3. Vectors

Unit Vectors

# Unit Vectors

Patrick Ford

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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.

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