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Pre-Calculus: Adding Vectors & Multiplying Scalars

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About this Lesson

  • Type: Video Tutorial
  • Length: 9:26
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 102 MB
  • Posted: 01/23/2009

This lesson is part of the following series:

Trigonometry: Full Course (152 lessons, $148.50)
Pre-Calculus Review (31 lessons, $61.38)
Trigonometry: Applications of Trigonometry (14 lessons, $26.73)
Trigonometry: Vector Basics (3 lessons, $4.95)

Professor Burger shows you how to add and subtract vectors and use scalar multiplication to elongate or shrink vectors while maintaining their direction angle. The magnitude of a vector can be altered with scalar multiplication. A scalar is simply a number (positive or negative or a fraction) used to multiply a vector by, with the vector keeping its same direction and changing magnitude. Vectors can also be added and subtracted by simply adding or subtracting the components. It is also simple to find the answer graphically by creating a parallelogram with the two vectors, which Professor Burger demonstrates.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, 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 UT-Austin 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, p-adic 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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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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Applications of Trigonometry
Vector Basics
Vector Addition and Scalar Multiplication Page [1 of 2]
So let’s see how we can actually take a look at vectors and start to manipulate them in some way. So first off, let me
just start with a simple example. Suppose I have a vector v! and that equals, let’s just say—oh, I don’t know, <1, 2 >.
So graphically, what does that look like? Well, graphically it would look like this. I would go 1 unit over and then 2
units up, and that would be where the end of my vector is, and then the vector would look like that. That would be the
vector v! .
Now what if I wanted to elongate it? Well, one way of doing that is by a technique known as scalar multiplication.
Now, scalar multiplication is the following. I just take a scalar—and by the way, a scalar is just a fancy name for a
number versus a vector, so a scalar is just a number, like 5 or -2 or ½ or anything. And I multiply a scalar by a vector
by doing exactly what you think scalar multiplication would do, and in fact exactly the same thing you do with matrices.
You just multiply every component by that. So for example, if I considered now, let’s say, 2 v! , scalar multiplication
would just take each component and multiply it by 2. So it would be now at < 2, 4 >. And what would that look like
graphically? Well, graphically I would now be at 2, one, two, and then one, two, three, four. So graphically I’d now be
right here, and this new vector would be way out here. Look at that, it’s a huge vector. Oh la la, that’s a big one.
Now, that vector you’ll notice, actually is in the exact same direction as the original v! , so this is original v! here. The
only thing that’s changed is the magnitude. This is now 2 v! . So that’s the only difference. And you can start to see
what’s going to happen. If I multiply by a number that’s really big, it’s going to make the vector really, really long. Oh
la la! But what happens if I multiply by something that’s sort of small? For example, what if I took half of v! , what do
you think would happen? It would take this vector and shrink it by half, so now it would be a little teeny vector.
What about negative? What if I were to multiply by a negative number? For example, what about -3 v! ? Well, then I
multiply through by -3. I see <?3,?6 >, and what would the effect of that be? Well, the effect of that actually, the
negative sign is going to actually flip how the arrow is pointing. So now the arrow is going to go this way. It’s going to
go this way and the -3 means I go down three times the length of this. In fact, it’s going to be off the page. You have
to use your imagination because my screen is not big enough here, but it would go all the way down to -3 over and -6
down. It’s in the exact same direction but now it’s three times as long and the negative sign means I’ve flipped over
the origin. You see that? So in fact, scalar multiplication just requires you to multiply each of the components through
by that scalar, so it’s easy to do arithmetically, and the geometric interpretation is also clear. It just elongates or
shrinks or flips, if the number is negative, by that scalar. Nothing else changes.
Okay, now one thing we’d like to do is if we have a couple of vectors, you want to put them together and combine
them. So what would it mean to add two vectors? Let’s think about that for a second. Suppose I have two vectors.
Let’s say I’ve got <1,3 >, so that’s some vector, and I now want to figure out what it means to add it to another vector
< 2, 2 >. Let’s think about this graphically at first and see what would be a reasonable thing to think about here. So
the first one is <1,3 >, so one; one, two three. So the first vector comes up to there, <1,3 >. And the next vector is
just < 2, 2 >, so we have one, two; comma, two. So the next vector is over here. So notice those are different vectors
that are pointing in different directions, and they potentially have different magnitudes. And I want to ask, what does
the sum of those two vectors equal?
Well, if you think about it analytically, it’s pretty easy to add two vectors. What you do is just treat them like matrices
basically and you add the appropriate components together. So I just add the first components together, 1 and 2, and
that makes 3. And then I take 3 and 2 and add those together and I get 5. But what does this mean geometrically?
So what’s actually happening is something really pretty cool. What’s actually going on is, I take these two vectors and
I consider them as being adjacent sides of a parallelogram. So if you now put in the rest of the parallelogram, which is
something like this, and now I can just fill in the rest of the parallelogram. All I did was take this thing and copy it up
here; took this thing and copied it up there, so now it’s a parallelogram and these are two sides. It turns out that this
vertex right there is going to represent the end of the sum of those two vectors. Isn’t that cool? So it’s way out there.
And if you look, even visually this looks pretty good, because if you look, that point seems to be at 3; one, two, three,
four, five. I mean, visually it even looks really good.
Applications of Trigonometry
Vector Basics
Vector Addition and Scalar Multiplication Page [2 of 2]
So adding vectors in fact is no big deal. You just take the two vectors and graphically what you do is you just consider
the parallelogram that they form, and then you look at that far vertex and that tells you. So for example, what’s
another problem here? In fact, let’s do another one on another page. So that’s the answer.
Subtraction works the exact same way. You have < 3, 2 > and you want to subtract <1, 4 >. Well, what you do is you
just subtract 3 minus 1 would be 2; 2 minus 4 would be -2. And what would that look like? Well, subtracting is just
adding in reverse. You’re sort of doing a reverse process here. So what would it look like? You have < 3, 2 >, so one,
two, three; one, two. So you’ve got one vector right up there. That’s that first vector, < 3, 2 >. And now I want to
subtract this vector, which is <1, 4 >. And so <1, 4 >, one; one, two, three, four. It’s way up here. And I want to
subtract that vector. Since I’m subtracting, geometrically what’s going on? Well, if you think of subtraction as just
adding a negative—if I think of this as < 3, 2 > plus negative this, then I can use the adding method. So let’s do that.
Let’s look at negative <1, 4 >. So what would negative <1, 4 > be? Well, that would be <?1,?4 > just taking -1 times
that vector in scalar multiplication. And what do I see? I see -1, and then -4 is way down here all by itself, the lonely
shepherd, right here. Isn’t that sad? It’s all by itself. The lonely shepherd vector.
Now, that vector represents <?1,?4 >. So now to combine this, all I’ve got to do is add these two things. So if I add
these two things now, what do I do? Well, I make the parallelogram. This is always a little tricky here. So if you make
the parallelogram carefully, it should look like this. And now I’ll make it a little darker so you people on the information
superhighway can see it. So this is actually—if I call this vector w ! and this vector v! , this vector here is negative v! .
Because that’s what I’m doing. I’m taking w ! plus negative v! . And this is w ! . And then the answer, the resultant
vector, would be this one right here. And what is that? Well, notice it is one, two, negative 2.
So in fact, you subtract vectors by subtracting components, and visually, if you want to visualize it, if you take the two
vectors, you take the second one—the one you’re subtracting—look at its negative, and then add its negative to the
vector, and that gives you this. Anyway, you can see now how to add and subtract vectors together—it’s this
parallelogram business and just component-wise combining. And scalar multiplication, you multiply everything
through by scalar. It either elongates—oh la la—or shrinks. If you want to add vectors, you just add up the
components and make a parallelogram, fine. If you want to subtract vectors, you take the one you’re subtracting and
look at the negative of that, so it’s all by itself—it’s a lonely shepherd—and then you do the little parallelogram adding
to it.
Up next we’ll take a look at some more of both visual and algebraic issues involving vectors. I’ll see you there.

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