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Pre-Calculus: Finding Vector Magnitude & Direction


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About this Lesson

  • Type: Video Tutorial
  • Length: 6:43
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 72 MB
  • Posted: 01/22/2009

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)

In this lesson, Professor Burger will show you how to find the magnitude and direction angle of a vector provided in standard form (and how vector notation associated with standard form looks and should be interpreted). He will also review how to depict a vector graphically that we have described by standard form. The magnitude is the length of a vector (reminder: it must be positive), and we use the Pythagorean theorem to calculate the vector's magnitude. The direction angle is always measured counter-clockwise from the positive side of the x-axis. Once we know the vector's length, we can use trigonometric functions to calculate the direction angle of the vector. Last, Professor Burger solves for the magnitude and direction of some of the vectors using a calculator.

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

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 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

2174 lessons

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 or visit Thinkwell's Video Lesson Store at

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...


Recent Reviews

Wonderful - very simple explanation!
~ hmt79

This really helped me to understand the ins and outs of vectors, their magnitude and their direction. Thanks!

Wonderful - very simple explanation!
~ hmt79

This really helped me to understand the ins and outs of vectors, their magnitude and their direction. Thanks!

Applications of Trigonometry
Vector Basics
Finding the Magnitude and Direction of a Vector Page [1 of 2]
So let’s take a look at vectors that are given to us in the standard form and see if we can figure out both the
magnitude and in some sense the direction, which is given by what’s called the “direction angle,” and that’s just the
angle you make starting at the positive x-axis and going out to your vector. So let’s just look at an example and start
to warm up to this.
So I have the vector < 3,1>, let’s call that v! . So first of all, let’s look at this visually. What does that look like?
Well, that means that I go over 3 and I go up 1. So that vector is this. This is v! . So what I’d like to find out first of
all is its magnitude. That’s just the length of that arrow. And then I want to figure out its direction angle. And the
direction angle is always given by starting, with the positive x-axis, and then moving up in a counter-clockwise
direction, like that, as you normally measure angles.
Well, how does this look? Well, first of all, what’s it length? Well, to find its length is pretty easy. I mean, you could
think about that formula for magnitude or you could just think about the Pythagorean theorem, which is the way that I
always operate, by the way. This is 3 and this is 1. So now I use the Pythagorean theorem to find the magnitude
of v! . I know that this squared—so 3 squared, which is just 3; plus 1 squared, which is just 1—that has to equal the
magnitude of the vector squared. So that means that I see that the magnitude of the vector squared equals 4. So
what’s the magnitude of the vector? Now remember the magnitude of the vector is a length, so when I take plus or
minus the square root, I throw away the negative square root since this is actually measuring a length, and I just
would see 2. So this thing would have length 2.
Now, what’s the angle or what’s the direction angle? Well, the direction angle, if I call that ? , let’s see what that
would be. How could I find the direction angle? Well, I’ve got to use some trig function of some kind, right? So what
could I possibly use? Well, anything will work fine. Let’s just use tangent just for fun. So tan? would equal what?
Well, it would equal
, so
. Okay, now the question is, what angle has tangent
? Well, you might
remember this one or you could use a calculator. Let me show you how you’d use a calculator really fast to do this if
you wanted to. So 1 divided by 3 , and you would see that ? is 30°, which is of course what you might have
remembered. Thirty degrees, or as I like to say it,
Okay, let’s try one more, < 3, 4 ? > is the vector I’ll call w ! now. I’m starting to introduce the notation, too. What would
this look like? Well, what you would have here would be the following. So we go ?3, one, two, three, and then +4 ,
one, two, three, four. So now here’s the point. The vector looks like this. Okay, so first of all let’s find the magnitude.
This is w ! . Let’s find the magnitude, the length of w ! . Well, you could actually use the Pythagorean theorem or you
could see that this is a 3, 4, and this must be therefore a 5, right triangle. It’s the famous right triangle, a 3,4,5. So
you could either verify that using the Pythagorean Theorem or you may just recognize that right triangle and see this
is actually 5, so that takes care of that.
And now what about the angle? Well, the angle—you see, I’ve got to go all the way around, so you go right around.
And if you have to swing way around—I mean, maybe your vector is way down here. Let me just show you. If your
vector was like way over here, you always start on the positive x-axis and you go counter-clockwise. So you go way
around. You just go right around the origin. Right around the origin. The origin is like a chewy nougat center in a way
in mathematics. You sort of go right around it.
So anyway, in this case I’m just going to go up here and go out this way, and now what would I see? Well, I’ve got to
find that angle. Well, how would you find that angle? Well, now we’re back to basic trig stuff, so what would I do? I’d
first find the reference angle. Let me call this angle ? , that’s the direction angle. Let me first find this reference
angle, which I’ll call ? for the time being. So I’m looking for, now, the angle such that—what I do know? Well, how
about—well, you could use anything, actually. In fact, we could just go right to using inverse trig function and say that
Applications of Trigonometry
Vector Basics
Finding the Magnitude and Direction of a Vector Page [2 of 2]
cos? would equal what? Well, cosine would be
, which would be
. So I just now use my
calculator—actually, it’s not my calculator, but I’ll just use someone else’s calculator, and we should be able to do this.
So I look at the inverse function of cosine, so inverse
cos 3
. And we know the answer should be something bigger
than 90 but less than 180, or bigger than
and less than ? . And so this equals—why is this not working? Let’s try
it again. This is what happens when you use someone else’s calculator. Oh, I think I pushed the wrong key, yes.
And this turns out to give me a directional angle of 126.869°. And that’s right because of course it has to be greater
than 90 and less than 180—and I’m using degrees now here just for fun, just to break up the monotony.
Anyway, you can see what’s going on here. What you see is, to find that directional angle, you go this way to the
angle; that tells you the direction. And then the magnitude you can figure out by looking at the length, and then you
know the vector exactly. Up next we’ll take a look at some more basic properties about vectors and how to put them
together. See you soon.

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