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About this Lesson
- Type: Video Tutorial
- Length: 12:05
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- Posted: 07/01/2009
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This lesson was selected from a broader, comprehensive course, Physics I. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/physics. The full course covers kinematics, dynamics, energy, momentum, the physics of extended objects, gravity, fluids, relativity, oscillatory motion, waves, and more. The course features two renowned professors: Steven Pollock, an associate professor of Physics at he University of Colorado at Boulder and Ephraim Fischbach, a professor of physics at Purdue University.
Steven Pollock earned a Bachelor of Science in physics from the Massachusetts Institute of Technology and a Ph.D. from Stanford University. Prof. Pollock wears two research hats: he studies theoretical nuclear physics, and does physics education research. Currently, his research activities focus on questions of replication and sustainability of reformed teaching techniques in (very) large introductory courses. He received an Alfred P. Sloan Research Fellowship in 1994 and a Boulder Faculty Assembly (CU campus-wide) Teaching Excellence Award in 1998. He is the author of two Teaching Company video courses: “Particle Physics for Non-Physicists: a Tour of the Microcosmos” and “The Great Ideas of Classical Physics”. Prof. Pollock regularly gives public presentations in which he brings physics alive at conferences, seminars, colloquia, and for community audiences.
Ephraim Fischbach earned a B.A. in physics from Columbia University and a Ph.D. from the University of Pennsylvania. In Thinkwell Physics I, he delivers the "Physics in Action" video lectures and demonstrates numerous laboratory techniques and real-world applications. As part of his mission to encourage an interest in physics wherever he goes, Prof. Fischbach coordinates Physics on the Road, an Outreach/Funfest program. He is the author or coauthor of more than 180 publications including a recent book, “The Search for Non-Newtonian Gravity”, and was made a Fellow of the American Physical Society in 2001. He also serves as a referee for a number of journals including “Physical Review” and “Physical Review Letters”.
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We use vectors whenever we have to talk about something physical that has a magnitude and a direction associated with it. I always like to think about velocity as kind of a typical example. Velocity is a vector. It has a magnitude, 55 miles per hour for instance, and a direction, northeast. When you have a vector as a physical quantity, you can do things with it. You can add vectors, you can scale vectors, and we've seen how you can do that graphically on a piece of paper, representing the vectors as arrows.
Now, if you want to add to vectors, or worse yet, subtract them, on a piece of paper, it can be a little bit tricky. You have to draw them very carefully and to scale and I'm a lousy artist, I hate to do that. So, it seems like there should be some way of manipulating vectors symbolically, mathematically in a kind of more elegant way. The key to doing that is thinking about the vector in a slightly different way. So, here's a vector, which is a classic vector. It's got a length, it's got a direction - you could characterize that direction as this angle, for instance, and that's a perfectly valid way of describing this vector.
But when I look at it, I can think about it in another way. I sort of see that this vector has an "acrossness" to it. It's sort of got this much across, and then it's got an "upness" to it. It's going about this much up. In fact, it's not about, it's exact. I can think about he vector in terms of its length and the angle, or I can think about the vector in terms of how far across does it go and how far up does it go? I still need two numbers, but they're different numbers and it's just another way of thinking about the vector.
Let me try to draw a little picture of this. Here's a vector A, so now I'm drawing it in the horizontal plane, one way to describe it is to give its magnitude and the angle. Another way, the new way would be to tell how much across is that vector, and how much up is it? You might think of how much across it as the projection of the arrow in this direction. Sometimes, I think about shining a light on this arrow and then I just look at the shadow of it. The shadow is the projection in this direction. If you want to label things, you'd probably want to pick a coordinate system. For instance, we decide to call right the X axis and up, the Y axis. We can pick any coordinate system we want. This is a pretty typical one. X is horizontal. So this piece of the arrow, we call A[X]. It's just a natural notation. It's the X of the vector A. It's now a number. It's the length of this dashed line. And we can project the arrow in the Y direction. That's this piece. How much of this vector is going up? And we would call that, naturally, A[Y]. And if you look at this picture, by the nature of these projections, this is a right angle.
Supposing that we had characterized the vector in the first place by angle and magnitude A. Look at the picture, think of a little trigonometry. A[X] is the adjacent side in a right triangle. A[X] is equal to the magnitude of A times the cosine of the angle. Hypotenuse times cosine is the adjacent side. And, similarly, A[Y] is equal to Asin. I'm just getting that from trigonometry. I'm looking at the picture. Think about what these formulas are telling us. If you know the magnitude and the angle, then these formulas immediately tell you numerically what is A[X] and what is A[Y]. So you can go from one language to another mathematically very easily. If I tell you A in , you just work out A[X] and A[Y] on a calculator. You can go backwards, too.
Supposing that in the first place, I had told you A[X] and A[Y] . If you look at the picture that uniquely identifies the vector and you can easily figure out, for instance, what's the length of the vector. It's a right triangle, You use the Pythagorean theorem. A is the square root of A[X] plus A[Y]. And, similarly, is the arced tangent of , opposite over adjacent. So, this is a neat way of describing vectors. And there's kind of a related way of thinking about the vector once we have introduced this A[X] and A[Y]. So let's just take a step back and let me look at our vector A again. Here is it's X and Y components. And now, instead of just thinking of this as a number, let me instead view it as a vector. It's always true that any vector A can always be thought of as a horizontal vector plus a vertical vector. In fact, it's unique. There's only one horizontal vector, this green one that I've laid down, and there's only one vertical vector. The vector A is the vector sum of this green vector and that blue one. They're tip to tail. Graphically, you can see that the vector sum of green plus blue is equal to A.
Let me try to write this down mathematically. In order to do that, I'm going to introduce a convention, which is very commonly used and very convenient, so first of all, remember we have a coordinate system. This is the X direction, that's the Y direction. Let me draw a unit vector. That's a vector of length exactly 1. The unit vector in the X direction, we're going to give a name. You might think we should call it X or something, but for some reason, lots of people call it i. And they put a hat on top of it to indicate that, very clearly, it's a vector with length 1. The hat is a special kind of vector. It's a unit vector. is the unit vector in the X direction; is the name that we give, just a convention, for the unit vector in the Y direction. So, let's look back at our vector A. I am arguing that it is a sum of two vectors. It's A[X] , that's the magnitude in the I direction, and we're adding A[Y, ]that's the magnitude of this arrow, in the J direction. Let me just say it again. Look at this expression - AX. That's a vector. It's a number times a vector. We've seen that last time. It represents a vector. The length of that vector is A[X] this just has unit length, and the direction is the direction of . That's the green one. So, this whole term here represents that green arrow. This whole term represents that blue arrow. There you go. The vector A can always be written in this way. It's component A[X] X plus it's component A[X]. That's a neat way of writing down a formula which describes vector.
So, supposing you've got this formula, A=A[X]+ A[Y]. It's just handed to you. And you want to do the various things that we've been doing with vectors. What would that mean? For example, supposing that you wanted to multiply A by two. That was scaler multiplication, it's 2A[X] +2A[Y]. How did I get that? I'm trying to double this vector. If you double a vector, you double it's X component and you double its Y component. Nice and easy. What if you wanted to find -A. It's -A[X] -A[Y]. What if you wanted to find some arbitrary constant K times A. It's just KA[X]+ KA[Y]. So mathematically, scaling a vector is very easy if you've got the components.
What about if you want to add vectors? We know how to do it graphically. It's the tip to tail method. If we've got the vector in components, it's really very easy to add A+B. If you just stop and think about it for a second, the vector A+B is just going to be A[X] +B[X], that's going to be the X component of the sum, and A[Y]+B[Y] and that's going to be the Y or component. That's it. It's a formula if you know A and B, these are just numbers. You add them up. It's really quite straightforward now to add vectors. I don't even have a picture here anymore of A and B. It really doesn't matter. I don't have to look at them in order figure out the sum. So, that's nice for somebody like me who can't draw vectors very well. If you want to subtract A minus B, it will be the same story, it will be A[X]-B[X]+A[Y]-B[Y] .
So, let me do one example. Supposing that I give you some specific vectors, A and B and here they are. Here's A, there's B. Now, the old way of describing A and B was magnitude and direction. Magnitude 3, thirty degrees from vertical. B is magnitude 5, 37 degrees from vertical. I don't think this picture is to scale, but never mind. That's the point here. All we need is a sketch and we can figure out the sum A+B. Well, we know how to do that graphically. They're already tip to tale, there it is. It's this dashed sum vector. The sum is A+B. It goes from the beginning to the end. But, if I really wanted to know that sum, what does that mean? It means I really need to know the magnitude and the angle or the components. If I haven't drawn this to scale, it would be awfully hard to figure out, but I'm all set up. If you just stare at this picture, all I need to do is figure out the numbers A[X], A[Y], B[X], B[Y] and we've got the formulas and we've got the sum. Let's just take a look at what they are. A[X] , look at that little right triangle that we've got. A[X] is the A times the sine of 30 degrees. Look at the picture and convince yourself that it really is the sine. A[X] is opposite from that angle. I just had a formula shortly ago, where I said A[X] is Acos. So how come I'm using sine? Well, because my angle there is defined in a funny way. In the picture, I've defined with respect to the vertical, whereas in my formula before, I had a different which respect to the horizontal. So, watch out. Look at the picture, use what's correct. In this case, it is Asin, calculate the numbers. AY, check the picture. A[Y] is down. The A vector is down and across. Down is in the negative Y direction, so the A[Y] component is negative. There's the B[X] and B[Y]; those are really straightforward. That B triangle is set up just the way we had set up before.
So, if you want to find the sum, the sum in the X direction is just A[X]+ B[X]. It's four plus one point five, the sum in the Y direction is just B[Y] + A[Y], 3 - 2.6 is 0.4 and we're done. We've got the X and Y components and the sum. If you want the magnitude, use the Pythagorean theorem.
So, what we've learned is, if you want to manipulate vectors, there's two different ways to think about them. You can think about them as having a magnitude and a direction. That's kind of a nice conceptual way. You draw pictures. You can also just think of the vector as having an X component and a Y component. That's really convenient when you're trying to calculate.
Vector Components and Unit Vectors Page [2 of 2]
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