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Vector examples

Visually understanding basic vector operations. Created by Sal Khan.

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Video transcript

In the last video I was a little formal in defining what Rn is, and what a vector is, and what vector addition or scalar multiplication is. In this video I want to kind of go back to basics and just give you a lot of examples. And give you a more tangible sense for what vectors are and how we operate with them. So let me define a couple of vectors here. And I'm going to do, most of my vectors I'm going to do in this video are going to be in R2. And that's because they're easy to draw. Remember R2 is the set of all 2-tuples. Ordered 2-tuples where each of the numbers, so you know you could have x1, my 1 looks like a comma, x1 and x2, where each of these are real numbers. So you each of them, x1 is a member of the reals, and x2 is a member of the reals. And just to give you a sense of what that means, if this right here is my coordinate axes, and I wanted a plot all my x1's, x2's. You know you could view this as the first coordinate. We always imagine that as our x-axis. And then our second coordinate we plotted on the vertical axis. That traditionally is our y-axis, but we'll just call that the second number axis, whatever. You could visually represent all of R2 by literally every single point on this plane if we were to continue off to infinity in every direction. That's what R2 is. R1 would just be points just along one of these number lines. That would be R1. So you could immediately see that R2 is kind of a bigger space. But anyway, I said that I wouldn't be too abstract, that I would show you examples. So let's get some vectors going in R2. So let me define my vector a. I'll make it nice and bold. My vector a is equal to, I'll make some numbers up, negative 1, 2. And my vector b, make it nice and bold, let me make that, I don't know, 3, 1. Those are my two vectors. Let's just add them up and see what we get. Just based on my definition of vector addition. I'll just stay in one color for now so I don't have to keep switching back and forth. So a, nice deep a, plus bolded b is equal to, I just add up each of those terms. Negative 1 plus 3. And then 2 plus 1. That was my definition of vector addition. So that is going to be equal to 2 and 3. Fair enough that just came out of my definition of vector addition. But how can we represent this vector? So we already know that if we have coordinates, you know, if I have the coordinate, and this is just a convention. It's just the way that we do it. The way we visualize things. If I wanted to plot the point 1, 1, I go to my coordinate axes. The first point I go along the horizontal, what we traditionally call our x-axis. And I go 1 in that direction. And then convention is, the second point I go 1 in the vertical direction. So the point 1, 1. Oh, sorry, let me be very clear. This is 2 and 2, so one is right here, and one is right there. So the point 1, 1 would be right there. That's just the standard convention. Now our convention for representing vectors are, you might be tempted to say, oh, maybe I just represent this vector at the point minus 1, 2. And on some level you can do that. I'll show you in a second. But the convention for vectors is that you can start at any point. Let's say we're dealing with two dimensional vectors. You can start at any point in R2. So let's say that you're starting at the point x1, and x2. This could be any point in R2. To represent the vector, what we do is we draw a line from that point to the point x1. And let me call this, let's say that we wanted to draw a. So x1 minus 1. So this is, I'm representing a. So this is, I want to represent the vector a. x1 minus 1, and then x1 plus 2. Now if that seems confusing to you, when I draw it, it'll be very obvious. So let's say I just want to start at the point, let's just say for quirky reasons, I just pick a random point here. I just pick a point. That one right there. That's my starting point. So minus 4, 4. Now if I want to represent my vector a, what I just said is that I add the first term in vector a to my first coordinate. So x1 plus minus 1 or x1 minus 1. So my new one is going to be, so this is my x1 minus 4. So now it's going to be, let's see, I'm starting at the point minus 4 comma 4. If I want to represent a, what I do is, I draw an arrow to minus 4 plus this first term, minus 1. And then 4 plus the second term. 4 plus 2. And so this is what? This is minus 5 comma 6. So I go to minus 5 comma 6. So I go to that point right there and I just draw a line. So my vector will look like this. I draw a line from there to there. And I draw an arrow at the end point. So that's one representation of the vector minus 1, 2. Actually let me do it a little bit better. Because minus 5 is actually more, a little closer to right here. Minus 5 comma 6 Is right there, so I draw my vector like that. But remember this point minus 4 comma 4 was an arbitrary place to draw my vector. I could have started at this point here. I could have started at the point 4 comma 6 and done the same thing. I could have gone minus 1 in the horizontal direction, that's my movement in the horizontal direction. And then plus 2 in the vertical direction. So I could have drawn, so minus 1 in the horizontal and plus 2 in the vertical gets me right there. So I could have just as easily drawn my vector like that. These are both interpretations of the same vector a. I should draw them in the color of vector a. So vector a was this light blue color right there. So this is vector a. This is vector a. Sometimes there'll be a little arrow notation over the vector. But either of those vectors. I could draw an infinite number of vector a's. I could draw vector a here. I could draw it like that. Vector a, it goes back 1 and up 2. So vector a could be right there. Similarly vector b. What does vector b do? I could pick some arbitrary point for vector b. It goes to the right 3, so it goes to the right 1, 2, 3 and then it goes up 1. So vector b, one representation of vector b, looks like this. Another represention. I can start it right here. I could go to the right 3, 1, 2, 3, and then up 1. This would be another representation of my vector b. There's an infinite number of representations of them. But the convention is to often put them in what's called the standard position. And that's to start them off at 0, 0. So your initial point, let me write this down. Standard position is just to start the vectors at 0, 0 and then draw them. So vector a in standard position, I'd start at 0, 0 like that and I would go back 1 and then up 2. So this is vector a in standard position right there. And then vector b in standard position. Let me write that. That's a. And then vector b in standard position is 3, go to the 3 right and then up 1. These are the vectors in standard position, but any of these other things we drew are just as valid. Now let's see if we can get an interpretation of what happened when we added a plus b. Well if I draw that vector in standard position, I just calculated, it's 2, 3. So I go to the right 2 and I go up 3. So if I just draw it in standard position it looks like this. This vector right there. And at first when you look at it, this vector right here is the vector a plus b in standard position. When you draw it like that, it's not clear what the relationship is when we added a and b. But to see the relationship what you do is, you put a and b head to tails. What that means is, you put the tail end of b to the front end of a. Because remember, all of these are valid representations of b. All of the representations of the vector b. They all have, they're all parallel to each other, but they can start from anywhere. So another equally valid representation of vector b is to start at this point right here, kind of the end point of vector a in standard position, and then draw vector b starting from there. So you go 3 to the right. So you go 1, 2, 3. And then you go up 1. So vector b could also be drawn just like that. And then you should see something interesting had happened. And remember, this vector b representation is not in standard position, but it's just an equally valid way to represent my vector. Now what do you see? When I add a, which is right here, to b what do I get if I connect the starting point of a with the end point of b? I get the addition. I have added the two vectors. And I could have done that anywhere. I could have started with a here. And then I could have done the end point. I could have started b here and gone 3 to the right, 1, 2, 3 and then up 1. And I could have drawn b right there like that. And then if I were to add a plus b, I go to the starting point of a, and then the end point of b. And that should also be the visual representation of a plus b. Just to make sure it confirms with this number, what I did here was I went 2 to the right, 1, 2 and then I went 3 up. 1, 2, 3 and I got a plus b. Now let's think about what happens when we scale our vectors. When we multiply it times some scalar factor. So let me pick new vectors. Those have gotten monotonous. Let me define vector v. v for vector. Let's say that it is equal to 1, 2. So if I just wanted to draw vector v in standard position, I would just go 1 to the horizontal and then 2 to the vertical. That's it. That's the vector in standard position. If I wanted to do it in a non standard position, I could do it right here. 1 to the right up 2, just like that. Equally valid way of drawing vector v. Equally valid way of doing it. Now what happens if I multiply vector v. What if I have, I don't know, what if I have 2 times v? 2 times my vector v is now going to be equal to 2 times each of these terms. So it's going to be 2 times 1 which is 2, and then 2 times 2 which is 4. Now what does 2 times vector v look like? Well let me just start from an arbitrary position. Let me just start right over here. So I'm going to go 2 to the right, 1, 2. And I go up 4. 1, 2, 3, 4. So this is what 2 times vector v looks like. This is 2 times my vector v. And if you look at it, it's pointing in the exact same direction but now it's twice as long. And that makes sense because we scaled it by a factor of 2. When you multiply it by a scalar, or you're not changing its direction. Its direction is the exact same thing as it was before. You're just scaling it by that amount. And I could draw this anywhere. I could have drawn it right here. I could have drawn 2v right on top of v. Then you would have seen it, I don't want to cover it. You would have seen that it goes, it's exactly, in this case when I draw it in standard position, it's colinear. It's along the same line, it's just twice as far. it's just twice as long but they have the exact same direction. Now what happens if I were to multiply minus 4 times our vector v? Well then that will be equal to minus 4 times 1, which is minus 4. And then minus 4 times 2, which is minus 8. So this is on my new vector. Minus 4, minus 8. This is minus 4 times our vector v. So let's just start at some arbitrary point. Let's just do it in standard position. So you go to the right 4. Or you go to the left 4. So so you go to the left 4, 1, 2, 3, 4. And then down 8. Looks like that. So this new vector is going to look like this. Let me try and draw a relatively straight line. There you go. So this is minus 4 times our vector v. I'll draw a little arrow on it to make sure you know it's a vector. Now what happened? Well we're kind of in the same direction. Actually we're in the exact opposite direction. But we're still along the same line, right? But we're just in the exact opposite direction. And it's this negative right there that flipped us around. If we just multiplied negative 1 times this, we would have just flipped around to right there, right? But we multiplied it by negative 4. So we scaled it by 4, so you make it 4 times as long, and then it's negative, so then it flips around. It flips backwards. So now that we have that notion, we can kind of start understanding the idea of subtracting vectors. Let me make up 2 new vectors right now. Let's say my vector x, nice and bold x, is equal to, and I'm doing everything in R2, but in the last part of this video I'll make a few examples in R3 or R4. Let's say my vector x is equal to 2, 4. And let's say I have a vector y. y, make it nice and bold. And then that is equal to negative 1, minus 2. And I want to think about the notion of what x minus y is equal to. Well we can say that this is the same thing as x plus minus 1 times our vector y. Right? So x plus minus 1 times our vector y. Now we can use our definitions. We know how to multiply by a scalar. So we'll say that this is equal to, let me switch colors. I don't like this color. This is equal to our x vector is 2, 4. And then what's minus 1 times y? So minus 1 times y is minus 1 times minus 1 is 1. And then minus 1 times minus 2 is 2. So x minus y is going to be these two vectors added to each other, right? I'm just adding the minus of y. This is minus vector y. So this x minus y is going to be equal to 3 and 3 and 6. So let's see what that looks like when we visually represent them. Our vector x was 2, 4. So 2, 4 in standard position it looks like this. That's my vector x. And then vector y in standard position, let me do it in a different color, I'll do y in green. Vector y is minus 1, minus 2. It looks just like this. And actually I ended up inadvertently doing collinear vectors, but, hey, this is interesting too. So this is vector y. So then what's their difference? This is 3, 6. So it's the vector 3, 6. So it's this vector. Let me draw it someplace else. If I start here I go 1, 2, 3. And then I go up 6. So then up 6. It's a vector that looks like this. That's the difference between the two vectors. So at first you say, this is x minus y. Hey, how is this the difference of these two? Well if you overlay this. If you just shift this over this, you could actually just start here and go straight up. And you'll see that it's really the difference between the end points. You're kind of connecting the end points. I actually didn't want to draw collinear vectors. Let me do another example. Although that one's kind of interesting. You often don't see that one in a book. Let me to define vector x in this case to be 2, 3. And let me define vector y to be minus 4, minus 2. So what would be x in standard position? It would be 2, 3. It'd look like that. That is our vector x if we start at the origin. So this is x. And then what does vector y look like? I'll do y in orange. Minus 4, minus 2. So vector y looks like this. Now what is x minus y? Well you know, we could view this, 2 plus minus 1 times this. We could just say 2 minus minus 4. I think you get the idea now. But we just did it the first way the last time because I wanted to go from my basic definitions of scalar multiplication. So x minus y is just going to be equal to 2 plus minus 1 times minus 4, or 2 minus minus 4. That's the same thing as 2 plus 4, so it's 6. And then it's 3 minus minus 2, so it's 5. Right? So the difference between the two is the vector 6, 5. So you could draw it out here again. So you could go, add 6 to 4, go up there, then to 5, you'd go like that. So the vector would look something like this. It shouldn't curve like that, so that's x minus y. But if we drew them between, like in the last example, I showed that you could draw it between their two heads. So if you do it here, what does it look like? Well if you start at this point right there and you go 6 to the right and then up 5, you end up right there. So the difference between the two vectors, let me make sure I get it, the difference between the two vectors looks like that. It looks just like that. Which kind of should make sense intuitively. x minus y. That's the difference between the two vectors. You can view the difference as, how do you get from one vector to another vector, right? Like if, you know, let's go back to our kind of second grade world of just scalars. If I say what 7 minus 5 is, and you say it's equal to 2, well that just tells you that 5 plus 2 is equal to 7. Or the difference between 5 and 7 is 2. And here you're saying, look the difference between x and y is this vector right there. It's equal to that vector right there. Or you could say look, if I take 5 and add 2 I get 7. Or you could say, look, if I take vector y, and I add vector x minus y, then I get vector x. Now let's do something else that's interesting. Let's do what y minus x is equal to. y minus x. What is that equal to? Do it in another color right here. Well we'll take minus 4, minus 2 which is minus 6. And then you have minus 2, minus 3. It's minus 5. So y minus x is going to be, let's see, if we start here we're going to go down 6. 1, 2, 3, 4, 5, 6. And then back 5. So back 2, 4, 5. So y minus x looks like this. It's really the exact same vector. Remember, it doesn't matter where we start. It's just pointing in the opposite direction. So if we shifted it here. I could draw it right on top of this. It would be the exact as x minus y, but just in the opposite direction. Which is just a general good thing to know. So you can kind of do them as the negatives of each other. And actually let me make that point very clear. You know we drew y. Actually let me draw x, x we could draw as 2, 3. So you go to the right 2 and then up 3. I've done this before. This is x in non standard position. That's x as well. What is negative x? Negative x is minus 2 minus 3. So if I were to start here, I'd go to minus 2, then I'd go minus 3. So minus x would look just like this. Minus x. It looks just like x. It's parallel. It has the same magnitude. It's just pointing in the exact opposite direction. And this is just a good thing to kind of really get seared into your brain is to have an intuition for these things. Now just to kind of finish up this kind of idea of adding and subtracting vectors. Everything I did so far was in R2. But I want to show you that we can generalize them. And we can even generalize them to vector spaces that aren't normally intuitive for us to actually visualize. So let me define a couple of vectors. Let me define vector a to be equal to 0, minus 1, 2, and 3. Let me define vector b to be equal to 4, minus 2, 0, 5. We can do the same addition and subtraction operations with them. It's just it'll be hard to visualize. We can keep them in just vector form. So that it's still useful to think in four dimensions. So if I were to say 4 times a. This is the vector a minus 2 times b. What is this going to be equal to? This is a vector. What is this going to be equal to? Well we could rewrite this as 4 times this whole column vector, 0, minus 1, 2, and 3. Minus 2 times b. Minus 2 times 4, minus 2, 0, 5. And what is this going to be equal to? This term right here, 4 times this, you're going to get, the pen tablet seems to not work well there, so I'm going to do it right here. 4 times this, you're going to get 4 times 0, 0, minus 4, 8. 4 times 2 is 8. 4 times 3 is 12. And then minus, I'll do it in yellow, minus 2 times 4 is 8. 2 times minus 2 is minus 4. 2 times 0 is 0. 2 times 5 is 10. This isn't a good part of my board, so let me just. It doesn't write well right over there. I haven't figured out the problem, but if I were just right it over here, what do we get? With 0 minus 8? Minus 8. Minus 4, minus 4. Minus negative 4. So that's minus 4 plus 4, so that's 0. 8 minus 0 is 8. 12 minus, what was this? I can't even read it, what it says. Oh, this is a 10. Now you can see it again. Something is very bizarre. 2 times 5 is 10. So it's 12 minus 10, so it's 2. So when we take this vector and multiply it by 4, and subtract 2 times this vector, we just get this vector. And even though you can't represent this in kind of an easy kind of graph-able format, this is a useful concept. And we're going to see this later when we apply some of these vectors to multi-dimensional spaces.