call again theta because that's the angle between A and B. Because you got here a triangle for which this is 90 degrees. So you can find this side by taking this side which is the length of B and multiplying by the sign of this angle. So H is this and therefore the length of A cross B is the length of A times H which is length of A times the length of B times the sign of this angle. So to repeat the cross product of two vectors is a vector perpendicular to the plane spanned by these two vectors which satisfies the orientation rule with respect to those two vectors and whose length is equal to the length of A length of B times the sign of the angle between them. This is the analog of the first definition of the dot product which also involves nothing but the length and the cosine of the angle. In this case, there is a little more because I also specify a particular direction which is determined by the plane. Now, what about the second definition? In the second definition, I'm only allowed to use the components X1, Y1, Z1 of the first vector and X2, Y2 and Z2 of the second vector. So our priority is not clear what the formula should be. So I'll just give you the answer and then it's an interesting problem which you can read about in the book to see that indeed the two definitions are equivalent. Last time I explained to you how to show that one definition implies the other. Did not explain how the first implies the second which you can also read in the book. So the second definition is the following. It involves a certain, a new piece of notation which is called the determinant. The determinant is given by a certain formula. The determinant, you apply the determinant to this table or three by three table of symbols which you can multiply and add to each other as is the case here. And specifically the formula in this case, more concretely, the formula is the following. There's a simple rule how to calculate it. And the rule is that you have to go along the first row so you have I, then you have J, and then you have K. So the formula will have three terms. One corresponding to I, one corresponding to J and one corresponding to K. The term corresponding to I is obtained as follows. You simply cross the colon and the row where you have I. So what remains is this two by two square. And in this two by two square, you take a product along this diagonal minus the product along this diagonal. So this minus this. In other words, you have is I multiplied by Y one Z two minus Z one Y two. This is the first term. For the second term, we take J and to get the corresponding term, we cross out the row and the colon where J is positioned. So what you end up with again is a two by two square. You kind of put them next to each other. So you get X one Z one X two and Z two. And you take this diagonal product minus this diagonal product. So the result is J times X one Z two. Minus Z one X two. And finally, you do the same to K. So the result is this square. Then you get X one Y two minus Y one X two. And there is one more thing you have to remember. You have to remember to alternate the sides. So you put plus here, minus here and plus here. Well, this you don't have to put because it's the first meaning that I'm continuing that formula, okay? So very explicit. Something you have to play with it to memorize. Not too difficult, yes? What about? Yeah, you can use, question is what about some other rule which I couldn't hear. But you can use whatever, or you can use whatever rule you like as long as you get the right answer. It's fine. I'm sure some people can just look at it and tell you the answer. I think it's much easier than doing Rubik's cube, you know, to be honest. Some people can do it in like 20 seconds. So not me. All right. So I'm sure each of you will have their own way to calculate this. I just told you mine, which, well, not it's not mine, of course, but the one I use and which I think most people use. Okay. So that's a cross product. And so again, you have two different definitions. One is geometric. It's about the vectors and their positions and the angle between them and the lengths and so on. Geometric characteristics. And the second one is algebraic in terms of just the components of these vectors. And so what it means is that you can also apply this to calculate various things. For example, you can calculate this way, the area of the parallel gram, which is spanned by these two vectors because the area of the parallel gram is just the magnitude of this vector, right? So we can use it. Use it. To compute of the parallel gram by simply taking the magnitude of this vector, which is to say taking the square of this coordinate plus the square of this coordinate plus the square of this coordinate and taking the square root of the result. Let's do some calculation. Let's say, example, find area of the parallel gram. Let me just save time, right? Parallel gram like this, spanned by the vectors A, which is equal to one, two, three, and B, which is equal to negative two, three, and one. Just random numbers. Okay, very easy. So first of all, we calculate the gross product between these two vectors by using this formula. So we put i, j, k, one, two, three, negative two, three, one. It would be embarrassing if the answer is zero. So they will actually be parallel, although that would be a good illustration of that case. So i is equal to, in front of i, we have this one. So it's two minus nine. And then you have, you have to put minus. Remember, there is a minus. Maybe I should emphasize it. There is a minus here in the middle. So I put a minus. Then you have j, so you have one plus six. And finally, you have, in front of k, you have this, three plus four. So that's minus seven, negative seven i, minus seven j plus seven k, okay? So that's the gross product. And you want to calculate the area. To calculate the area, I have to take the magnitude of this vector. Please check my calculations because, hold on. So now the magnitude is just the square root of the sum of the squares, which is seven squared plus seven squared plus seven squared, which is three times seven squared, which is seven times square root of c. So that's the final answer. Any questions? So that's the gross product. And yes, that gives us the area of that parallel ground. I'm sorry? Does it look like distance formula? Well, it looks like distance formula for a reason because what we are calculating is the length of this vector. And the length of the vector is given by the same formula as a distance formula for the distance between the initial point and the endpoint, right? So in this case, the initial point is zero, zero, zero, and the endpoint is negative seven, negative seven, seven. So that's why we simply have to do, apply the distance formula. We are applying the distance formula, right? So let me just say this since the question was asked. If this is a vector and just any vector, not necessarily coming from this problem could be from any problem. If I have a vector, the length of this vector is just the distance between the initial point and the endpoint. So if this is point A and this is point B, we'll be the distance between A and B. And so it would be equal to the square root of the squares of the distances, of the differences of the coordinates. In the case of a dot product, there was a special case when the cosine was zero. And cosine is zero when two vectors are perpendicular. In the case of cross product, there is also a special case when sine is zero. But unlike the cosine, sine is equal to zero, theta is equal to zero, or pi, which is to say that the two vectors either point in the same direction so that the angle between them is zero, right? Or they point in opposite directions. So in this case, the cross product according to the first formula is the zero vector, right? Because the first formula tells us that you get a vector whose magnitude is zero. There's only one vector whose magnitude is zero and that's the zero vector. And for a good reason because in this case, if the two vectors a and b are parallel to each other, there is no plane which they span. So we cannot speak of a plane, we cannot speak of a vector perpendicular to this plane. Therefore, the only answer that we could possibly get is the answer zero, which incidentally is a very good opportunity to talk about the vector zero. Because this is something which people might find confusing. Partly, the confusion is due to a very unfortunate choice of notation that we have. So I'd like to say it once and for all, kind of to explain this once and for all. There are three different objects which are denoted in almost the same way. We have to know the difference between them. One is zero, one is zero like this, and there is also O. So how do you distinguish between this? So I'll kind of just to emphasize that these are numbers zero and this is, okay? So what are these objects? This is the simplest one. This is just number zero. This is something which you don't need to take, to understand this, you don't need to take multivariable calculus. You don't need to take calculus at all, right? It's something which we all know what zero means, hopefully. Now this is something which we are, which is very much to the point here. Amongst all the vectors, we have vector zero. And the easiest way to explain what vector zero is, is to think about vectors the way I explained last time. To think about vectors as a shift, to think of a vector as a shift in a three-dimensional space. As an operation of a parallel transport of all points in space. Applied to each point, it gives you another point which we illustrate in general, we illustrate by a vector, by arrow like this. So this vector is equal to this vector, to this vector and so on. Because we shift the entire blackboard this way. So each point gets shifted in the same direction with the same magnitude. But once we start talking about shifts, we have to include a very particular shift. Namely, a shift by nothing. The identical transformation, we do nothing. If we talk about, if we allow to take any shifts whatsoever, we certainly should be allowed to take a trivial shift. A shift by zero. That's the vector which is represented in this way. Now, we can't really draw it because see, we usually draw it when, because to draw it, we use the initial point and the final point. And then we connect them by segment and then we place an arrow to indicate from where to where we are going. If there is no shift, we start and end at the same point. And then there is no place to put an arrow. So it starts looking like a point, which it is not, okay? Because as we discussed, that the point is just a static object in space which is, doesn't know about anything else, just about itself, right? And the vector is a totally different concept. The vector is not a point, vector rather is an operation which you apply to all points at the same time. And that's why we can represent each vector by an arrow starting at each point. So this is a special vector which responds to the transformation by nothing, okay? That's what it is. Now you can ask, why do we need it? Why do we need such a transformation? If it's nothing, it's not going to help us. It is important because we have to include it for consistency because we know that vectors, we want to have various operations on vectors. For example, addition of vectors, multiplication by scalar. So for instance, we can take two vectors which are pointing in the opposite directions and which have the same length. Such vectors we can call a and negative a. Negative a simply means that it has the same magnitude but points in the opposite direction. Now we would like to have a consistent system of vectors in the sense that if we take the sum of two vectors we are going to get another vector. So in particular, we should be able to take the sum of these two vectors. What's the answer? We have to have a vector which is the sum of them but the sum of them is precisely this transformation which I'm talking about. You first shift the blackboard this way but then somebody comes and says, no, no, no, put it back. So you shift it back. The net result is that you did nothing but sort of nothing came out as a result of something. So to be able to have a sum of these two vectors we have to have this vector on our books. That's why this vector is important. That's one of the reasons. We would not have a consistent system of vectors if we did not include this but note that this is not the number zero because the sum of two vectors is not a number. In fact, if we want to represent it in coordinates, we would have to write it like this. And again, I emphasize that these are not letters but numbers zero. So that's the difference. This is just one zero and these are three zeros. So it's different but of course the meaning of these three zeros is very special because for a general vector like vector a, this one, two, three would correspond to the components of this vector. Here, this is a vector which has component zero in each place. So I hope that the difference between these two is clear. And finally, we also have this as if this was not enough. We have one more object which almost looks the same or notation for which looks almost the same. And that is actually the point which is the origin. When we draw our coordinate system, we oftentimes, so we like this, right? X, Y. We oftentimes need some notation for this special point because there's this very special point in the three dimensional space once we introduce the coordinate system, which is at the center of the coordinate system. We call it the origin of the coordinate system. So we oftentimes need a letter for it. And because it's called origin, it's natural to call it by the letter O, which is very unfortunate because it looks like zero. But it's not zero. The point is, however, that you can represent, so this is a letter O, but just like every point, P, you can write it in coordinates. And the coordinates would be 0, 0, 0, you see? So now it looks very much like this, but not the same because if we use the round brackets, we are talking about points. And if we use the angle brackets, we are talking about vectors. So this corresponds to the actual point, right? Actual point, which just sits here and it doesn't know anything about other points. Whereas this represents a vector, which you can think of as a transformation of the entire space, which you can apply to any point. It's just that the result will be that same point. You can apply it to this point, but you can also apply it to this point. And that's what we, the difference between these two we indicate by different notation. So this is something to chew on, I guess, if you're still confused about this, just think about it. I think it's fairly self-explanatory, but I think you will not be able to make much progress in this area, which we are discussing now, if you don't clearly understand the distinction between these three objects. So this is a very good test, how well you understand vector analysis and coordinate system in three dimensional space. I really wanted to emphasize this because this is one of the most common points of confusion. All right, are there any questions about this? Okay, so what do we do next? Next, actually, we are ready to do some fun stuff in space. We have basically laid the groundwork for it. We have developed the formalism we need, by which I mean finding vectors and studying some of their properties. And now we can actually use this formalism to start representing various geometric objects in space. So what kind of objects am I talking about? Sorry? Say again? Conics. Conics we will also represent but there are simpler ones. So we will start with the simplest ones and we will progress to the harder ones, right? Which seems like a good strategy. Conics will be sort of the next level of difficulty. That's right, lines and planes. This is something I already told you about earlier. The simplest objects are linear objects, which means in dimension one, lines, and in dimension two, planes. It's sort of, I guess it's sort of a philosophical question as to why these really are simplest objects. But mathematically it's quite clear because the equations and formulas which you use to represent them are the simplest possible ones, as you will see. So when we try to build a formalism for representing general curves and general surfaces in three-dimensional space, it's natural to start with lines as the simplest curves and planes as the simplest surfaces. So let's start with lines and let's talk about ways of representing lines in three-dimensional space. And here the idea that will be very useful is the idea which we learned when we studied parametric curves on the plane. The idea we learned back then is that to understand the general curve, it is a good way to understand the general curve is to parameterize it by an auxiliary parameter. So here you have a line and by this I mean a general line, not necessarily in this blackboard but could be on one of those tilted blackboards. So you have somewhere in the background a three-dimensional coordinate system which you are going to use to represent this line. So we have to decide how best to parameterize it. So the first point is that what information do we need to describe this line? And here is one convenient way to describe it. To determine a line, I have to tell you a particular point on that line and I have to tell you a direction of this line. And directions we have now learned to describe in terms of vectors. So what I'm saying is that in order to describe a line, it is sufficient to pick a point, let's call it L, pick a point of L and second, pick a direction vector. We are going to call this vector V and let's call this point P. Now, neither of these objects is unique. I could have chosen this point or this point or this point, anything from the infinitely many points which belong to this line. Likewise, the direction vector is not unique either. I could take any multiple of this vector. I could take this vector, this vector or going in the opposite direction like a vector. For example, this one would work also. So this is not unique, but I have to make some choices. Once I make these choices, once I make these choices, we will have determined completely what the line is. So all we need to do now is to give a description, a parameterization of other points on this line. We already have one of them, so we need to find other points. And doing this is actually very simple because we can now take advantage of the notion of addition of vectors. So here, let me explain this. Let me draw the position vector of this point. I recall that the position vector of a point is a vector which has the origin, the letter O as the initial point and our point P as the final point. So that is the position vector of this point. So now I would like to find other points on this line. Finding other points is the same as finding position vectors of those other points. So for example, I know another point on this line, which is this point, the end point of this vector. So the question is, how do I find the position vector of this point? Well, for this, we can use addition of vectors. And I recall that if you have two vectors, you can use a triangle rule to find the sum between them. Here, they're perfectly arranged for that. I first move from the origin to this point and then I move from this point to this point. So what's the net result? The net result is I move from the origin to this point. So let's call this position vector by R zero and let's call this one, this vector of R one. Then what I'm saying is that R one is equal to R zero plus V. So now by using addition of vectors, I have been able to find a different point on the same line. What about the rest of the line? Well, to get to other points, I can argue it's follows. If I take any other point, say this one, the vector which connects my initial point P to this one, let's call it P prime, can be expressed as a multiple of the vector V, right? Because parallel vectors are all proportional to each other. Certainly the vector P prime is proportional to V. So there is some number T one, say, for which we'll have this formula for some number. Actually, this simply means product of V and the scalar T. We can write it like this or we can write it in the reverse order T one times V, it doesn't matter for some number T one, right? But if so, then I can find the point, the vector P prime or P prime as R zero plus V times T one. To show it, to see it on the diagram, simply take this vector. You find the position vector for this point by taking the sum of the initial one and this one which is V times T one. But in fact, this is true for any point P prime. So for each point P prime, there will be a particular number T one. Or maybe it's better to just call it T so that we don't worry about the indices, right? And so we see that this way, we can actually parameterize, connection parameterize all points on this line. So in other words, we can write the position vector, say R, for a general point on this line as R zero plus VT. And that's going to be the crucial formula. So you see, we have exploited the notion of addition of vectors. This is why it was important to talk about vectors and about addition of these vectors because otherwise it wouldn't be so clear how to get to those other points which belong to the same line. Now let's write this formula in components. Let's write it in components. So we'll say R is X, Y, Z and R zero is X zero, Y zero, Z zero and V is A, B, C. These are three vectors. So now we just write this formula out in components. So we get the following. For the first one, we get X is equal to X zero plus AT. For the second one, we get this. For the third one, we get this. And now we look at this and we recognize that it looks very similar to parametric representation of curves on the plane which we have talked about for two weeks or a week and a half. And for a very good reason because in fact this is a parametric representation of the line, right? Because I have now parametrized general points of this line in terms of some auxiliary coordinate T. It certainly is very reminiscent of the general formula for a parametric curve on the plane. The difference is that when we talk about curves on the plane we only have two coordinates to describe X and Y. That's why there are only two lines instead of three lines here. Now we work in the three dimensional space so we have to specify the dependence of each of the three coordinates. But otherwise it looks very similar because this is some function. You can call it F of T if you want. This is another function called G of T and this is a third function which we can call H of T. Are there any questions about this? So I explained to you how to derive it by using vectors. But once you have the end result you recognize that this is simply parametric representation of the line. So in practice on homework and on various tests you're going to be asked to find parametric representation for a line like this. And this is really easy because all you need to know is a particular point on this line and a particular vector on this line or a particular vector which goes along this line like this vector V on this picture. That's all you need to know. And you can use the data which you are given in a given problem to find a point and a vector and then you just combine this information in a formula like this. So here is an example. It's a very typical question. Find the line which passes through the points three, one, negative one and three, two, negative six. So that's another way to describe a line because the line is also determined if you specify two points on it, right? There is a unique line which passes through two different points. The points should be different. Otherwise if two points coincide there are many lines which pass through one point. But if points are different then there's going to be a unique line which passes through it. But we would like to convert this information into the information that we need for this parametric representation. And so we need to know a point and the direction vector. So the point will represent by this position vector r zero and the direction vector will be our V. In this case we actually have two points. So you can choose either one. It's your choice. So pick one. Let's say the first one, three, one, negative one. So your r zero is, so you see I'm being really rigorous and pedantic here. I am distinguishing between the point and the position vector of this point by using different brackets, round brackets for points, angle brackets for vectors. It's very important as I already explained. So this is your x zero, y zero and z zero. And now to find V we can simply take the difference between the position vectors of these two points. You have two points on the line. You can surely find this vector by taking the difference between this vector and this vector. So this is V, you can write V OP prime minus OP. So I simply take the difference from three to negative six. I subtract three, one. What's the result? The result is zero, one and negative five. So now I have found what I needed, which is I have found a point on this curve, on this line and I have found the direction vector. And now I just put all this data into this formula. So the end result is x is equal to, this am I, x zero, y zero and z zero. So x is three plus, this is V, plus zero times T, which I can just erase just like this. Y is equal to one plus one times T. Z is equal to negative one minus five T. So that's the answer. When I say it's not the answer, it's one possible answer because someone else could instead choose as a reference point, not the first one by the second one or take the vector V, the difference not from this to this, but from this to this or even multiple of that. So in other words, you have to realize that there is not a unique parametric representation for a line. There are many different ways. So if you get a different answer from your friend, it doesn't mean that one of you made a mistake. It could be that both of you have the right answer. Any questions about this? Now there is another way to write the equation of a line. In general, we're going to have this formula, right? So let's focus on this formula. Suppose that A, B and C are all non-zero. Then we can express T from each of these formulas by dividing by A, B and C. So we can find that T is equal to X minus X zero over A. T is Y minus Y zero over B. And T is also Z minus Z zero over C. But it's all the same T. So we could write the following formulas, X minus X zero divided by A is equal to Y minus Y zero divided by B equals Z minus Z zero divided by C. So that's another way to represent a line. This is the first way, and this is the second way. I would like to explain what happens if one of these numbers, A, B, C is actually equal to zero, which is the case we have in this particular problem. So if A is equal to zero, then we have the equation X equals X zero instead of X equals X zero plus A T, right? So in this case, we cannot express T from the first equation because T is not in this equation. But we can express T still from the second and third equations provided B and C are non-zero. So if A is zero, but B and C are non-zero, we can write X equals X zero, Y minus Y zero over B equals Z minus Z zero over C. So we get two equations like this. It's not surprising that we get two equations. In fact, here it looks like one equation, but that's because of a sort of a nice way in which I wrote it. It's not really one equation that two equality signs. This line actually represents two equations. This is equal to this and also this is equal to this. These are two separate equations. And someone can say, wait a minute, it's actually three different equations because it says this is equal to this, this is equal to this and also this is equal to this. But the third equation follows from the first two because if this is equal to this and this is equal to this, then the first and the third are equal. That's the usual logical implication that we have for the equality. So this really corresponds to two equations. Two equations. And we can see it more clearly in the case when A is equal to zero. There are indeed two equations in this case. So in both cases, in the general case when A, B and C are all non-zero or in a special case when A is equal to zero but B and C are non-zero, we have two equations describing a line. There is also a special case when both A and B are equal to zero but C is not equal to zero. I'm not going to tell you what the answer is. You'll have to figure it out on your own but you will see that it's also two equations. And A and B and C cannot all be equal to zero because that would mean that this vector V is equal to zero but I insisted that this is a non-zero vector. Pick a direction vector. I have to make the same. For otherwise, we wouldn't be able to do this to argue in this way. What's your question? I didn't know that. Are you explaining? Oh, great. But sorry. If A is equal to zero, B is equal to zero and C equals equal to zero, then V is not defined. Well, V is defined but V cannot be used in this case. V has to be a non-zero vector. Now, so you see what happened is that we found actually two different ways to represent the line. This is the first way and an important feature of this first way is that we have one auxiliary parameter which we use to parameterize it, one parameter. Or we can write it in this way which really means that we write two equations for this line. And now I want to remind you that when we talked about curves on the plane, I explained that there are two different ways to represent a curve. This is the first way when we have one parameter but then there is also the second way in which you have one equation like y equals f of x, maybe let's call it f capital to not distinguish with that, or the favorite equation for the circle, x squared plus y squared equals one. So remember, there was exactly the same picture in the case of curves on the plane. We learned two different ways to represent such a curve. One is a parametric way where we use one parameter and another one is by means of an equation and then we have one equation. So what I want to emphasize now is the dimension count. Here we talk about curves, which means dimension one. Curves in the plane, the dimension of a curve, dimension of the curve is one and the dimension of the ambient space is two in this case because we talk about curves. This is about curves on the plane. I'm not going to write it down. I'm talking, ah, yes I did. Curve on the plane. So the object itself has dimension one and it's immersed, it's put in the ambient space which has dimension two. So when you parameterize it, the number of parameters is going to be equal to the dimension of your object, of your object. In this case, dimension is one, so there is one parameter. But the dimension of your object, dimension of the object, as I explained earlier, is equal also to the dimension of the ambient space, ambient space minus the number of equations. In this case, we have dimension one which is equal to two which is number of variables X and Y, the dimension of the ambient space, the plane, so we have to subtract one to get from down from two to one. That means that number of equations for a curve on the plane has to be one. That's why a circle is given by one equation, X squared plus Y squared equals one. That's why graph of a curve is given by one equation, Y equals F of X. What happens now is that instead of working on the plane, we work in space and in space, the dimension of is three and not two. So dimension of the ambient space now becomes three. But we are still talking about curves. Well, more specifically, we are talking about lines. But lines are special examples of curves. So our object is a curve, but now, so the dimension of the curve is still one, but the dimension of the ambient space is three. If you want to parameterize it, you have to still use one parameter, just like for curves on the plane. That's why it's not surprising that we end up with a parametric representation with one parameter. But if you want to describe your line by equations, you have to use two equations now because you have to be able to come down from three to one. You have to drop the dimension from three to one, so you have to impose two independent equations. So one is equal to three minus two. That's the number of equations. That's why it's not surprising that we really have two equations describing this line. Well, in this case, I arranged them neatly in sort of one equation, but as I just explained to you, it's not one equation, but it's two equations. Or in that case, also clearly two equations. So the same principle that I was talking about earlier of dimension count, dimension of an object being the dimension of the ambient space minus the number of equations still works here. And it will be interesting to see now how it works for planes. Before I get to planes, I just want to mention one thing which is that you can be asked various questions about lines. Once you learn how to represent them, you can be asked about relative positions of lines. And what can happen with lines is that lines can intersect or not. That's one possible question. Do these two lines intersect or not? Another question is, are they parallel or not? And so in fact, there are different relative positions that can happen. First of all, two lines can just intersect. That's a possibility. And you can easily find out whether it is the case. You simply write the equation for the first line with using parameter t and then you write the equation for the second line using a different parameter s because these are two different lines. Our priority don't talk to each other. So they have two different parameters. So what we need to know is whether for a special value, let's say this one is parameterized by parameter t but this one is parameterized by parameter s. So you want to know whether for some t equals t equals some special value t zero, there exists some special value s which is equal to s zero so that the resulting point is the same. And if you write this down, you will get a system of equations which may or may not have solutions. And that's how you know whether they intersect or not. The second issue which arises here is whether these two lines are parallel to each other or not. And that's real easy because part of a parameterization of the line is the choice of a direction vector. Two lines are parallel if and only if the direction vectors are proportional. How do you know if they're proportional? Well, you just see if one of them can be obtained by multiplying the other one by a scalar. That scalar will be immediately determined by the first coordinates and then you'll check whether it works for the second and third coordinates. That's how you see if they're parallel or not. So questions like this but once you have this parameterization, you can handle them. Finally, it's possible that the lines do not intersect and they're not parallel to each other. It's also possible. And here's an example. So let me keep this one, this line. So let's suppose I have this line. If I have also this line, I mean, first of all, this line can intersect this one easily. Even if it's not in this plane, that could be some in another plane, it could be a more general line. It can also be parallel, right? But these are not the only two choices because it can be parallel to this plane but not parallel to the line. So it can be like this. So if you kind of look like in this way, you see that they kind of intersect. But actually in three space, they don't intersect because there's some distance between this line and this plane. So in this case, they're called skew. And again, it's very easy to find out whether they're skew or not by using parameterization. So in the remaining minutes, I would like to talk about planes. So I talked extensively about lines, but now I want to talk about planes. Which means that now these are objects of a different kind, right? Because now these are objects which are two-dimensional. In the case of curves, we had analogous objects on the plane because the plane was two-dimensional. So the plane fits one-dimensional objects curves in a nice way. But the plane cannot fit any two-dimensional objects except itself, right? Or maybe some, you can cut some piece of a plane. It's also two-dimensional because it is already two-dimensional itself. But in three space, we can now have all kinds of surfaces. And again, the simplest surfaces which we study are the kind of linear surfaces, the planes. The question is, what is the best way to represent a plane in a three-dimensional space? So we look again at these two options. One is to parameterize. The other one is to write equations. If we were to parameterize a plane, we would need two parameters. Two parameters, not just one, because it's two-dimensional. But if we were to write down equations, we only need one equation, right? Because for a plane, the dimension of a plane is two, which is three minus one. So that's the number of equations. So we need to find a nice way to write an equation. Clearly, it is more economical to represent a plane by an equation than to try to represent it in a parametric form. Because in parametric form, we would need two parameters. Formals would be rather complicated. But we only need one equation. So let's just find out. Let's find a good way to write down equations for these planes. And the point is that there is a very nice, there is a very nice way to get to this equation because a plane can also be determined by a vector and a point. So remember, we discussed that a curve, a curve can be, sorry, a line can be determined by a point on this line and a direction vector. In the case of a plane, there is no direction vector. If there were, it would be a curve. It would be one-dimensional. So instead of talking about the direction vector of a plane, it is much wiser to talk about perpendicular vector to a plane. So a plane, unlike a curve, is determined by a point on the plane and a vector which is perpendicular to it. So instead of vector v, which is sort of like direction vector in the case of lines, we are going to have a perpendicular vector, determined by, and I will just draw it as a picture, by a point v and a vector n. Point vector n perpendicular. So now how do we write down an equation? Here we use the cross product. So that's where the cross product really comes into play because if you have a different point p prime and you connect these two points, this vector n is going to be perpendicular to this vector. You can easily see it on that, or using that board, just say. So this is your vector from p to p prime. This is your plane. This vector is perpendicular to any vector that you can have inside this plane. But the condition of being perpendicular can be expressed by using the dot product. I'm sorry, I said cross product, I meant the dot product. So I'll just write down this equation. So the equation is n dot p prime, sorry, dot pp prime is equal to zero. This is a dot product. Because remember we decided, we discussed and we agreed that two vectors are perpendicular if and only if the dot product between them is zero. That's what I'm writing. Now if I write n as a, b, c, and I write pp prime as before as x minus x zero, y minus y zero, and z minus z zero, and use the rule of dot product, I will find the following equation. So here you have one equation in which a, b, c, x zero, y zero, and z zero are given x, y, z are the three variables. And all x, y, z which satisfy this equation correspond to the totality of all points on this plane. So as promised, a plane is given by one equation and here's a way to derive this equation. So once you have this equation, there are all kinds of questions that can be asked. For example, if you have two planes, are they parallel to each other or not? What is the angle between these planes? But all of this can be immediately learned from this equation by looking at it and interpreting the coefficients a, b, c, and x zero, y zero, and z zero as this data. Okay, so I stop here and I want to say one thing which is that I have posted the solutions for the homework, for the first homework set on the b space, and I will be posting them regularly every Tuesday, Tuesday night when the last of the sections, just when the last of the sections is over, okay? So I'll see you next Tuesday.