Visualizing scalar-valued functions
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Representing points in 3d
- [Voiceover] So a lot of the ways that we represent multivariable functions assume that you're fluent with understanding how to represent points in three-dimensions and also how to represent vectors in three-dimensions. So I thought I'd make a little video here to, spell out exactly how it is that we describe points and vectors in three-dimensions. And before we do that, I think it will be valuable if we start off by describing points and vectors in two-dimensions. And, I'm assuming if you're learning about multivariable calculus, that a lot of you have already learned about this. And you might be saying what's the point? I already know how to represent, points and vectors in two-dimensions. But there is a huge value in analogy here, because as soon as you start to compare two dimensions and three-dimensions, you start to see patterns for how it could extend to other dimensions that you can't necessarily visualize, or when it might be useful to think about one dimension versus another. So in two-dimensions, if you have some kind of point just, you know, off sitting there. We typically represent it, you've got an x-axis and a y-axis that a perpendicular to each other. And we represent this number with a pair. Sorry, we represent this point with a pair of numbers. So in this case, I don't know, it might be something like one, three. And what that would represent, is it's saying that you have to move a distance of one along the x-axis and then a distance of three up along the y-axis. So you know this, let's say that's a distance of one, that's a distance of three, it might not be exactly that the way I drew it, but let's say that those are the coordinates. What this means is that every point in two dimensional space can be given a pair of numbers like this and you think of them as instructions where it's kind of telling you how far to walk in one way, how far to walk in another. But you can also think of the reverse, right? Every time that you have a pair of things you know that you should be thinking two-dimensionaly and that's actually, a surprisingly powerful idea that I don't think I appreciated for a long time how there's this back and forth between pairs of numbers and points in space and it lets you visualize things you didn't think you could visualize, or lets you understand things that are inherently visual just by kinda going back and forth. And in three-two-dimensions there's a similar dichotomy, but between triplets of points and points in three-dimensional space. So, let me just plop down a point in three-dimensional space here, and it's hard to get a feel for exactly where it is until you move things around. This is one thing that makes three-dimensions hard is you can't really draw it without moving it around or showing, showing a difference in perspective in various ways. But we describe points like this, again with a set of coordinates, but this time it's a triplet. And this particular point, I happen to know is one, two, five. And what those numbers are telling you is how far to move, parallel to each axis. So just like with two dimensions, we have an x-axis, and a y-axis. But now there's a third axis that's perpendicular to both of them, and moves us into that third-dimension, the z-axis. And the first number in our coordinate is gonna tell us how far, whoop, can't move those guys, how far we need to move in the x direction as our first step. The second number, two in this case, tells us how far we need to move parallel to the y-axis, for our second step. And then the third number tells us how far up we have to go to get to that point. And you can do this for any point in three-dimensional space, right? Any point that you have you can give the instructions for how to move along the x, and then how to move parallel to the y and how to move parallel to the z to get to that point, which means there's this back and forth between triplets of numbers and points in 3-D. So whenever you come across a triplet of things, and you'll see this in the next video when we start talking about three-dimensional graphs, you'll know, just by virtue of the fact that it's a triplet "Ah, yes, I should be thinking in three-dimensions somehow" just in the same way whenever you have pairs you should be thinking ah, this is a very two-dimensional thing. So, there's another context though where pairs of numbers come up and that would be vectors. So a vector you might represent, you know you typically it with an arrow. Oh, ahh. Help, help! (chuckling) So vectors, So vectors we typically represent some kind of arrow, let's, this arrow nice color. An arrow. And if it's a vector from the origin to a simple point, the coordinates of that vector are just the same as those of the point. And the convention is to write those coordinates in a column. You know, it's not set in stone, but typically if you see numbers in a column you should be thinking about it as a vector, some kind of arrow. And if it's a pair with parenthesis around it you just think about it as a point. And even though, you know, both of these are ways of representing the same pair of numbers, the main difference is that a vector you could have started at any point in space it didn't have to start in the origin. So if we have that same guy, but you know if he starts here and he still has a rightward component of one and an upward component of three, we think of that as the same vector. And typically these are representing motion of some kind whereas points are just representing like actual points in space. And the other big thing that you can do is you can add vectors together. So, you know, if you had another, let's say you have another vector that has a large x component but a small negative y component, like this guy. And what that means is that you can kind of add like, imagining that second vector started at the tip of the first one and then however you get from the origin to the new tip there, that's gonna be the resulting vector, so I'd say this is, this is the sum of those two vectors. And you can't really do that with points as much. In order to think about adding points you end up thinking about them as vectors. And the same goes with three-dimensions. For a given point, if you draw an arrow from the origin up to that point, this arrow would be represented with that same triplet of numbers, but you typically do it in a column, I call this a column vector. That's not three, that's five. And the difference between the point and the arrow is you can think of, you know the arrow or the vector is starting anywhere in space, it doesn't really matter, as long as it's got those same components for how far does it move parallel to the x, for how far does it move parallel to the y-axis, and for how far does it move parallel to the z-axis. So in the next video, I'll show how we use these three-dimensions to start graphing multivariable functions.