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Expressing a vector as the scaled sum of unit vectors. Created by Sal Khan.
Video transcript
Good afternoon. We've done a lot of work with vectors. In a lot of the problems, when we launch something into—In the projectile motion problems, or when you were doing the incline plane problems. I always gave you a vector, like I would draw a vector like this. I would say something has a velocity of 10 meters per second. It's at a 30 degree angle. And then I would break it up into the x and y components. So if I called this vector v, I would use a notation, v sub x, and the v sub x would have been this vector right here. v sub x would've been this vector down here. The x component of the vector. And then v sub y would have been the y component of the vector, and it would have been this vector. So this was v sub x, this was v sub y. And hopefully by now, it's second nature of how we would figure these things out. v sub x would be 10 times cosine of this angle. 10 cosine of 30 degrees, which I think is square root of 3/2, but we're not worried about that right now. And v sub y would be 10 times the sine of that angle. This hopefully should be second nature to you. If it's not, you can just go through SOH-CAH-TOA and say, well, the sine of 30 degrees is the opposite of the hypotenuse. And you would get back to this. But we've reviewed all of that, and you should review the initial vector videos. But what I want you to do now, because this is useful for simple projectile motion problems-- But once we start dealing with more complicated vectors-- and maybe we're dealing with multi-dimensional of vectors, three-dimensional vectors, or we start doing linear algebra, where we do n-dimensional vectors—we need a coherent way, an analytical way, instead of having to always draw a picture of representing vectors. So what we do is, we use something I call, and I think everyone calls it, unit vector notation. So what does that mean? So we define these unit vectors. Let me draw some axes. And it's important to keep in mind, this might seem a little confusing at first, but this is no different than what we've been doing in our physics problem so far. Let me draw the axes right there. Let's say that this is 1, this is 0, this is 2. 0, 1, 2. I don't know if must been writing an Arabic or something, going backwards. This is 0, 1, 2, that's not 20. And then let's say this is 1, this is 2, in the y direction. I'm going to define what I call the unit vectors in two dimensions. So I'm going to first define a vector. I'll call this vector i. And this is the vector. It just goes straight in the x direction, has no y component, and it has the magnitude of 1. And so this is i. We denote the unit vector by putting this little cap on top of it. There's multiple notations. Sometimes in the book, you'll see this i without the cap, and it's just boldface. There's some other notations. But if you see i, and not in the imaginary number sense, you should realize that that's the unit vector. It has magnitude 1 and it's completely in the x direction. And I'm going to define another vector, and that one is called j. And that is the same thing but in the y direction. That is the vector j. You put a little cap over it. So why did I do this? Well, if I'm dealing with two dimensions. And as later we'll see in three dimensions, so there will actually be a third dimension and we'll call that k, but don't worry about that right now. But if we're dealing in two dimensions, we can define any vector in terms of some sum of these two vectors. So how does that work? Well, this vector here, let's call it v. This vector, v, is the sum of its x component plus its y component. When you add vectors, you can put them head to tail like this. And that's the sum. So hopefully knowing what we already know, we knew that the vector, v, is equal to its x component plus its y component. When you add vectors, you essentially just put them head to tails. And then the resulting sum is where you end up. It would be if you added this vector, and then you put this tail to this head. And you end up there. So you end up there. So that's the vector. So can we define v sub x as some multiple of i, of this unit vector? Well, sure. v sub x completely goes in the x direction. But it doesn't have a magnitude of 1. It has a magnitude of 10 cosine 30 degrees. So its magnitude is ten. Let me draw the unit vector up here. This is the unit vector i. It's going to look something like this and this. So v sub x is in the exact same direction, and it's just a scaled version of this unit vector. And what multiple is it of that unit vector? Well, the unit vector has a magnitude of 1. This has a magnitude of 10 cosine of 30 degrees. I think that's like, 5 square roots of 3, or something like that. So we can write v sub x-- I keep switching colors to keep things interesting. We can write v sub x is equal to 10 cosine of 30 degrees times-- that's the degrees --times the unit vector i-- let me stay in that color, so you don't confused --times the unit vector i. Does that make sense? Well, the unit vector i goes in the exact same direction. But the x component of this vector is just a lot longer. It's 10 cosine 30 degrees long. And that's equal to-- cosine of 30 degrees is square root of 3/2 --so that's 5 square roots of 3 i. Similary, we can write the y component of this vector as some multiple of j. So we could say v sub y, the y component-- Well, what is sine of 30 degrees? Sine of 30 degrees is 1/2. 1/2 times 10, so this is 5. So the y component goes completely in the y direction. So it's just going to be a multiple of this vector j, of the unit vector j. And what multiple is it? Well, it has length 5, while the unit vector has just length 1. So it's just 5 times the unit vector j. So how can we write vector v? Well, we know the vector v is the sum of its x component and its y component. And we also know, so this is a whole vector v. What's its x component? Its x component can be written as a multiple of the x unit vector. That's that right there. So you can write it as 5 square roots of 3 i plus its y component. So what's its y component? Well, its y component is just a multiple of the y unit vector, which is called j, with the little funny hat on top. And that's just this. It's 5 times j. So what we've done now, by defining these unit vectors-- And I can switch this color just so you remember this is i. This unit vector is this. Using unit vectors in two dimensions, and we can eventually do them in multiple dimensions, we can analytically express any two dimensional vector. Instead of having to always draw it like we did before, and having to break out its components and always do it visually. We can stay in analytical mode and non graphical mode. And what makes this very useful is that if I can write a vector in this format, I can add them and subtract them without having to resort to visual means. And what do I mean by that? So if I had to find some vector a, is equal to, I don't know, 2i plus 3j. And I have some other vector b. This little arrow just means it's a vector. Sometimes you'll see it as a whole arrow. As, I don't know, 10i plus 2j. If I were to say what's the sum of these two vectors a plus b? Before we had this unit vector notation, we would have to draw them, and put them heads to tails. And you had to do it visually, and it would take you a lot of time. But once you have it broken up into the x and y components, you can just separately add the x and y components. So vector a plus vector b, that's just 2 plus 10 times i plus 3 plus 2 times j. And that's equal to 12i plus 5j. And something you might want to do, maybe I'll do it in the future video, is actually draw out these two vectors and add them visually. And you'll see that you get this exact answer. And as we go into further videos, or future videos, you'll see how this is super useful once we start doing more complicated physics problems, or once we start doing physics with calculus. Anyway, I'm about to run out of time on the ten minutes. So I'll see you in the next video.