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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'll 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 you might be saying what's the point I already I already know how to represent points and vectors in two dimensions but there's a huge value in analogy here because as soon as you start to compare two dimensions in three dimensions you start to see patterns for how it could extend to other dimensions that you can't necessarily visualize or 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 are we typically represent it you've got an x-axis and a y-axis that are 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 1 3 and what that would represent is it saying that you have to move a distance of 1 along the x axis and then a distance of 3 up along the y axis so you know this let's say that's the distance of 1 that's the distance of 3 might not be exactly that the way I drew it but let's let's say that those are the coordinates what this means is 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 this of the reverse right every every time that you have a pair of things you know that you should be thinking two dimensionally and that's actually a surprisingly powerful idea that I don't think I appreciate it 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 kind of going back and forth and in three dimensions there's a similar dichotomy but between Triplets of points and points in three-dimensional space so let me just 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 as you can't really draw it without moving it around or showing showing a difference in perspective in various ways but we described 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 going to tell us how far we can't move those guys how far we need to move in the x-direction as our first step the second number to in this case tells us how far we have 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 3d 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 that 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 represent it with an arrow whoa-ohh so vectors so vectors we typically represent some kind of arrow that'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 as a vector some kind of arrow and if it's a pair with parentheses 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 it 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 where as 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 you can kind of add by imagining that 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 going to be the resulting vector so you might 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 it'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 how far does it move parallel to the y axis and 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