 Hey everyone, welcome to Tutor Terrific. Today, we're going to start my third unit on physics. We are going to be looking at kinematics in two dimensions. In the last unit, as you know, we were looking at kinematics in one dimension. So, we're going to be studying motions now in two dimensions. We're going to look at the 2D system, our two-dimensional world, and how it's based. We're going to look at vectors after that, all the combinations and all the components and how we resolve those in two dimensions. So, let's start first with this idea of the Cartesian coordinate plane. This is our two-dimensional world. What we're used to, guys, is this, the number line. The number line was our one-dimensional world. We could go right or left, and if we rotated it so it was vertical, we could go up or down, and those were our only options. We were quite restricted. Well, that world is gone. We have now replaced it with this. Two number lines, like this, where their respective reference points or their zeros are at the same spot. This is the Cartesian coordinate plane. This is now the world in which we live. So, let's first talk about how position is mapped in this plane. As you are well familiar with from algebra, we have two coordinates for our numbers for each position. The x and the y coordinate. We need both of those to specify a specific location in two dimensions. These are also known as ordered pairs. Now, a very special point will be zero comma zero. So, zero on the x and zero on the y, that would be the origin, as you already know. That's the two-dimensional reference point. These axes are both spatial, and I know we looked at what looked like two-dimensional graphing in the previous chapter, but in that instance, one of those dimensions was actually time, and the other one, the vertical axis, was your one-dimensional world as far as position was concerned. This is no time information on this graph at all. This is just two-dimensional position right now. If we wanted to include time in our analysis, we would show objects moving, and we would draw a trajectory into dimensions like this. And as time goes, sort of like a parametric curve, for those of you who are familiar with that from pre-calculus, this would be your trajectory, and as time goes on, the object moves to a different ordered pair or coordinate, a pair of coordinates in this plane. Now, vectors, okay? Vectors, we are used to now in one dimension that kind of look like this, that could be positive moving to the left, excuse me, positive moving to the right, or negative moving to the left, and we were very limited as far as which way we could go. And if there is up or down, the vectors could point up or down with different lengths. Well, things are a little bit more complicated in two dimensions, and some people say it's more free. We could actually graph position with vectors, okay? So let's say we've got this vector over here, vector one, it's at 3 comma 1. If I draw a vector from the origin, so its tail is at the origin, and the tip is at that point, this could be a vector for position. Just like we did when we were in one dimension where we would start at zero on the number line and go to the location of our particular point. And here's another vector for position V2, okay? Negative one comma negative two. So position vectors start at the origin and end at the location, that's one way to do that. Well, here's another example, 3 comma 2 for this point P. But what about this vector over here? What's going on there? It doesn't start at the origin, okay? We're not going to talk about that. Well, vectors do not have to start at the origin. Position vectors do, but displacement vectors do not. They can end and begin at any coordinates. And the numbers you show, such as right here, negative one comma negative two, illustrate the motion from start to finish. So here I start at negative two comma zero, and I move down to negative three comma negative two. So that's the change of negative one in the x direction and negative two in the y direction. These would work as displacement vectors, one position to another. All right, now these vectors, as you can tell, are at an angle. How do we deal with that? How big is the angle? Is it pointing up and to the left, down and to the right? There's all different combinations now. It seems like an infinite array of possibilities for the way these vectors could look. Well, there's ways to resolve those into components. Now, this process is to break down a single vector at an angle into a horizontal and vertical vector that make it up. These horizontal and vertical vectors are called components, and the process of getting them is called resolving. So I have here a vector I call A, okay? Now, we can look at A, and it's at some angle. Now, we would draw the components by looking at how far it moves horizontally separate from how it moves vertically. So it moves a decent amount from the tail to the tip, from the left to the right this much, okay? So I just draw that section, that's the horizontal component. Then, if I look vertically, it moves quite a bit vertically from this position lower to this position higher over here. So this vertical vector takes that into account. So now that I've drawn the components, how I find their magnitudes horizontally and vertically as far as length is concerned. Well, for the vertical component I call A sub y, that would be the magnitude of the entire vector A times the sine of this angle theta. This angle theta represents the angle between the horizontal component and the vector itself. The horizontal component, specifically A sub x, can be found by taking the magnitude of A and multiplying it by the cosine of this angle. Again, the angle must be defined with respect to the horizontal. So basically this little set of phrases here makes it clear what to do. To get the horizontal component, multiply the magnitude of the original by cosine of theta. For the vertical component, multiply the original magnitude by the sine of theta. Make sure that this angle that you're using is defined with respect to the horizontal and is not this angle up here, which is defined with respect to the vertical. If you were to do that, you know from trig, for those of you who are experienced with trig, you should be in physics, at least at the Algebra 2 level, you would flip these two formulas. So if the angle was up here, the A y would be A cos theta and the A x would be A sine theta. But usually what we do for a simplicity and sort of consistency is we would define the angle with respect to horizontal by default. All right, so let's practice. We've got this angle, 50 meters per second, so it's a velocity vector at an angle of 60 degrees. I want you to resolve this vector v into components. First we draw the components. So here's the horizontal component and here's the vertical component. So we can see this right triangle we're going to use. And the horizontal component would be 50 meters per second, the angles, I mean, the magnitude of the vector times the cosine of 60. Okay, 50 times the cosine of 60 degrees meters per second, which would give you 25 meters per second. V y, the vertical component would be 50 times the sine of 60 degrees in meters per second. That would give you 43 meters per second. Now, if these particular vectors were drawn and labeled, they'd look like this. 25 meters per second for the horizontal component and 43 meters per second for the vertical component of this vector. Now, what if we have a vector who's not in the first quadrant? Okay, we've got angles and we know from a little bit of studies of the unit circle or from the Cartesian coordinate plane that each section is a quadrant. This is quadrant one, this is quadrant two, this is quadrant three, and this over here is quadrant four. So clearly this vector is in the second quadrant, not in the first. So the angle that we are used to using with respect to the positive horizontal x-axis is larger than 90 degrees. Well, there's another angle here that is less than 90 degrees, called the reference angle, that is related between the vector and the nearest horizontal line, which would be the negative x direction. So how could we find the components dx and dy of this vector, which are labeled in black? Let me give you some measurements to start. Let's say that the magnitude of the blue vector d is 253 meters, so it's a position vector, and the black angle here is 55 degrees. We could find dy and dx, the components, the following way. We could use the black angle. So dx would be the magnitude of d 253 times cosine of 55 degrees, and dy would be the magnitude of d times the sine of 55 degrees. That's fine, but because we're in the second quadrant, dx would have to have a negative sine slapped onto it, because we know that in the second quadrant our x component is negative and our y component is positive. We could do that. Now, another way to do it would just be use the original red angle, even though it's bigger than 90 degrees. In your calculators, the way cosine is defined, if you would do the cosine of 180 minus 55, 125 degrees actually, you would get a negative value for dx by default, and that's what's meant by this statement here. 253 sine of 125 degrees would continue to be positive. So if you use the red angle instead of the black angle, in other words, you found the red angle and then used it, you would get the proper signs by default. Now, you might come across what are called bearings, and there's many ways bearings are used. This is not the way that air traffic controllers use bearings or pilots, but this is another way to write bearings, something like 20 degrees south of east or 47 degrees west of north. What in the world is this garbage? What am I talking about? Well, let me explain. In order to figure out the direction of a vector using this bearing language, we would start with the second word as our home base. Here, east, for example, we'd start with a vector pointing directly east, and we'd go in the angle and the direction of the first word, so 20 degrees south. We'd start at east and go 23 degrees in the southern direction, so it'd be a vector that would point from the origin to about here. Nice compass. Well, you may be given an angle with respect to the vertical, and we'll need to find its complement so that you can have an angle with respect to the horizontal so we can use our formulas. 47 degrees west of north is an example of that since we would start at north and we would go 47 degrees west. This angle would be defined with respect to the vertical. I'm going to give you another one here. 72 meters at 55 degrees west of south. I don't want you to resolve that into components, so we have its magnitude and we have its angle west of south. We start at south and we go 55 degrees west. I'm going to call this vector s. This angle right here is 55 degrees. That's not defined with respect to the horizontal, so we would find its complement. 90 minus 55 would give us this angle here. It'll be 35 degrees, and here I'm going to draw it for you. 35 degrees. Notice how this vector is in the third quadrant. It's in the third quadrant. That means that both the sx, s sub x, x component of s, and the y component of s are both negative. If we had been in the first quadrant, we would have north of east or east of north type bearing language used. I'm going to slap those negatives on and use this 35 degree angle here, not the angle with respect to the positive x axis in the easterly direction. I have a negative 72 times the cosine of 35 degrees. It gives me negative 59 meters, and then sy would be negative 72, the original magnitude, times the sine of 35 degrees, which would give me negative 41 meters. Those are my two components. If I were to draw those, I'd draw the sx from the origin going all the way out to the same horizontal position as s. Then I'd start there and go straight down for sy to the tip of the original vector s. You might see a phrase like to the horizon instead of this bearing language. That means we start in the positive x direction, so east, and we go north. It would be the same as north of east. If you saw 20 degrees to the horizon, that means 20 degrees north of east. Let's look at combining vectors. There's many ways to do it. Let's refresh ourselves with some addition and subtraction of vectors in one dimension. Addition. Vectors in the same direction combine to make one vector whose length is the sum of the vector's individual lengths, such as these first two examples here. Vector with magnitude 5 plus vector with magnitude 5 in the same direction equals a vector in that same direction with a magnitude equal to the sum of the originals 10. These two vectors, which point in opposite directions with the same magnitude, 5 plus vector negative 5, would be a zero vector, so it would have no length. Then we have these two vectors, 5 plus 10. Well, it would be a very long vector with a magnitude 15. What about 5 plus negative 10? Now we've got adding vectors in opposite directions. We combine to make one vector whose length is the difference in the individual lengths, the difference in the individual lengths. For here, we have 5, the vector 5, and a vector negative 10. Well, the difference in those lengths is 5, and it points in the direction of left because the larger vector points in that direction. This is just like adding integers. Here's another one. 5 plus negative 15, the difference in those lengths is 10, but we'll point it to the left again because the larger vector is negative 15 and wins out. What about these two? Vector 10 up plus negative 5, which would point down, the difference is 5, and it would point up because we have a vector 10 that's larger, longer than the vector negative 5. Now let's look at subtraction. When we're subtracting vectors in the same direction, we subtract by mentally switching the direction of the second vector and following addition rules. You could look at this vector here and treat it as this vector here, negative 5. 5 plus negative 5 would be 0. That's one way to do subtraction. In this case, you would write a plus here and turn this vector around, so it would be 5 plus negative 10, like this situation over here. The result should be negative 5. What about this one, where it's already pointing in the opposite direction? When you subtract vectors in the opposite direction, you would subtract by mentally switching the direction again of the second vector and then following the addition rules this time. It would be 5 plus positive 10, which would give you 15. This one, 10 up minus negative 5 down would be the same as 10 up plus 5 up, which would give you positive 15 upwards. This is how we would add vectors in one dimension. Now the reason I showed you that is because it is important when we relate to two dimensional adding. But first, before I leave this, I want to remind you that when you combine vectors, the result is called the resultant. All of these answers over here on the right side are called resulted vectors. Now let's go to two dimensions with that one dimensional foundation. Addition and subtraction, there are multiple ways to do both. To add two vectors in two dimensions, we can do the tailed tip method. This is the favored method by most students. What you do is translate one of the vectors so that its tail is at the tip of the other. Look at B right here. What I do is I move B until it's at the tail of A. The tip of A. Right now it's at the tail of A and it could be anywhere, but I'm going to move it so that it starts at the tip of A. I'm translating it so I don't change its magnitude or direction. I just move the whole thing isometrically, which means I don't change anything about it. It's congruent to the original value of itself. Now B is at the tip of A. Tail to tip. What you do is you draw a vector from the first vector's tail to the second vector that you translate its tip. This vector C would be A plus B. Here's another example. I have A lined up here and I put B right at its tip. The tail is right at A's tip and I keep B the way it was before where it originally was located. I draw another vector from the tail of the first vector to the tip of the second. In this case it's called R, but in this case it was called C. This vector R is vector A plus B. As you might have guessed the length of R is not equal to the length of A plus the length of B unless the vectors were originally parallel and that's how 2D vector adding works. You can add in either direction. I could have put A at the tip of B and I've gotten the same magnitude and direction for my resultant. It doesn't matter the order. The addition is commutative for two dimensional vectors just like it is for one dimensional vectors. Well there happens to be another way to add two vectors. It's called the parallelogram method. What you do in this case is translate one of the vectors so that it's at the tail of the other vector. So they start at the same position and go in the respective magnitudes and directions. What you do then is copy the lengths of these two vectors at the tips of the other vectors respectively and you create in this case a parallelogram. Now the diagonal from the original starting point of both vectors where their tails are to the opposite vertex of the parallelogram is your resultant in this method. So that's another way to do it but some find it hard to create this parallelogram. Now I'm going to teach you two ways to subtract vectors. The first one is called the triangle method. What you do here is you translate one of the vectors so that its tails are at the same location. So it's kind of like the parallelogram method for adding at the start. But then comes the tricky part. You create the resultant of the subtraction such as vector a minus vector b by putting the tail at the head of the subtracted vector. Okay that's the hard part. You put the tail of your resultant at the head of the subtracted vector. That's this position and you finish your resultant at the head of the first vector. So this is a minus b. Some would really think to put it backwards but that's incorrect. So you take a basically and you move its tail so that its tail starts at the tip of the vector you're subtracting. Now it's called the triangle method. You'll always create a triangle with it. Okay but there's another way to do that which involves something we did on the last slide. The opposite method. So what we can do is multiply the vector we subtract by negative one which essentially just switches it around. So it points the opposite direction. So now I have v2 here plus negative v1 which points to the left instead of to the right. And then I can add by the tail to tip method. So I move v1 so that it starts at the tip of v2, translate it so it looks like this now. And then I would create the resultant by starting at v2 and ending at the tip of v1. So this would be v2 plus negative v1 or v2 minus v1. This is another illustration that shows that if we're doing a minus b what we would do is we could put turn b around. So it's negative b. Put it in the proper tail to tip position and then create the resultant at this yellow vector from the start of a to the end of negative b. Alright, now let's look at multiplication. First by a scalar. So a single number multiplied by a vector. What is that? We're combining the two types of measurements. Oh no, can we do that? Yes. Here's one example. Okay, when you multiply by a positive scalar what that's going to do is it's going to magnify the length of the vector. So here's v, the original, here's 1.5 v. It's one and a half times as long. If you multiply a vector by a negative scalar it turns it around and then magnifies the length. So if we multiply by negative one all it's going to do is turn the vector around and it's going to have the same magnitude as the original. But if I multiply the original by negative two, which is this one over here, it not only turns it around but doubles its length. Okay, so this is how we multiply by scalars. You can imagine what 5 v would look like. It would be a vector in the same direction as this one five times as long. Okay, now multiplication of vectors together. There are actually two ways we can multiply two vectors together. The dot product and the cross product. Let's look at the dot product first. A dot product with two vectors, this is the formula we use in general. A dot product, the scalar is an output. Okay, it is not another vector that you get when you do a dot product. So basically it's like you're dissolving the two vectors into a scalar. And here's the formula for doing so. Let me explain this formula. To multiply two vectors together by dot product, what we would do is we'd find the magnitude of the components of each vector. So AX, AY, and BX and BY. Then we multiply like components together, such as AX times BX and AY times BY. Then we add, we finish by adding those two products together. AX BX plus AY BY. Okay, there's another way to actually do dot products. And that is to find the projection of one vector onto the other. What you would need to do to do this is to first find the angle between the two vectors and then multiply the magnitudes of the two vectors together times the cosine of the angle between them. This effectively takes the projection of B onto A. So it's like the shadow of angle B onto surface vector A. And then you'd multiply those two lengths together. Alternatively, this formula works great. The magnitude of each vector times the cosine of the angle between them. Now I know there are two bars on either side in this picture, but I want you to understand that this is what some images and textbooks use for magnitude. I like two bars, but this image was really nice. So I plugged it in anyway. Now the cross product is a vector product. The output is a vector, not a scalar. This dot product gives a single number as its answer, but a vector is the output of a cross product. The way we talk about cross products is A cross B. When we talk about dot products is A dot B. So here's a visualization. I'm not going to do the math for the cross product. It's beyond the scope of this physics course. But I want to show you that if I have two vectors A and B, I put their tails together like I do in the parallelogram method for adding. And A cross B is another vector that is perpendicular mutually to the original two. I'm not going to get into how it's like this defined, but it's definitely longer than the either of the originals. But its direction is mutually perpendicular to the other two. So we get into really three dimensions here. And there are rules for determining whether it points up. Let's say you've got a plane for your vector A and B. A cross B would point up, and B cross A would actually point downwards in this situation. And that's because the cross product is not commutative. You create a separate vector. And that's what we're used to in matrices as well. But it's a little simpler with vector cross products. A cross B would point in one direction. B cross A would point in the exact opposite direction. There is a special rule called a right hand rule that you can use to determine that. But I'm not going to get too much into that now. I just wanted to introduce you to all the ways you can combine vectors with addition and multiplication. That was a long one, guys. Thanks for sticking with it. I hope that was a good introduction to two-dimensional kinematics for you so you could see the world we live in and how we treat and handle vectors. All right, guys. Thanks so much for watching. It's the end of this video. Stay tuned for less of two. This is Falconator signing out.