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Physics: The Basics of Vectors

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  • Type: Video Tutorial
  • Length: 12:42
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 136 MB
  • Posted: 07/01/2009

This lesson is part of the following series:

Physics (147 lessons, $198.00)
Physics: Preliminaries (8 lessons, $11.88)
Physics: Vectors (2 lessons, $4.95)

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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Physics is about describing and predicting the motion and the behavior of physical systems. So, what does that mean? You've got some ugly fly here and you want to describe it to a physicist. You might characterize this by certain measurements. The width, the height, how much it weighs, or its mass. These are all numbers. There's kind of a fancy word for a number that's used to describe a system. It's called a scalar. It's a mathematical term.
So, so far you might think that that's all you need. If you want to describe physical systems, all you need is numbers. But that's not quite correct. Imagine that this fly is sitting here and it moves from one point to another. That's a physical motion. That's something that we would like to be able to describe and explain in the language of Physics. So, think about how you have to describe that motion. You might say it went a foot. But that's incomplete. It's more than just going a foot that you have to tell. You also have to explain that it was traveling in a diagonal line, straight path, at an angle of 45 degrees. You have to tell the direction of travel as well as the distance of travel to fully describe this motion. You might kind of abstractly imagine describing this motion, it's a physical quantity, with this arrow. The arrow is a mathematical representation of the physical quantity displacement. This arrow is given a name. We don't' want to call it a scalar because it's got more than just a length. It's given a length and a direction. So the mathematical name for such a quantity that has both direction and magnitude or length, is vector A. You put this little arrow, some people will put a full arrow on top and that's the symbol that you use. I've just given it the name "A", that's an arbitrary name. Vectors have properties and a magnitude. It's usually written as these are absolute value signs, so you might say, the absolute value of A, but people would usually read this as "magnitude of A." And oftentimes, you'll just write it for simplicity like this. You use the same symbol, it's the same name, but you just don't write the arrows on the top, and that's the magnitude of this vector.
You can describe certain physical properties with vectors. We've just talked about displacement, but there's more. For example, velocity. If you're driving your car and you want to explain to somebody else about the velocity of your car, if you just say, "I was going 55 miles an hour," that's not enough. They need to know, 55 miles an hour northeast, 55 miles an hour northwest? In which direction were you going? It matters. You're going to end up in a different place. It's a different velocity even though it's got the same magnitude. So, to fully describe a vector, you need to give these two quantities.
So, you've got vectors in Physics. There's vectors all over the place. We'll be using them throughout this course. You're going to want to be able to manipulate them. You want to ask questions. For example, if you have a vector which represents displacement, as a specific example. Here I am, sitting at home, and I head over to the physics building on campus and I want to specify my displacement. I say my displacement vector - I'll give it the name A - let me write it like this. So, that's a good characterization of a vector. Magnitude and a direction. And there's the picture of it. Then, maybe it's lunchtime and I decide to go have some lunch, so I have a second displacement vector and I might give it another name, maybe B, and I might characterize that as four miles north.
You could ask, what's my net displacement. I might care about this. For instance, somebody at my house calls me up on the phone while I'm eating lunch and says, "I want to meet you, where should I go?" I don't want to tell them to do A and then B, I want to tell them what's the sum, A+B. I want to tell them about this vector. And we call it A+B. It's a manipulation of two vectors. It's the vector sum. And graphically, it's very obvious. You start at the beginning when you want to add two vectors. Notice what I've done. I've got the first vector and then I put the second one right here with the tip of the first pointing right at the tail of the second. Tip to tail. And the result - the sum. A+B starts at the beginning and goes directly towards the end.
We might want to give that resulting sum a name. We might want to talk about the sum A+B and call it C. Now, this is a vector equation. Vector A + vector B = vector C. You might be tempted, if you see an equation like that, to write down A+B=C. It's very tempting and it's wrong. A+B is not equal to C. Look at the picture. If this is a nice right triangle, like I've drawn, remember A represents the magnitude of the vector A. It's three miles. And if you add three miles and four miles, that's seven miles. This is certainly not seven miles. In fact, it's a nice little right triangle, three, four, it's five miles. So, just something to bear in mind. When you're adding vectors, you're not adding their magnitudes.
I took a particularly simple example, a nice right triangle here. One could add arbitrary vectors, for instance, here's some funky A pointing at some random angle, here's B pointing also at some random angle. A+B, it's already tip to tail, and the sum A+B starts at the beginning, ends at the end. So that's the graphical way that you would add vectors.
I've got an equation here. A+B=C. There's an equal sign with vectors. We need to think a little bit about what "equals" means in the world of vectors. These are not numbers. So, if you've got a vector here and you want to talk about another vector that's equal to it, here's one, it's clearly equal, they're equal lengths, equal direction. How about this one? That's, I think, clearly not equal. These two vectors are pointing different directions. It's like we discussed velocities. If you're going northeast, if you're going southeast, it's totally different velocities. These vectors are not equal. These vectors are equal. When vectors have the same magnitude and the same direction, they're equal, because that's all there is to a vector, magnitude and direction. These two are equal. These two are equal. You can slide a vector around the page as long as you don't rotate it accidentally while you're sliding it. You can't stretch it because that would change the vector, but as long as you move it like so, parallel to itself, these two vectors are equal.
Once you've got a vector, you may want to think about other things that you can do besides adding two vectors. For example, supposing that you have a vector like so, and this vector describes, say, a displacement. I went from here to there and now what if I want to describe half of that displacement, how do I do that? This is half of that displacement. It's a vector which is one-half of the original one. So, it's a slightly funny notation. I'm multiplying a vector times a number. It's physically, I think, pretty obvious what you mean by that. Same direction, but half as long. You could talk about twice a vector. It's doesn't have to be a nice simple number like this, but this is called scalar multiplication because I'm multiplying a scalar. Remember, that's just a fancy word for any old number times a vector. So, you can do scalar multiplication and it's a useful thing, to be able to scale the length of a vector without changing its direction.
You might ask yourself whether the following makes any sense: I can multiply by numbers. Would it make sense to multiply a vector by a negative number? How about negative A, -1 x A. So, let's draw vector A and ask, what could we mean by this, -A. What do I mean by that? Well, the answer is simple. It's the same length, the same magnitude, but opposite in direction. How do I know that that's really what I mean by -A? Remember, sliding a vector around doesn't change it. This is the same situation and what I've really got here is A, -A, and look, I'm adding them, it's tip to tail. I started here, I went up, I came back, I ended right back where I started. There's no net displacement. The sum of these two vectors is equal to zero. And that's exactly what adding a negative should do. A+-A should equal to zero. That's what happens with numbers. That's what happens with vectors.
So, I know how to subtract A from A. Can I subtract A from B? What would that mean? So, let me draw some random vector A, here, and some other random vector B, here. What do I mean by B-A? What could this mathematical expression represent? It turns out there are two easy ways to find the difference of two vectors. The first is to just rewrite this like B+-A. B+-A. Let's see, here's A, and here's A. Here's -A. I can put this arrow anywhere I want as long as I don't change its direction. If I want to add them, here's B and here's -A. B+-A. They're already tip to tail now. And I get the result. There it is B-A.
There's another way of adding B with A. Let me go back. Sorry, it's subtracting. Another second way to subtract B-A and it's just graphical. It's purely graphical. Look at the two vectors, put them tail to tail. There's the answer, from one tip to the other. It's jut a trick. Notice that vector and that vector are the same, so you can see geometrically that it works. If I hadn't worked it out the other way, so I've just got B and A and I want to form B-A. I want to use this trick. I put their tails together - oh, I forget, does it point this way or this way? I know it's going to go from one tip to the other, but I can't remember. The way I remember is the following. I write down A+B-A. Why would I do that? I'm just staring at this expression and I say, "If I take A and I add B-A, what am I going to get?" The A's cancel. I should get B. If you take A and you add B-A, take this vector, it's already tip to tail, take A, add B-A, you get B. So these are the kinds of things you can do with vectors. Vectors are physical quantities that have direction and magnitude. You can manipulate them graphically like we've been doing. You can scale them, you can add them. The golden rule for the addition of vectors is very simple. Tip to tail. Take A, add B, and you get A+B. And the way to subtract vectors is, if you've got A and B and you want to form B-A, you put them now tail to tail. And that's the golden rule for subtraction. And you start at the tip of A and you end at the tip of B.
Preliminaries
Vectors
The Basics of Vectors Page [2 of 2]

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