 Hello. Today, we're going to talk a little bit about vector addition and how to add vectors. We know how to add numbers. 2 plus 3 equals 5. And we also know how to add scalar values. 2 hours plus 3 minutes are 2 scalar values that we have to do a little bit of conversion in order to add them first. For example, we can convert our hours into minutes, noting that there are 60 minutes in one hour, canceling, canceling. And we get 120 minutes plus 3 minutes, which is 123 minutes. But even if you can add 2 scalar values, the resulting value may not have a useful physical meaning. For example, let's say I take my age of 47 years and my son's age of six years. If we add them together, we can do so. We get a sum that is 53 years. So what? Is that meaningful? What does a number like 53 years mean when it was the age of the two people? That may not have a meaning, but the addition might still be useful. For example, I might want to know the average of our two ages. So the sum might be useful of those two things, even if the sum itself doesn't have a real physical meaning. Notice that the usefulness of the sum depends on the basis of the scalar values. In other words, how we defined it, i.e. the quantity, the dimension, and the entity that are being considered. For example, let's say we want to add a 2 meter length of a rectangle to a 3 meter width of the rectangle. But we can definitely add 2 meters to 3 meters. That gives us 5 meters. But what is that 5 meters? Well, that 5 meters is half the perimeter of the rectangle. Or they say that you can't add apples and oranges. Well, we can. 2 apples plus 3 oranges. We can add the 2 and the 3. We get 5. But 5 what? Apples? No. Oranges? No. What we end up with is 5 fruit. So it can make sense if we play around a little bit with our basis, what our definition is. If we have different entities, apples and oranges, but we redefine them all as fruit. Or if we have different dimensions to lengthen the width. But then we talk about what the combination of the dimension might mean, half of the perimeter. Vector addition presumes that the vectors and the sum are all expressed in the same basis, so that the components can be added individually. For example, if I have the distance along a path and the time spent along the path, I could express the first part as the first component in the vector, and the second part as the second component of the vector. Perhaps that's the first leg of my trip. And then I could consider the second leg of the trip. Looks like I might be running some sort of sprint here. And if I do add them together, I can add each component. 25 and 50 is 75 meters. 30 and 55 is 85 seconds. And this final vector would represent my entire trip as short as it was. So it's meaningful with this set of vectors, but it might not be meaningful if I add something else. For example, if my vector is age and heart rate, I might have one vector of 47 years or one set of components of 47 years and 64 beats per minute and a second vector of six years and 78 beats per minute, where the first one represents the statistics for an adult and the second one for a child. Yes, I could add those together. But when I do so, the values don't really have a meaning. Unless, of course, I again divide them to maybe find an average or there might be some other use for those. So the sum may not have a useful interpretation. For graphical vectors, components of a Cartesian basis can be added meaningfully. For example, if I have a vector that represents three meters east and four meters north, and I want to add another vector representing one meter east and three meters north, well, I can just add the components. Three meters plus one meter is four meters, and four meters plus three meters is seven meters east and north, respectively. I can sort of think about this as if we were adding along a number line. If I think about going east here and I create my basis and I think about going north and make that perpendicular and I add a basis, and if I start here at the origin, in the first case, I can think about going three meters east, one, two, three, and four meters north, one, two, three, four. That puts me at a location right there. Notice my scale is a little bit longer in the east direction than it is in the north direction. So there, that vector there can represent my first vector. Well, if I add the eastern parts, I can think about going one more vector unit east, and I can think about going three more vector units to the north and consider that to be my second vector. And notice the sum in the east direction is the three plus the one, whereas the sum in the northern direction is the four plus the three. And if I add those together, I get the four. Let's do this in purple. One, two, three, four plus the seven. One, two, three, four, five, six, seven. And there's the vector representation of the sum of those two. This leads to this head to tail representation of graphical vectors. You take each of the vectors. You start at some starting point. You move along until you get to the head of the first vector. Then you use that as the starting point for the tail of the second vector. And you move along to the head of that vector. And then your resultant vector, the purple one here, known as the resultant, which is the sum of the two, comes from connecting the tail of the very first one to the head of the very last one. So this works in our Cartesian basis, where we have two components, two basis axes that are perpendicular. However, if we try a spherical basis or a polar basis, since we're in two dimensions, notice that if I take similar vectors, I could take a vector here that I'm describing. One, two, three, four, five. Well, it's described this as being five meters with an angle of 53 degrees. And I could decide to add it to a second vector. One, two, three meters with a magnitude of three meters at 72 degrees. I can still do a vector addition adding components here. If I take five meters and three meters, I get 5 plus 3 is equal to 8 meters. And if I add the two angles, 53 degrees and 72 degrees, I get an angle measurement of 125 degrees. We can still do that measurement. Let's go ahead and draw it. Here's roughly 125 degrees. One, two, three, four, five, six, seven, eight. And there's eight meters. That is still a vector sum. However, that particular vector sum is not particularly meaningful. It's not straightforward what that represents physically. And so even though we can do the sum, we don't choose to do the sum in spherical or polar coordinates because it doesn't have a physical meaning to us. Here's some spatial quantities where addition does make sense. Displacement, if I want to walk to some place and then from there I'll walk to another place, I can talk about how much I displace myself with each of those vectors to find the resultant where I am. A second example might be velocity in a moving medium. For example, if I'm in a rowboat and I'm rowing across a river, but the river is flowing in a particular direction and I'm rowing across that flow, my actual velocity or my speed is going to be some combination of those two. And it makes sense to add those two vectors. And our third example, the one we're going to use most in something like civil engineering, is going to be an example of forces. If I have some system, for example our cart from horse and cart, I'm interested in the forces that are acting on that system. And they can be added together to determine whether or not the system is in static equilibrium or is going to move. There are a couple of ways to add vectors using graphical methods. One way, called the parallelogram method, can be done using a compass. So the compass tool, we take a line of a particular length and create a circle of radius with that length. Well, I'm going to take this particular compass tool and I'm going to stretch it out and I'm going to measure the length along this first vector. So I have the magnitude of this first vector. And if I take the magnitude of that first vector, move the compass to the head of the second vector. Now I'm going to go ahead and draw a little arc here, which means any point along that arc is going to have the same length from the head of F2 as that length of F1. I'm going to repeat that process, take the compass and stretch it out along the length of F2. Hopefully I can get it long enough here. And similarly, draw an arc that's the length of F2. That intersection point represents the head of the two vectors combined. Because essentially what we've done is we've taken vector F1 and applied it to the head of F2 or vector F2 and applied it to the head of F1. Either one is fine, but the resulting vector and this parallelogram is created graphically by this mean. Notice again, that's similar to this idea of head detail. The other way that you can simply do something like this is you can take one of the vectors and move it so that it's head detail with the other vector. It's a little harder to do that graphically because you want to keep the same direction, but you could measure an angle and measure an angle to make sure that you're moving it head to tail in that fashion. That head to tail method is also a very common method of adding vectors graphically. However, this graphic method doesn't usually give us good numbers. We can get numbers by using measurements, but they may not be as accurate as we might like, depending on how accurate our methods are with our compass and our straight edge and our measuring tools. The other way that we can go about adding vectors is using components. In order to add vectors using components, there's a few things we have to do. Let's go through the process. First, we define our basis because we just saw that we can add vectors in a Cartesian basis. So we define a Cartesian basis here. You see there is an x-axis and a y-axis that are perpendicular defined here. We express the components in a new basis. In this case, let's take a look at force 1 here. Force 1 here is at a 45-degree angle, and we can consider that as being two components, an x-component, f1x, and a y-component, f1y. If we use a little bit of trigonometry here, we see that f1x is equal to, if force 1 is the hypotenuse, we see that f1x is equal to f1 cosine theta 1 for 50 pounds times the cosine of 45 degrees, which, if we do the math, is 35 pounds. For the y-component, we recognize that f1y is equal to f1 times the sine of our angle 1 there, which is 50 pounds times the sine of 45 degrees, which also happens to be 35 pounds. We apply this same process to f2, recognizing that it has a different angle of 30 degrees. So now f2 of x is equal to f2 cosine of that angle, or 100 pounds times the cosine of 30 degrees, which is 87 pounds. And our y-component is similarly the sine of the angle. So 100 pounds times the sine of 30 degrees, which is going to give us 50 pounds. So now that we've found the components of both of the forces we want to add, we can just simply add those two components together. For example, the total force in the x direction is going to be equal to the 35 pounds from force 1x plus the 87 pounds from force 2x. We add the two of them together, and that's 122 total pounds in the x direction. Similarly, if we add the components in the y direction, we can find the 35 pounds contributed by force vector 1 and the 50 pounds contributed by force vector 2 for a total of 85 pounds in the y direction. Our last step here, we have found the vector here. We have both of the components. And that may be all that you want, but usually we are going to be interested in re-expressing the magnitude and direction of the resultant vector in polar or spherical coordinates. So we take our x component of 122 pounds, and our y component of 85 pounds. And we recognize that they create a right triangle. So to find our resultant, our resultant force, first we use the Pythagorean theorem. f squared equals fx squared plus fy squared. So we take fx squared of 122 pounds and fy squared, 85 pounds, and we add the two together. And then we take the square root of the entire sum. And when we do so, we get a result of 149 pounds magnitude for the resultant force. We're also interested in knowing what this angle is. We'll call this angle theta. Well, we know that the tangent of that angle is equal to opposite fy over adjacent fx, which in this case, so if we take the inverse tangent of both sides there, the angle is equal to the inverse tangent of the force y over the force x, which is the 122 over the 85. In this case, that gives us 34.8 degrees. Notice we can look at this graphically in a coordinate system. And you can again see how the addition worked. Our first vector, we had an x component. And our second vector had its own x component. The x components got added along the x axis, whereas the first vector had its own y component. And the second vector had a y component that got added along the y axis. And then you see that the resultant vector is the sum of both.