 Let us begin to learn to describe motion in all spatial dimensions. This lecture is simply part one of a much larger lecture on vectors, but we won't need to know everything about how to use vectors right now. We will merely define them and learn how to handle some basic operations. We want to go from one dimension to all dimensions. We live in a three-dimensional world, in space. Now there is an additional dimension for time, and we'll think about that some more later. But so far all we've been doing is describing motion in only one dimension. Again, think about the train moving forward or backward along a track. The track is the dimension and the train is constrained to only move on that dimension. But we need to do better, because in reality motion is complex and it takes place in at least one dimension, but it can be two or three, and of course the most realistic motions take place in three dimensions. But to get there we have to have a language in mathematics to develop such a description of the world around us, and for that we need vectors. Now in one dimension we can describe a quantity, like displacement or velocity, as a positive number which indicates motion in the forward direction along that direction or coordinate axis, or a negative number, which indicates backward motion along that same axis. It's the mathematical sign, whether it's positive or negative, that is in front of such numbers that carries the information about the direction of motion. So we've already learned something about how to indicate the magnitude or length of a movement and the direction of that movement. But how can we incorporate more detailed information about movement in a specific direction? That direction may not be only along one coordinate axis. We want to be able to generalize the details of motion so that we can discuss motion, be it displacement in space or velocity, or changes in velocity with time, so acceleration, in any direction in a three dimensional space. This can include quite complex motions of course. So how do we do that? Well the way we're going to do that is to use a kind of number called a vector. Vectors are a numerical system and they are really a simple expansion of already familiar numbers. But these kinds of numbers allow us to encode information about both length and direction. Now let's take a small aside here. Did you know that there are different kinds of numbers? Maybe this is the first time it's ever been laid out plainly to you. But in reality the kinds of numbers that are possible in mathematics are not limited only to the ones that we're most familiar with. So let's explore some of the ones that are more common in physics. There are scalars. Now scalars are the kind of number that you have really spent your whole life primarily using. So for instance, five is a scalar. So if you say I have five dollars, you are making a scalar statement. Now you can think of a scalar as a directionless number. That is a number with magnitude but no other information. If you say that you have five dollars, nobody can infer from that statement what the next amount of money you might have in time could be. So a scalar is a directionless number. It doesn't tell you anything about what the trend is or where to go next. Vectors on the other hand carry this kind of directional information. A vector is merely a number with direction. For our purposes in this physics course, a vector can be constructed from scalars. And we'll explore these next. Now let me just note that there are other interesting kinds of numbers but this is by no means an exhaustive list. This is just an example list and one of the interesting kinds of numbers are objects called tensors. In fact scalars and vectors are a kind of tensor. Tensors are a more general form of scalars and vectors. In fact vectors and scalars are a subset of all possible tensors that you can write down. So one example of a tensor that you might have seen before is a mathematical object called a matrix. This is merely an array of numbers and it can have rows and columns. So you can have a two by two array of numbers, a three by two array of numbers, etc. This is a kind of tensor. But tensors are by no means limited only to the number of dimensions that you can physically write down on a piece of paper or imagine in your mind. They can have unlimited rank. And rank is an indication of the kind of structure that you're describing with a specific tensor like whether or not it's a matrix or a vector or a scalar. And then there's an interesting kind of numbers that I feel is worth mentioning here only because it's a strange little beast and you may encounter it later if you continue in mathematics or physics. And those are numbers known as Grassman numbers. These are numbers where the order of multiplication matters. Did you ever think about that with scalars before? Whether or not the order of multiplication matters you're used to thinking that it doesn't. For instance five times two is ten and two times five is ten. The order in which you multiply five and two has no effect on the outcome. But for Grassman numbers that is no longer necessarily true. So for instance if A and B are two Grassman numbers A times B is not necessarily equal to B times A. And this may seem crazy but in fact this kind of number has all kinds of uses in mathematics and in physics it's found a home in advanced topics like supersymmetry and string theory which if you want to know more about that ask about it in the special topics portion of the course. So this is just a wet your appetite on numbers and their applications and the kinds of interesting numbers that are out there that are totally self consistent in a mathematical sense. But let's focus on vectors. Here are the tools that you are going to need to understand and use vectors. You're going to need algebra. Algebra is a symbolic system of representing things like numbers and you need to be comfortable with symbolic representations of numbers or other objects in order to be able to manipulate and get answers from vectors. You need to be comfortable handling arithmetic operations like addition, subtraction, multiplication and division between those numbers and in fact in this lecture we're going to learn how to apply algebra to vectors using already familiar concepts from scalar algebra. You also need to be comfortable with geometry. You need to be confident with concepts like lines, angles, squares, triangles and especially with the suite of tools that is referred to as trigonometry, the study of triangles and the relationships between angles and sides and so forth in such an object. Be comfortable with the relationships between angles and sides in a triangle especially right triangles. Triangle where one of the angles is a 90 degree angle interior inside the triangle. And be comfortable with trigonometric functions like the sine, the cosine and the tangent. I'm not going to go into details about these here. I've written them down on the slide at the bottom. You can see here the familiar definitions of sine, cosine and tangent with respect to the labeled sides of the triangle. Here is an angle which is denoted by the lower case Greek letter theta. It has an adjacent side. It has an opposite side. And then there's the hypotenuse of the right triangle. And here are the definitions of these trigonometric functions and how you can in fact relate them to one another, for instance. Be comfortable with these ideas because whether we use them today in this lecture or later in the course, you're going to exercise them over and over and over again. So let's begin to think about representing length or magnitude. These words are interchangeable as well as direction using numbers. So consider the following arrow indicated in blue on this two-dimensional coordinate system that I have drawn here on the slide. The horizontal axis denotes distances in the x direction in centimeters. The vertical axis indicates distances in the y direction, also in centimeters. And the blue arrow is the arrow I want you to think about for a good bit of the next part of this lecture. So indicating motion of some kind can be done with an arrow just like this blue arrow. It has a starting location, in this case it's 0, 0 in the coordinate system or the origin of the coordinate system. And it indicates where you should go to get to the endpoint of the displacement or the motion, whatever we're describing here. And that's indicated by where the head of the arrow is, the arrow head. Okay, so we have a graphical representation of displacement and space. In this case it seems to be entirely contained along only the x axis. So consider the following questions about this arrow. What is the length of the arrow? What is the direction of the arrow? If the arrow were longer or shorter, would that affect its direction? If the arrow changed direction alone, would that affect its length? So ponder these questions for a moment, get out some paper, get out a writing implement, jot some notes down, see if you can come up with answers to each of these four questions. What is the length of the arrow? What is the direction of the arrow? If the arrow were longer or shorter, would that affect its direction? If the arrow changed direction alone, would that affect its length? See if you can come up with answers to these questions, pause the video and we'll resume in a moment and see what the answers are to each of these four questions. Considering the first question, what is the length of the arrow, we can begin by simply reading the graph. And I've already indicated this a few moments ago. We can see that the arrow begins at zero centimeters along the x axis. So its initial x position is zero centimeters, and it ends at something like, and in fact the way I drew this, it's exactly like 2.5 centimeters along the x direction. So we can say that the length of the arrow is 2.5 centimeters. Now recall from earlier discussions of length that it's always supposed to be a positive number. Whether we flip this arrow around to the left or keep it where it is, as long as we preserve this 2.5 centimeter displacement from the origin, we've preserved the length of the arrow. So we could write that delta x, the displacement, is equal to 2.5 centimeters. But remember, displacement can be negative. If I put that arrow head way over here on the left side of zero along the x axis, then of course the displacement would flip sign, it would become negative 2.5 centimeters. We want a definition for length that enforces its positivity. It must always be a positive number. And for that, we can enclose displacement in a special mathematical operator known as the absolute value symbols. And this is just a pair of vertical lines that enclose the displacement, and whatever the value of the displacement is, the absolute value renders it positive. So even if the displacement were negative, enclosing a negative number in the absolute value symbols forces it to become a positive number, and this gets us what we want. So this is one definition of length mathematically. The absolute value of displacement is length, and that will be 2.5 centimeters for the specific example I've drawn here. So we've answered the first question. What is the length of the arrow? Let's consider the second question. What is the direction of the arrow? Again, reading the graph, the arrow points from left to right, from left where it starts to right where it ends. It points in the direction of increasingly positive x axis values. So you might have said something like the direction is positive, or perhaps the arrow points in the positive x direction, something like that, even to the right would work in this case. But you see the problem here. There are many perfectly valid, but individually subjective ways of indicating directionality, because we have not yet defined a firm mathematical basis that we can use to consistently communicate direction from one person to another, independent of the vagaries or nuances of language. We need a way to represent that sense mathematically, and we're going to develop that in just a few moments. We can't keep using arbitrary words to denote direction, so let's fix that situation. But before we do that, let's consider the other two questions. The third question was, if the arrow were longer or shorter, would that affect its direction? You might have had to think about this one for a bit. I'll have a corollary to this question in a moment, because some of you might have been overly clever in thinking about the answer to this question. But if you merely think about grabbing the arrowhead and compressing the arrow, or grabbing the arrowhead and stretching the arrow, we would be able to change the length of the arrow by doing that. You know, imagine that the arrowhead is made of a springy material that can be easily compressed or easily stretched. Fine, it still indicates direction, it still has length. You can shift the length around, but at no point does that cause the arrowhead to point in a different direction. So shrinking the length of the arrow, the arrow still points to the right, growing the length of the arrow, the arrow still points to the right. So it seems that length doesn't affect direction. Now some of you who are considering this question might have thought about pushing the arrow so far that it turns around and points the other way. But that's not the same as changing the length of the arrow. Changing the length of the arrow would be changing it from zero up to infinity, because length is a positive number. So whatever you do to this arrow to change its length, all you're doing is either compressing it down to zero or stretching it out to infinity. But flipping it around is considered changing the direction of the arrow. I want to emphasize that point now, because it will come in handy when you think more about the mathematical definitions of vectors in a moment. And finally, the fourth question was, if the arrow changed direction alone, would that affect its length? So imagine grabbing the arrowhead and just kind of rotating it around so it points in the other direction, but it's now made of an incompressible substance. You can't compress it, you can't stretch it. That arrow is really fixed in length. And so you flip it all the way around, does that change its length? And the answer is, well, no. And you might have had to think about this one for a bit as well. So we could have imagined something simple, like tilting the arrow until it points up along only the vertical axis, and it would still top out at 2.5 centimeters, but this time along the y direction, not the x direction. So indeed, it seems that turning the arrow in any direction preserves its length as long as you don't monkey with its length at the same time. So it seems that direction doesn't affect length. So let's sum up what we've learned by considering these four questions. An arrow is characterized by two quantities, length or magnitude, and direction. Length doesn't affect direction, and direction doesn't affect length. So however it is that we choose to represent arrows, we must have two pieces, two components to each arrow to represent its length and its direction. The mathematical representations of geometric arrows are what we call vectors. Now vectors in general don't have to have a physical interpretation. In a more advanced physics course or in a more advanced mathematics course, you might be given vectors where it's impossible for you to represent them on a piece of paper with some physical line. For our purposes, that won't be too much of a challenge. But I want to make the point that while it is true that we can use geometric arrows to motivate vectors, or vectors to motivate geometric arrows, they don't necessarily have to be connected directly to one another. But for our purposes, and especially because somebody like me is a visual thinker or a visual learner, I like to be able to ground my mathematical exercises in pictures, and I like to be able to take pictures and turn them into mathematics. And so for me, it's very important to emphasize both of these things. But for those of you who may have been exposed to the subject before or to more advanced mathematical or physics topics, don't think that you always have to connect a geometric or pictorial representation to the mathematics of vectors and vice versa. You can have very abstract vectors that cannot be represented pictorially. So let's continue with this idea. Let's begin by defining the mathematical notation for a vector. I will denote the arrow to the left, this blue arrow, as v with a little hat over its head. And that little hat is shaped like an arrow to indicate that it has direction. So this is the notation for the phrase the vector v. So when I say the vector v or the vector z, I mean the letter v or the letter z with this symbol over its top, a little arrow. Now that arrow might be a full arrow, it might be a line with an arrow head on it, or it might be a line with only the top arrow on it. You'll see these notations both used and they're both perfectly acceptable. But either of these is the notation for the vector v. So let me go ahead and add that label to my line. So now I've taken and put that symbol over the top of the blue line. So this is my vector v and there it is represented graphically. So now we need to be able to write an equation for v. The vector v must be equal to something mathematical. So we're going to try to relate it to its length and its direction. Now we've already kind of touched on length. The length of a vector is usually written simply using the letter without the hat. So for instance, if I want to denote the length of v, I can simply remove the hat and write the letter v and that is going to indicate its length. So anytime I see on a piece of paper somebody writing a letter with no arrow hat over its head, I take that to mean length or magnitude. If you meant vector, you need to put that symbol, the hat, the line with the arrow on the end over the letter. Okay, so that's an important thing to keep in mind. Notation matters because notation conveys meaning and meaning conveys intent in this subject. So you can explicitly denote the length using a more complex notation. You could, for instance, take the vector and enclose it in absolute value signs. For a vector, absolute value signs return not only a positive number, but that number is the length of the vector. So if I want to say mathematically the length of vector v, then this would be equivalent to either the absolute value symbol enclosing the vector notation itself or just the letter without the vector hat over it. Either of these two are acceptable notations for magnitude or length. And sometimes you'll find when you're solving a problem, it's more convenient to use one or the other to avoid confusion in the way you're symbolically representing things in your equations. So I leave it up to you. We will exercise this a little bit as we go through the course. You'll see why some notations are better in certain situations than others to avoid confusion, but either of these are acceptable ways of writing length or magnitude. So in our example to the left, this blue arrow over here, denoted by the v with the arrow hat over it, the length v with no arrow hat over it is 2.5 centimeters. Now I'm not gonna keep writing 2.5 centimeters. The reason we have algebraic systems is so that we don't have to write cumbersome numbers with decimal places and units all the time. And in fact, I'm gonna emphasize this point now, and it will be re-emphasized throughout the course, if not already in class time. It is important, if not wise, that when you are solving a mathematical problem, you keep things as symbolic as possible for as long as possible and put numbers in at the end. Because if you find yourself writing cumbersome numbers, even ones that don't seem so cumbersome like 2.5 cm, over and over and over and over again, you are bound to make transcription mistakes. It is easier to check symbolic representations of numbers and look for errors than it is to constantly keep big cumbersome numbers on a piece of paper and find all the little mistakes you made in writing down the decimal places. If this number were 2.5179, I'm guaranteed to screw up the order of the one and the seven at some point, if I have to keep writing that down over and over again. Symbols are your friend. Don't be afraid of them. Use them as much as you can. It's why we have algebra. Now let's see where we are. We can certainly write a partial equation now for the vector v, because we know how to write its length. So we can say that the vector v is equal to its length and then something else has to go in here to represent its direction. Now I've written multiplication and I'll motivate the multiplication sign in a moment, but just what the heck are we supposed to put in here to indicate the direction of this vector? We've only indicated its length. We have not indicated its direction. And for that, we need to exercise a new concept, unit vectors. To motivate unit vectors, consider the arrow shown at the left. Note now that I have a red arrow that starts at the origin and points to the right. Now this is not the arrow we've been looking at. The arrow we've been looking at was blue and it went all the way out to 2.5 centimeters before it stopped. This arrow has a different length. It is a length of one in units of centimeters. If I asked you, how might we manipulate this arrow to get it back to the arrow we had before with the blue arrow of length 2.5 centimeters? How would you answer that? Think for a moment about this question. How would you manipulate this arrow to restore it to the blue arrow we saw in previous slides? So think about that for a moment. Maybe jot some pictures down on paper, jot some sentences down on paper about how you might do this. Pause the video and we'll get to the answer in a moment. If you said something like, well, we could scale the length of this new arrow by a factor of 2.5 to get back to our old arrow. In other words, grab the arrow head, imagine the arrow is made of stretchy material, pull on the arrow head to the right until its length is now 2.5. That's equivalent to multiplying a one centimeter length object by a factor of 2.5 to make it a 2.5 centimeter length object. If you said something like that or something similar to that, then you are absolutely on the right track. The red arrow that I've drawn here is what is known as a unit vector. And the reason it's called a unit vector is because it has a length of one. Unit means one in mathematics and a unit vector is a vector with a length one. And the reason that these are so insanely useful is that simply by multiplying a unit vector by any real positive number, you can get an arrow of any desired length that points in the same direction. And if you want, you can flip the direction of the unit vector by multiplying by a real negative number. So that would be a way to flip the direction of the arrow from pointing to the right to pointing to the left. So instead of multiplying by 2.5 centimeters, you could multiply this unit vector by negative 2.5 centimeters, and that would flip its direction as well as stretch its length. And this can be illustrated quite simply. So I've taken the red arrow on the previous slide and I've multiplied its length by 2.5 and voila, we are back to our 2.5 centimeter length blue arrow. I could have instead of multiplying the length by 2.5 multiplied it by 0.5, so cut its length in half. And now we have an arrow that's half the length of the unit vector and still points in the same direction. So this just indicates the multiplicative action of stretching or squashing the length of the unit vector by a real number. Now unit vectors have a special notation all their own. They are represented by a letter or some other alphanumeric symbol with a little half triangular hat over their heads. So for instance, if I wanna write the unit vector that points with vector V, I would denote that as V with this little hat that looks like 2 thirds of a triangle over its head. So that special little hat always indicates that the length of this is one and its direction points in the direction of the original vector V. So let's put all of this together. We have the length of the vector V, which is given by just the letter V with no hat over it. We have the direction that the vector V points. We call this the unit vector V with this little triangular hat symbol over it. And verbally the way we say that is V hat or the unit vector V. Either of those are acceptable short hands for pronouncing this weird symbol. So V hat is how I like to say these things. So we have the length V. We have the direction V hat. The length scales the unit vector and so it must multiply the unit vector. And so we finally arrive at the final form of our vector V in this mathematical notation. The vector V is equal to the length of V times the unit vector V hat. And now we have an equation. It's complete. It contains the only two elements that we were able to identify that were needed to specify a vector, a geometric arrow, length, direction. Now let me make a comment on direction and positive signs and negative signs. We could flip the vector V around by multiplying the right hand side by a negative one. I like to group the direction together with the unit vector. So for instance, I might write this as the length V out in front and then enclose in parentheses negative V hat. That's just my taste. The rules of algebra, well specifically, the rules of the distribution of multiplication in a product state that it doesn't matter where we put that minus sign. I could put the minus sign in front of the V before multiplying the V hat. I could put the minus sign in front of the V hat and then multiply it by V. I still get the same answer at the end. It doesn't change the vector V. So I'll leave it up to you as to where you wanna put the minus sign, but if there is a minus sign present, I like to group it with the directional part of the arrow to help remind me that this is where the arrow points. And you'll see why this is important in a moment when we define the conventional unit vectors for coordinate axes. So for our original vector, we would write it in this notation as the vector V equals 2.5 centimeters, V hat, where V hat is a unit vector that points in the direction of the vector V. And there we go. I've labeled my blue arrow now with that information. Now, one last bit of notation, terminology, however you wanna frame this. And these are unit vectors that indicate motion along a specific coordinate axis like X or Y or Z. A unit vector that points only along the positive X axis is denoted by the special symbol I hat. A unit vector that similarly points only along the positive Y axis is denoted by the special symbol J hat. And a unit vector that points only along the positive Z axis is denoted by the special symbol K hat. Now, let me make a comment here. It was easy on a flat screen to draw an arrow to the right and an arrow pointing up. But we have a third dimension, which is Z. And the Z dimension, its positive direction points out of the screen toward our eyes. So I can't draw an arrow that does that. We don't have a 3D TV that I can play with here and I wouldn't use that technology anyway because it's cumbersome and complicated. So what we do instead is we have a sort of graphical notational shortcut. When an arrow points up and out of the screen or up and out of a piece of paper, we draw it as a circle with a thick dot in the middle. And that indicates the head of the arrow pointing out toward us. Think about a game of darts. You would not want to see the point of the arrow coming at you, but that point might look like the cylinder of the dart with the tip as a point coming at your eye. And that's kind of what this is meant to represent. Similarly, if an arrow points into the page, you would see its aerodynamic stabilizing tail feathers heading away from you, which is good for you. And that would be represented by a cross if this was heading into the page. We'll exercise that more when we need three-dimensional vectors later. So since our original vector pointed only along the x-axis, we can write it in a more common and final form. The vector v equals 2.5 centimeters times i hat because it only pointed along the x-axis. Now, remember the comment I made about liking to group the negative sign if one is present with the unit vector? You can kind of see why now. If this vector actually pointed to the left, i hat would still indicate positive directions along x, but I would flip that around by multiplying it by a minus sign. So negative i hat points in the negative x direction or in the direction of increasingly negative numbers to the left. So that's why I like to group the minus sign with the unit vector. Typically the unit vectors that you'll be dealing with are i hat for the x direction, j hat for the y direction, and k hat for the z direction. And you'll want to indicate whether you're moving along positive x or negative x. And so positive x would just be i hat, negative x would be negative i hat. That's why I like to group the sign with that symbol. Let's now consider the two vectors that are drawn over here in the graph on the left side of the slide. We have our original vector in blue, and we have this new vector here in red. I want you to study this, and I want you to try to answer the following question. Are these vectors, the blue arrow and the red arrow, the same or different? So pause the video, ponder this question, jot some notes down on a piece of paper and see what you can come up with, and try to answer this question. Are these two vectors shown here the same or different? Now let me nudge you a little bit. To help you with this question in case you've been struggling with it, or would like to check your answer with some additional information provided for you, consider the following guiding questions. Do they point in the same direction? Do these two vectors, these two arrows, point in the same direction? If so, they must be represented by the same unit vector. A second question you might ask is, do they have the same length? And if you judge that to be the case, then the number multiplying the unit vector must be the same. So, with those guiding questions in mind, are these two arrows, these two vectors, depicted here on the left the same or different? Pause the video, check your answer, think about it a little bit more, and when you're ready, resume the video. These two arrows definitely point in the same direction. After all, the arrows themselves lie entirely along horizontal lines, which would coincide with the x-axis or x-direction in this graph. They also have the same length. This original arrow had a length of 2.5 centimeters, and if you look very carefully, you'll see while this arrow starts at a half a centimeter, it ends at three centimeters. And so again, its length is 2.5 centimeters. So these two arrows point in the same direction, they have the same length, and since length and direction are the only two things that can describe a vector, then you must conclude that they are in fact the same vector. But how can this be? I mean, it's incredibly obvious, I would hope, that these two arrows do not occupy the same physical location in space. One is down here definitely occupying part of the x-axis itself, and the other one here is two centimeters up higher above it and shifted off to the right by half a centimeter. The answer to this is in the math. Because a vector is uniquely defined by only two things, its length and its direction, then because the length and direction of these two arrows is not unique, it must also be the case that these two arrows are not unique. They are in fact the same arrow shifted around, but with direction and length preserved. And by this definition, by this mathematical understanding of vectors and the arrows that represent them, it must be the case that in fact, they're the same arrow just shuffled around in space. So it turns out that this realization makes for one of the most useful properties of vectors. You can transport them anywhere you like, and so long as you do not alter the length or the direction of the vector, you have in fact preserved the vector. The vector can be said to be invariant under transformations of its location in space that do not alter its direction or its length. All of the vectors shown in the picture to the left over here are in fact the same vector, the vector v equals 2.5 centimeters v hat. It's the same vector we've been looking at since I drew this blue line on the x-axis to begin with. It's just transported around in space, but with no alterations to either its length or its direction. And that means in fact, mathematically, it is the same thing. Now things get more interesting once we expand our thinking to two dimensions. We've really been focused on vectors so far that lie along only one coordinate axis. And it's time to shuffle off the careful training wheels of this exercise and really try to master this concept a little bit more deeply. The arrow shown here at the left on the same coordinate grid, it in fact has the same length, 2.5 centimeters, as the arrow that we've so far been considering, but it certainly does not point in the same direction. So if you want, reverse the video a little bit, compare this arrow to the arrow we've been looking at previously, and you'll see that while the length may have been preserved, indeed the direction is definitely not the same. This arrow no longer only points along the x-axis. It points somewhat along y and somewhat along x and we have to find a way to think about that. So how can we write this in mathematical notation? How can we turn this graphical representation of an arrow, a vector into a mathematical vector? Now this is where the first signs of trigonometry come into play. We'll use more aspects of trigonometry going forward, but in order to make the arrow shown here, you can start to think about where triangles can play a role in understanding a more complex vector like the one shown left. For instance, we can think of this arrow as representing a journey, and that journey begins at the origin, zero comma zero, and it ends where the arrow head points. Now there are of course many ways one could make this journey. One could simply walk along the arrow. That's the most obvious thing you could do. You could take some really winding path that goes into all sorts of crazy places, but eventually comes back and settles here on the tip of the arrow. And in the end, the displacement would be the same, no matter what route we took, so long as the starting point is the same and the ending point is the same. So let's pick a convenient choice of journey that allows us to begin to think about how we represent this kind of two-dimensional arrow in a mathematical notation. Imagine first that we're going to make the journey by only traveling along the x-axis in the positive direction and going no further than where the tip of the arrow would, if it cast a shadow, fall down here on the x-axis. And then we're going to start from that point and only make a journey vertically along the y-direction until we reach the tip of the arrow and we will stop. The displacement is the same. The starting location and the ending location are the same as that represented by the blue arrow, but we did it in two convenient steps, one only along the x-axis, one only along the y-axis. So what we see by doing this journey from the origin, zero, zero, only along the x-axis, is that we would represent this part of the journey by another arrow. And I've drawn it here as a dashed red arrow, so as to distinguish it somewhat from the arrow we really care about, the vector we care about, which is shown in blue and is solid. So the red dashed arrow represents that first stage of the journey along the x-axis. Now this takes us to a coordinate location that happens to be 1.5 comma zero. We have now, as I mentioned before, traveled to the farthest point along the x-axis without going past the tip of the arrow. We can then continue the journey as shown in this purple dashed arrow by going from the tip of the red arrow to the tip of the blue arrow. And this takes us to our final point. This second phase of the journey is only taken along the y-direction, so this is a perfectly vertical arrow shown here in purple with the dashed line. So what shape does this whole thing, the blue arrow, the red arrow, and the purple arrow, what shape does it represent? Well, if you said a right triangle, you're absolutely correct. This is a triangle where one angle, in fact, this one down here where the red arrow meets the purple arrow, is 90 degrees. Now the long side opposite the 90 degree angle is the blue line, our original vector, and it is the hypotenuse of this right triangle. The two remaining sides of this triangle that we have constructed here by imagining this journey, they are what are referred to as the components of the blue vector, the vector v, the original vector. One component lies only along the x-axis, one component lies only along the y-axis. So how would you write the x-component shown in red with the dashed line? How would you write the x-component of this vector? Use what you have learned about vector mathematical notation and try to do this. Get out a piece of paper, get out a writing implement, and try to write an equation for this horizontal component of the blue vector in the mathematical notation we've been exploring so far. Keep in mind that the length of this horizontal side here is 1.5 centimeters. Pause the video now and try to work on this question on your own for a little bit, and resume the video when you're ready to go further. The answer is that the x-component can be written thus. I've chosen to write the x-component as a vector but with a subscript x to remind me that this is a piece of the vector v that lies only along the x-coordinate axis. And it would be equal to the length times the direction. Well, the length is 1.5 centimeters. I could have also written this symbolically as just v sub x with no hat over its head. And it points in the positive x direction and that corresponds to a unit vector i hat. So using this lesson, how would you now write the y-component of this journey, of this blue vector, the purple dashed vector? How would you represent it in mathematical vector notation? Keep in mind that its length is two centimeters. So go ahead, work on this on paper for a moment, pause the video and resume the video when you're ready to move on. The answer is that the y-component is written in the following way and the notation should be clear at this point. The y-component of the vector v, which I've denoted as v with a subscript y and hat over it is equal to the length which is two centimeters, times the unit vector that points in the direction that the component points and that's entirely along the positive y-axis. So that's represented by j hat, the unit vector that only ever points along the positive y-direction. Since the total journey is accomplished by first, journeying along the x-axis, shown by the red dashed line, and then adding to that an additional journey only along the y-axis, how do you think that the whole vector, v, should now be written? You have the component along the x-axis, that represents the red dashed, you then add to that a journey along the y-axis, you have written that component down as well. How would you mathematically represent the entire vector v using this information and the components we have so far? Try to work this out on paper, pause the video while you're doing that, and resume the video when you're ready to move on. The answer is that the whole vector, v, with the hat over it, is written by adding together the two components. You simply sum them. You make the first journey and you add to it the second journey. You make the horizontal journey and then you make the vertical journey. And this is simply addition of two vectors. We have a vector who only lies along the x-axis, 1.5 centimeters i hat, and we have a vector that only lies along the y-axis, two centimeters j hat. They have no things in common that allow us to simplify this further. And so this is about the simplest that we can write this particular vector, and we're done. So the vector v is 1.5 centimeters i hat plus two centimeters j hat. And we're done. That's as far as we can go with the mathematics of this. And congratulations, you now have an equation that represents the graphical blue arrow shown over on the left. Now one last point can be made here. How do we know that this is all self-consistent? How do we know we've done a good job of representing the vector v with these components? After all, this thing we've drawn here is supposed to be a right triangle and that should give us a test that allows us to check our answer. So can I demonstrate that the answer that I have written down for the vector v makes sense? Here's a test. I can do so by employing the Pythagorean theorem. The Pythagorean theorem states that for a right triangle, the length of the hypotenuse, which can be written as h, and that would be the length of the blue arrow, when squared, so h squared, should be equal in length to the sum of the squares of the lengths of the other two sides of the triangle. So in other words, if I denote this side as l with a subscript one and this side is l with a subscript two in length, then if I square l one and I square l two and I add those together, the Pythagorean theorem states that the length of this side over here, the hypotenuse, squared, should be equal to that sum. Then you can just take the square root of h squared to get h. So let's check this. For our example, we know that the length of the hypotenuse was 2.5 centimeters because that was the vector we've been playing with this whole time, a vector of length 2.5 centimeters. So let's take the lengths of the sides, 1.5 centimeters on the horizontal, two centimeters on the vertical, we'll square 1.5 centimeters, which I've done here, we'll square two centimeters, which I've done here. So this gives us 2.25 centimeters squared and four centimeters squared, which when added together yields 6.25 centimeters squared. And if I now take the square root of 6.25 centimeters squared, you will find that h equals 2.5 centimeters. So let's go a little bit further and let's take this idea of adding two vectors together to a more general next step. In order to build the vector v, we had to add two vectors together, but each of those vectors laid only along an entirely one coordinate axis. One vector, the horizontal component, was only along the x-axis, and one vector, the vertical component, was only along the y-axis. But consider the vectors that are shown here in the graph on the left. We have the blue vector, which is the same vector that we've been looking at this whole time, but now we have a new vector, v2, which is shown in red, and I've drawn it so that it starts at the end of the first vector and then goes to its end point. But I could have also drawn this vector as starting at the origin and having the same length and direction. Remember, the vector is the same as long as you preserve its length and direction. So I can take the red vector and I can move it wholly as long as I preserve length and direction anywhere in this grid. So I could have easily drawn the red vector as also starting at the origin and going up to its end point in the direction it currently points. I chose to do it this way because it will be helpful with the next step. How might we add these two vectors together? I mean, vectors are just a kind of number and numbers have arithmetic operations defined for them. So how would we add together two vectors? We want a single vector that represents this entire journey. So let's see if we can work through this. We might imagine, writing out graphically, the components of these vectors as shown in the left. You see that I've done a horizontal journey and then a vertical journey to get to the end of the blue arrow and then I've done a second horizontal journey and then a vertical journey to get to the end of the red arrow and this makes two right triangles. So I can see that my two vectors, v1 and v2, have each x components and each y components. And we can see that to make the full journey, all I would really need to do is travel the v1 x component followed by the v2 x component and then take the v1 y component followed by the v2 y component and that would get me to the end of this two vectors. So all I need to do to get the total vector represented by adding v1 and v2 is to add the components of the individual vectors together, add the x components, add the y components and this will give me, as a result, the big vector, capital V with a hat over it that represents the sum of v1 and v2. So let's write each vector using components and then do that addition. So here we go. Here are the mathematical representations of our two vectors. v1 hasn't changed. It's still 1.5 centimeters i hat plus two centimeters j hat. Now there's v2. Well, if you had looked carefully at this, you might have drawn the conclusion that in order to do the horizontal walk, the x component of v2, I just have to go a distance of 1.5 centimeters from where I start to where I end along the x axis. So again, that's 1.5 centimeters i hat. And then to get to the tip of the red arrow, all I have to do is make a vertical journey of one centimeter along only the y direction, which is j hat. So v2 is 1.5 centimeters i hat plus one centimeter j hat. Now let's add the two vectors together mathematically. And before we do this, I want you to keep something in mind. Adding these two complex objects together, v1 and v2, is really no different than adding one algebraic expression like 5x plus 6y to another algebraic expression like 7x plus 2y. What you do is you group the coefficients of like terms. So if I'm gonna add this expression, 5x plus 6y, to 7x plus 2y, I see that they both have an x and an x term, and they both have a y and a y term. So I can group the x terms together and then add their coefficients, and then I get a single coefficient that multiplies x, and I can do the same thing with the y terms. And I've demonstrated that here in line. 5x plus 6y added to 7x plus 2y. Well, you have to add 5 and 7, and that gives you 12. So the resulting simplified expression is 12x. And then you have 6y and 2y. So you're gonna add 6 and 2, and the resulting term in this expression is 8y. So the final simplified sum is 12x plus 8y. In vectors, it's the unit vectors that take the place of the algebraic symbols like x and y, and you merely add their coefficients together. So group like unit vectors together and add their coefficients with signs included. So here we go. I want to add v1 and v2. I write out v1 in all of its glory, and I write out v2 in all of its glory, and I bracket them off to remind myself that originally these were two separate entities that I'm now summing together, shown here with the big plus sign. Now what I do in the next step is I group like terms. Well, I have a 1.5 centimeter i hat term, and I have a 1.5 centimeter i hat term, so I can add those coefficients together, and the resulting sum will wholly multiply i hat. And similarly with the j hat terms, I have two centimeters and one centimeters. Those can be added together, and that result wholly multiplies j hat. So let's do that. And this works out pretty nifty. Three centimeters i hat, three centimeters j hat, that is the vector capital V with the hat over it. So I could have just done the following. This is the graphical way of representing the sum of two vectors. We can see that the big vector V is just one long journey along the x-axis that happens to take three centimeters, followed by one long journey along the y-axis, which also happens to take three centimeters. And so we could have simply represented it by this purple arrow shown here in the graph on the left, labeled with the capital V with the hat over it. It starts at the origin where the blue arrow starts, and it ends at the unmatched arrowhead of the red arrow, which is where the second part of the journey ends. So to add vectors graphically like this, all you have to do is put the vectors tip to tail. The tip of V1 is located at the same place as the tail of V2, if I'm adding V2 to V1, and then connect the unmatched tail, which happens to be the one that belongs to the blue vector, to the unmatched arrowhead, and that happens to be the one that belongs to the red vector, and draw a line that connects them directly straight in the coordinate system. That is the resulting vector from summing, from adding these two vectors together, adding V2 to V1. Now, a slight twist on this is subtracting vectors. Mathematically, subtracting vectors should now seem much more straightforward. I mean, after all, adding V1 to V2 is not much different from multiplying V2 by a minus sign and redoing all your arithmetic operations, where instead of adding the coefficients of the V2 terms, you subtract them. So mathematically, this is not a big leap, but what I wanna emphasize is that there's a way to graphically do this as well that will help you to check your mathematical answers. So again, here are V1 and V2, and as I said before, I can draw V2 anywhere I like, and so for now I'm gonna draw V2 starting at the origin and ending up here at 1.5 centimeters and 1 centimeters in the coordinate grid. This has the same direction and same length as V2 did before. And I'm gonna try to subtract V2 from V1. So I'm gonna do V1 minus V2 and get some new vector capital V from this. So you should try this out. You should work this out on a piece of paper, and if you do this, you should find that mathematically the answer is what is shown here, that the new vector, capital V, is a X component that has no length. It's zero centimeters in length because 1.5 minus 1.5 is zero. And it's Y component is one centimeter in length because two centimeters minus one centimeter is one centimeter. So I wrote this out in kind of, an exceptionally long way of doing it, but there is no X component because it has zero length. And so really the resulting vector only has a Y component. It only points vertically and it's one centimeter in length. So I've represented this graphically over here with a vector that starts at the origin and goes up one centimeter in length and stops on the Y axis here. But keep in mind, I can transport this vector anywhere I like. And so long as I preserve the length and the direction, it's still the same vector. So let me do this in a suggestive way. Let me move that vector over here so that it starts on the tip of V two and it ends on the tip of V one. This is completely allowed by the rules that we explored before. And so a graphical interpretation of the vector subtraction now becomes clear. If I subtract V two from V one, the resulting vector can be represented graphically as a line that starts at the tip of V two and ends at V one. So all you have to do is to get the drawing of the resulting vector from the subtraction of two other vectors, you are going to start that third vector on the arrowhead of the second vector and draw it pointing to the arrowhead of the first in the subtraction terms. What would happen to the capital V vector if instead I had calculated V two minus V one? What I've shown you here is V one minus V two. But what would it look like if I did V two minus V one? Now I'm not gonna show you the answer to that. Why don't you think about that a little bit? Play around with it, do it mathematically, do it graphically and see if you can come to a comfortable answer to this that you're satisfied with. This is a good place to try this out as a bit of a leap off the known territory and see how comfortable you are with the answers that you get. Let's review the key ideas that have been introduced in part one of this lecture on vectors. The key ideas that we've learned in this section of the course are as follows. We have learned some of the kinds of numbers that are available in mathematics to describe physical quantities. We have learned to represent motion, that is displacement, velocity, acceleration, anything that can be represented by a vector in more than one dimension using the more general vector concept and its associated graphics and mathematical notation. And as a first foray into using vectors, we have learned in both a graphical sense and in a mathematical sense to add and subtract vectors. I want to comment here that in a later lecture, part two of the vector lecture, we will learn about additional vector operations, arithmetic operations that are needed to describe the physical world. But for now, what we have encapsulated in this discussion will suffice for our needs for multiple topics to come in the course.