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Studying for a test? Prepare with these 10 lessons on Vectors.
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Video transcript
We've already seen that you can visually represent a vector as an arrow, where the length of the arrow is the magnitude of the vector and the direction of the arrow is the direction of the vector. And if we want to represent this mathematically, we could just think about, well, starting from the tail of the vector, how far away is the head of the vector in the horizontal direction? And how far away is it in the vertical direction? So for example, in the horizontal direction, you would have to go this distance. And then in the vertical direction, you would have to go this distance. Let me do that in a different color. You would have to go this distance right over here. And so let's just say that this distance is 2 and that this distance is 3. We could represent this vector-- and let's call this vector v. We could represent vector v as an ordered list or a 2-tuple of-- so we could say we move 2 in the horizontal direction and 3 in the vertical direction. So you could represent it like that. You could represent vector v like this, where it is 2 comma 3, like that. And what I now want to introduce you to-- and we could come up with other ways of representing this 2-tuple-- is another notation. And this really comes out of the idea of what it means to add and scale vectors. And to do that, we're going to define what we call unit vectors. And if we're in two dimensions, we define a unit vector for each of the dimensions we're operating in. If we're in three dimensions, we would define a unit vector for each of the three dimensions that we're operating in. And so let's do that. So let's define a unit vector i. And the way that we denote that is the unit vector is, instead of putting an arrow on top, we put this hat on top of it. So the unit vector i, if we wanted to write it in this notation right over here, we would say it only goes 1 unit in the horizontal direction, and it doesn't go at all in the vertical direction. So it would look something like this. That is the unit vector i. And then we can define another unit vector. And let's call that unit vector-- or it's typically called j, which would go only in the vertical direction and not in the horizontal direction. And not in the horizontal direction, and it goes 1 unit in the vertical direction. So this went 1 unit in the horizontal. And now j is going to go 1 unit in the vertical. So j-- just like that. Now any vector, any two dimensional vector, we can now represent as a sum of scaled up versions of i and j. And you say, well, how do we do that? Well, you could imagine vector v right here is the sum of a vector that moves purely in the horizontal direction that has a length 2, and a vector that moves purely in the vertical direction that has length 3. So we could say that vector v-- let me do it in that same blue color-- is equal to-- so if we want a vector that has length 2 and it moves purely in the horizontal direction, well, we could just scale up the unit vector i. We could just multiply 2 times i. So let's do that-- is equal to 2 times our unit vector i. So 2i is going to be this whole thing right over here or this whole vector. Let me do it in this yellow color. This vector right over here, you could view as 2i. And then to that, we're going to add 3 times j-- so plus 3 times j. Let me write it like this. Let me get that color. Once again, 3 times j is going to be this vector right over here. And if you add this yellow vector right over here to the magenta vector, you're going to get-- notice, we're putting the tail of the magenta vector at the head of the yellow vector. And if you start at the tail of the yellow vector and you go all the way to the head of the magenta vector, you have now constructed vector v. So vector v, you could represent it as a column vector like this, 2 3. You could represent it as 2 comma 3, or you could represent it as 2 times i with this little hat over it, plus 3 times j, with this little hat over it. i is the unit vector in the horizontal direction, in the positive horizontal direction. If you want to go the other way, you would multiply it by a negative. And j is the unit vector in the vertical direction. As we'll see in future videos, once you go to three dimensions, you'll introduce a k. But it's very natural to translate between these two things. Notice, 2, 3-- 2, 3. And so with that, let's actually do some vector operations using this notation. So let's say that I define another vector. Let's say it is vector b. I'll just come up with some numbers here. Vector b is equal to negative 1 times i-- times the unit vector i-- plus 4 times the unit vector in the horizontal direction. So given these two vector definitions, what would the would be the vector v plus b be equal to? And I encourage you to pause the video and think about it. Well once again, we just literally have to add corresponding components. We could say, OK, well let's think about what we're doing in the horizontal direction. We're going 2 in the horizontal direction here, and now we're going negative 1. So our horizontal component is going to be 2 plus negative 1-- 2 plus negative 1 in the horizontal direction. And we're going to multiply that times the unit vector i. And this, once again, just goes back to adding the corresponding components of the vector. And then we're going to have plus 4, or plus 3 plus 4-- And let me write it that way-- times the unit vector j in the vertical direction. And so that's going to give us-- I'll do this all in this one color-- 2 plus negative 1 is 1i. And we could literally write that just as i. Actually, let's do that. Let's just write that as i. But we got that from 2 plus negative 1 is 1. 1 times the vector is just going to be that vector, plus 3 plus 4 is 7-- 7j. And you see, this is exactly how we saw vector addition in the past, is that we could also represent vector b like this. We could represent it like this-- negative 1, 4. And so if you were to add v to b, you add the corresponding terms. So if we were to add corresponding terms, looking at them as column vectors, that is going to be equal to 2 plus negative 1, which is 1. 3 plus 4 is 7. So this is the exact same representation as this. This is using unit vector notation, and this is representing it as a column vector.