Unit Vectors - Video Tutorials & Practice Problems

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concept

Unit Vectors

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5m

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Hey, guys, sometimes you gonna run to problems in which you see vectors represented using a special notation with a bunch of eyes and jays, and these are called unit vectors. So in this video, I'm gonna show you what these unit vectors are all about and more importantly, how they help us describe and do vector math with vectors. Ah, lot more simply, Let's check it out. So, guys, let's think of the easiest way to describe a vector. We have graphical, which is where we have already the grids and squares. We can take this vector and break it up into its legs and just count up the boxes. This is three and four and then we could use our equations like Pythagorean theorem to figure out that this five and the angle is 53 degrees. Now, another way we could describe the vector is by already giving the magnitude and the direction, like, five and 53 degrees, that we have to just draw this vector out So positive X and positive. Why? And then we would have our 5 m here at 53 degrees. Now if we one of the legs we have to just use our calculating or component equations or decomposition equations to figure out what the legs of the triangle are. A coastline data, a sign data. And we just get three and four. Now. The last way that you'll see vectors represented is using this weird notation with a bunch of eyes and jays and case these air called unit vectors. And what's going on here, guys, is that these unit vectors are just special kinds of vectors that point in a direction, and they also just have a magnitude or length of one. So here's what's going on. If you have a vector, that's a three I plus four plus four J. All it's really going on is that I points in the plus X Direction J points in the plus y and K points in the pluses eat. So any time you see I XJS and Kay's just think of X, Y and Z s physicists came up with the system a long time ago. They thought it was, You know, they thought you didn't have enough confusing letters, so they decided to throw a bunch of eyes J and K's in there. So, for example, if we've got three I and Forge A All that's really saying is go three in the eye direction. So we've got one, 23 So it's three of them right here. And then go four and the J or the Y direction. So from here gonna go 1234 And this is JJ. JJ, you're gonna have four of them. That's all that's going on. So if you wanted to construct the vector, this is just three and four, And so our results and it's just gonna be from tip to tail like this. And this is gonna be five because this is basically already giving us the legs of the triangle. Three in the X and four and the wise are magnitudes five, and our angle is 53 degrees. So, guys, all of these things here are just different ways to describe the exact same vector. So that's what's going on here. That's all unit vectors are. So you can think about these eyes and Jay's as basically just already being the legs of the triangle. That's what these things are telling you. Alright, guys. So what unit vectors are really, really helpful for is making vector edition Super straightforward. Let's check it out. Using this example here, we've got these two vectors A and B. We're gonna draw them and calculate these the resultant in unit vector form. So I've got four I and two j. So basically, I'm gonna go one, 234 and the X direction I is gonna be X and then two in the J or the Y direction. So one and then two. So this right here gives us my A And this here is my a X and my A y right. It's just the components of this vector here, but I have it in terms of Unit Vector. So this is just four I, and this is just too Jay. Now, let's do the same thing for B. B is negative, I So if positive I points in the positive extraction negative, I was gonna point to the left or the negative X direction. So we've got one to the left and we got to up. We're gonna go one and then two. So this right here is gonna be my B X, and this is gonna be my B y. And then the vector is just gonna point from start to finish like this. Now, if we wanted to do vector addition, we have to follow all the steps. You have to make the table and all that stuff and you have the decomposed these things and then add them together. But unit Vector Edition makes this stuff super straightforward. So, for example, if we had this a vector weaken, just describe it in terms of its I and J components or its X and Y components. And I could just say that this vector is Eyes four I plus two j and then vector B is gonna be negative. I plus two j So if we wanted to find the result in Vector, which is the addition of A and B, I already have my X and my Y components. And remember that when we have the X and Y components, we just add them downwards. We just add them vertically in the table. This is basically already doing that for us. So my a plus B, what we could think about this is I'm just doing a X plus b X, and this is gonna be in the new X direction, and then I'm gonna do my a y plus my B y, and that's gonna be in the J direction or the new Y direction. So, for example, this new results in Vector is if I want the resulting vector in unit vector form, I'm just gonna do four I plus negative J or negative I or negative one I and that's gonna be in the new I direction. Plus, actually, hold on a second. I've got these eyes. I'm basically just gonna do four minus one in the eye direction. Plus now I've got two plus two So two plus two in the J direction. So I've got three I plus four j And so what? That would look like my result in Vector because I would go three in the eye direction and then four up so that my new vector would look like this. This would be my resulting vector. And this makes perfect sense because if I were to add these two vectors A and B together tip to tail, I'd have to move the B vector over this way. It's one to the left and to up, and I would get the exact same direction, so basically I would get the same exact vector. So that's just another way to describe ah vector using unit vector components. Alright, guys, that's it for this one. Let me know if you have any questions.

2

Problem

Problem

A = (4.0 m)i + (3.0 m)j and B = (−13.0 m)i + (7.0 m)j. You add them together to produce another vector C. (a) Express this new vector C in unit-vector notation. (b) What are the magnitude and direction of C?

A

C = (9 m) i + (10 m) j = 13.5 m @ 48° above the +x-axis

B

C = -(9 m) i + (10 m) j = 13.5 m @ 48° above the -x-axis

C

C = (9 m) i + (10 m) j = 13.5 m @ 0.83° above the +x-axis

D

C = -(9 m) i + (10 m) j = 13.5 m @ 0.83° above the -x-axis

3

example

Adding 3 Vectors in Unit Vector Notation

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6m

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Hey, guys, let's check out this example. Together, we've got these three vectors A, B and C, and they're all written in their unit vector forms. And we're gonna calculate the magnitudes and directions of a bunch of these combinations of unit vectors. So let's check it out. So in the first part here, we got to calculate magnitude and direction of D, which is just the addition of all the three unit vectors. So when we're doing Unit Vector addition, all we have to do is this lineup three vectors on top of each other. So let's write them out. So my a vector is gonna be three in the eye. So it's basically three of the X minus three in the J. So it's basically three in the negative y direction. That's how you can think about that. My be vector is just going to be I minus four j and then my see vector we know is just negative two i plus five j. So if we want to calculate the resultant vector this vector d here by just adding all of them together, then I could just add up each of the components basically all the parallel components on the ice together all the jays together because it's basically is if I already had the legs of all the triangles, I just have them in numbers. So I've got this d vector here, which is just a plus B plus c. So what I do is just just add these things straight down. Eso basically just add three and then with one and then the negative, too. So that means that the X component of D is just gonna be if I add all the components together So three plus one minus two, that's gonna be in the eye direction plus And now it's gonna be negative. Three minus four plus five And that's gonna be in the J. And so if I just add up all these numbers together through plus one is four minus two is too I and then I've got, uh, negative three negative for is negative. Seven plus five is negative, too. So that means I'm gonna have to use, so I'm gonna have negative Sorry to I hat. I'm gonna have to I minus two j and that is my d vector here. But I'm not done because this is just the D vector in written in unit vector form. So what I have to do is I actually have to get that magnitude and the direction of D. So let's just go ahead and just sketch it out really quickly. What this would look like if you sort of like a sketch out on a diagram, it doesn't have to be super pretty because we're just using a kind of a sketch is we've got to in the X direction and then this minus sign is in the J directions. We've got minus two J So that just means that we're gonna go to in the X and then two in the why So I mean, our vector is gonna look something like this. So this is two and then to like this. So we're gonna calculate the magnitude and the direction of this vector over here, so D and then I'll call this just faded d So my d it's just gonna be if I use the Pythagorean theorem Two squared plus negative two squared. Since we've already got the legs of the triangle and if you work this out, you're gonna get 2.8 Now the direction faded T is gonna It's gonna be if I do the tangent in verse and then I do the y component over the X component. But these things are actually going to be the same. And by the way, this is technically supposed to be negative. So we've got to over two because we're always plug in the positive components. What you should get is you should get 45 degrees. That makes sense because we've just got, like, a perfect 45 degree angle, which means the components have to be the same. Whatever is over here is gonna be over here, so that is our answers. So we've got our magnitude and direction. So let's move on to part being now. So we're gonna calculate the magnitude direction of a different combination. But it's the same principle here. We just have to use negative signs for when we add these vectors together. So let's see how that works. So we're gonna do the same exact procedure here. So, for B, we've got our A vectors. Actually, we might be able to just copy this over, so just go ahead and copy this over on your papers. Just like that. Okay, so now we're gonna again. We're just gonna be adding these things straight down. We're just gonna be using a different formula. So instead of a plus B plus C over here Now, this new vector, which we're gonna call E is actually going to be if we flip a and we flipped B and then added to see So what happens when we actually make these vectors negative? Well, remember what happened. We did Vector edition. We would just reverse the actual direction of those vectors. Well, all that's gonna happen here for the legs of each of these triangles. Is that these vectors for A and B? We're gonna have to reverse the sign of the components. So reverse signs off components. So that just means that if we have instead of three I for our a vector and the X direction, we're gonna have to use negative three because we have to basically just insert this little negative sign into that. So now we gotta we gotta flip the sign of the B vector, which normally would just be one. But now it's gonna be negative one, and then we're gonna add C so we're just gonna do nothing to see. Just stays the same. So now this is my new I direction plus And now I've got to flip this over here. So if I flip this, this actually becomes a positive, then I've gotta flip this one as well. So this also becomes a positive, right? Because I'm doing negative a minus B, and then I'm gonna add it to five without doing anything. So that means that now, my e vector, if I write it out Negative three minus one minus two is gonna be negative. Six in the eye and then three plus four plus five is actually 12. So now might have got negative six I plus 12 j and this is my e vector here, so I could do the same thing. I could basically just sketch out what this would look like. And if I have this little diagram like this, then this vector would look like Well, I have a component that's negative. So it points to the left. So that means that my ex component here would look like this. And I know that this e X is equal to negative six and then I have my ey, which is equal to 12. It doesn't have to be necessarily to scale, but this is what our vector would look like just to get an idea of what's actually going on. So we want to calculate now the magnitude directions, we're gonna use the same equation that we did before. So if we want to calculate the magnitude of E, we just use the Pythagorean theorem. So you just use the Pythagorean theorem of six squared or actually negative six square. But it doesn't matter because you always get a positive number and then 12 squared and you get 13 points. Four. So that is the magnitude. As for the direction we're gonna get the tangent in verse. And now Whoops. So tangent in verse. And this is gonna be the Y component, which is 12/6 and always gonna be positive numbers and you're gonna get 63.4 degrees. Alright, guys, that's it for this one. Let me know if you have any questions