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About this Lesson
 Type: Video Tutorial
 Length: 5:26
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 58 MB
 Posted: 07/02/2009
This lesson is part of the following series:
Trigonometry: Full Course (152 lessons, $148.50)
Trigonometry: Applications of Trigonometry (14 lessons, $26.73)
Trigonometry: Components of Vectors & Unit Vectors (3 lessons, $4.95)
Taught by Professor Edward Burger, this lesson comes from a comprehensive course, Trigonometry. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/trigonometry. The full course covers trigonometric functions, trigonometric identities, application of trig, complex numbers, polar coordinates, exponential functions, logarithmic functions, conic sections, and more.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UTAustin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, padic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
About this Author
 Thinkwell
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11/13/2008
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Finding a Unit Vector
So a vector is just a mathematical object that has both magnitude and direction. Magnitude and direction. Sometimes, though, in actual applications all we care about is the direction and we don't want the magnitude to enter into it. For example, suppose someone says, "Which direction are you hiking?" Well, they're not asking you how fast you're hiking in that direction. They just want to know what direction you're hiking. So a lot of times what people would like is just a way of expressing the direction and foregoing magnitude. In that case, what we tend to do is just take the vector that has length 1 that's in the same direction. Such a vector is called a "unit vector." So a unit vector is just a vector that has length 1. It's just a vector whose magnitude is 1.
So now suppose that someone gives us a vector and we want to find the unit vector that's pointing in the same direction as . So maybe is a really long vector. Maybe is quite obscene, like this. And that's maybe, let's say, length 3 or something. And what we want is just a unit vector that's in that same direction. Maybe that has length 1. So I want to figure out that vector. Just knowing that this vector is , how could I find that vector?
Well, what I want to do is I want to scale this vector so it has length 1. So what should I do? Well, l should take this vector and divide it by the scalar that represents its length. Because if it has length 3, then if I divide by 3, the new vector will have length 1. Do you see that? If this thing has length 3 and if I divide by that length, then it will all of a sudden have length 1. So in fact if I want to find a unit vector that's in the same direction as , and I usually write for unit vector, I just take the scalar 1 over the length of now that's just a number; that's just the reciprocal of the lengthand I multiply it by . And that will give a vector that's in the same direction as but now has length exactly 1. And you can sort of see that, right? If you take the length of this, you see the length of over the length of , and that's just 1. So in fact this has length 1. So these are called unit vectors in a particular direction. Let's do a couple of examples so you can see how you would actually find a unit vector. It's not a big deal. I didn't say it was a big deal; I just said it was an interesting deal if you want to know direction.
Now, suppose you take the vector , and suppose that equals <> and I want to find the unit vector that's pointing in the same direction. Well, what I do is I have to find the length of this vector. So I could use the Pythagorean Theorem, and I'll do that right here, and say, well, okayin fact, you might want to draw a little picture of this in fact. That wouldn't be a bad idea. Here we go. So we go one, two, three in the negative direction, and then 4 up, one, two, three, four. The vector is this. And I want that same exact direction but only length 1. You can see that length is really long, right? This is 3 and that's 4. How long is this?
Well, you could use the Pythagorean Theorem, this is 4 and this is 3, or you could just recognize that this is a special right triangle. It's a 3,4 right triangle, so in fact that length is 5. So I should divide through by the length, and scalar multiply that vector by that scalar. So what I see is that the unit vector would be just <>. And that's this vector right here. It's pointing in the exact same direction but now the length of that is 1. And if you actually checked the length of this, you would see what? The square root of this squared plus that squared. This squared is . This is . , which is 1. So in fact, that's the way to find unit vectors. It's real easy.
Suppose I give you this vector. Suppose that is the vector <1,1>. Find the unit vector in the same direction. By the way, that vector is easy to see. It's the diagonal one right here, <1,1>, that's . You may think that it has length 1, but actually no. If you use the Pythagorean Theorem, is actually 1+1, which is 2, so this has length . It's a little bigger than 1. So therefore the unit vector in that direction will be 1 over the length of this vector, scalar multiplied by the vector itself. And so the unit vector would be <>. And where does that look like? Well, that would be just a little bit shy of this. This is a little bit too long. That would be the unit vector right there.
So a unit vector is just a vector that's pointing in the exact same direction as some given vector but has length exactly 1. How do you find it, given a vector? All you do is take the magnitude of that given vector, take your vector and scalar, multiply it by the reciprocal of that magnitude; that gives you the unit vector. That's all there is to it.
Up next we'll take a look at an application and move on. Talk to you there.
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