Controlling a plane in space

Hello everyone! So I'm talking about how to find the tangent plane to a graph, and I think the first step of that is to just figure out how we control planes in three dimensions in the first place.

What I have pictured here is a red dot representing a point in three dimensions, and the coordinates of that point easily enough are 1, 2, 3. So the x coordinate is 1, the y coordinate is 2, and the z coordinate is 3.

Then I have a plane that passes through it, and the goal of the video is going to be to find a function, a function that I'll call l, that takes in a two-dimensional input x and y. This function l should have this plane as its graph.

Now the first thing to notice is that there's lots of different planes that could be passing through this point. Right at the moment, it's one that's got a certain kind of angle; you could think of it going up in one direction, but you could give this a lot of different directions and get a lot of different planes that all pass through that one point.

So we're going to need to find some way of distinguishing the specific one that we're looking at, which is this one right here, from other possible planes that can pass through it. As we work through, you'll see how this is done in terms of partial derivatives. But as we are getting our head around what the formula for this graph can be, let's just start observing properties that it has.

The first property is that the graph actually passes through this point (1, 2, 3). What that means, in terms of, you know, functions over here, is that if you evaluate it at the point (1, 2), the input pair where both x is 1 and y is 2, then it should equal 3. It should equal 3 because that's telling you that when you go to x equals 1 and y equals 2, and then you say what's the height of the graph above that point, it should be the z coordinate of the desired point.

So this right here is kind of fact number one that we can take into consideration. Beyond that, let's start thinking about what makes planes, what makes these kind of flat graphs different from the sort of curvy graphs that you might be used to in other multivariable functions.

The main idea is that if you intersect it with another plane, so here I'm going to intersect it with y equals 2, this kind of constant plane. So I'll go ahead and write that down; that plane that you're looking at is y equals 2. You can think of this as representing, you know, what does movement in the x direction look like. As we move along the x direction, this kind of has a slope; the two planes intersect along a line, and that's one of the crucial features of a plane.

If you intersect it with another plane, you just get a straight line, meaning the slope is constant as you move in the x direction, but it's also constant; that same slope if you move in the y direction. If I had chosen a different plane, you know, if instead I had chosen y equals 1, which looks like this, then you get a line with the same slope.

No matter what constant value of y you choose, it's always intersecting that plane with a line that has the same slope. Now if you look back to the videos on partial derivatives, and in particular on how you interpret the partial derivative of a function with respect to its graph, what this is telling you is that when we take the partial derivative of l with respect to x—because constant y means you're moving in the x direction—this should just be some kind of constant, some kind of constant a. I'll kind of emphasize that as a constant value here.

The same goes in the other direction, right? Let's say instead of intersecting it with constant values of y, you say, well, what if you intersected it with a constant value of x, like x equals 1? Well, in that case, what you should get, because you're intersecting it with a plane, is another straight line. So these two planes are intersecting along a straight line, which means as you move in the y direction, this slope won't change.

Also, as you move in the x direction, if you imagine slicing it with a bunch of different planes all representing different constant values of x, you would be getting a line with that same slope. This is telling you another powerful fact that the partial derivative of l with respect to y, you know, if you're moving in the y direction, that's equal to some other constant that I'm going to call b.

Now keep in mind these are very powerful statements because the partial derivative of l with respect to x is a function. This is a function of x and y, and that might actually be worth emphasizing here. This partial derivative of l with respect to y is something that you evaluate at, you know, a point in two-dimensional space, and we're saying that that's equal to some kind of constant value.

Now that's a pretty powerful thing, right? Because it's telling you—it's giving you control over the function at all possible input points, you know, for movement in a specified direction. The same goes over here; this is telling you that a function is constantly equal to, you know, some value b. We're not sure what this value b is, but just geometrically, we can kind of estimate what these things should be.

So if I take back the plane representing a constant y value and we say, what's this slope? You know, you're moving in the x direction; we've got a constant y value. What is the slope at which this plane intersects our graph? I would estimate this as about a slope of 2. You know, you kind of go over one and it goes up two, so what that would tell you is that, at least in the specific graph that we're looking at, this is at least approximately equal to 2.

Then similarly, if we look at a constant x value and we say that represents movement in the y direction, what is the slope there? This looks to me like about one as a slope; you kind of move over one unit and you go up one unit. So I'd say down here that the constant value of the partial derivative with respect to y is about equal to one.

So we have three different facts here: the value of the function at the point (1, 2), the value of its partial derivative with respect to x everywhere, and the partial derivative with respect to y everywhere. This information is actually going to be enough to tell us what the function as a whole should equal!

Now specifically, this idea that the partial derivative with respect to x is constant tells us that the function function l(x, y) is going to equal 2 times x plus something that doesn't have any x's in it—something that, as far as x is concerned, is a constant because the only thing whose derivative with respect to x is the constant 2 is 2x plus something that's constant as far as x is concerned.

Then similarly over here, if the partial with respect to y is the constant 1, then that tells you that the whole function looks like, you know, this looks like a constant as far as y is concerned. So once we bring in y, it's going to be 1 times y plus something that's constant as far as y is concerned.

You know, this part is already constant as far as y is concerned, so anything that I add beyond here has to be constant as far as both x and y is concerned. So that part has to actually literally be a constant. So I'm just going to put in, I'm just going to put in c for that to represent constant.

So, you can see this is a very restrictive property on our function because the only place x can show up is as this linear term, and the only place y can show up is as this linear term. When I use the word linear, you can pretty much interpret it as saying the term x shows up without an exponent or without anything fancy happening to it—it's just x times a constant. That's pretty much what I mean by linear.

It's got more precise formulations in other contexts, but as far as we're concerned here, you can just think of it as meaning variable times a constant. So the question is, what should this c be? You can imagine that based on this property, based on the value at the point (1, 2), we can uniquely determine c.

You can plug in x equals 1, y equals 2, know that this has to equal 3, and solve for c, which we could do, but I'm going to actually do something a little bit more convenient. I'm going to kind of shift around where the constants show up, and I'm going to say that the whole function should equal 2 times, and then I'm going to put a constant in with x.

I'm going to say x minus 1, and then I'm going to do the same thing with y; I'm going to say plus 1. Here's the partial derivative with respect to y, y minus, and then I'm going to say 2. The reason I'm doing this? Notice this doesn't change the partial derivative information. It's just if we take the partial derivative with respect to x, this will still be 2, and when we take it with respect to y, this will still be 1.

But the reason I'm putting these in here is because we're going to evaluate it at the point (1, 2), so I want to make it look as easy as possible to evaluate at the point (1, 2). And then from here, I'm just going to add another constant. So instead of saying c, because this is going to be slightly different from c, I'll call it k, but the idea is I'm just moving around constants. If you imagine distributing the multiplication here and having, you know, 2 times that negative 1 and 1 times that negative 2, you're just changing what the value of the constant stuck on the end here is.

Now the important part, the reason that I'm writing it this way, which is only slightly different, is because then when I evaluate this at l(1, 2), this whole first part cancels out because plugging in x equals 1 means this whole part goes to 0. Same with the second part, because when I plug in y equals 2, this part goes to 0. So k, this other constant that I'm tagging on the end, is going to completely specify what happens when I evaluate this at the point (1, 2).

Of course, I want it to be the case that when I evaluate it at (1, 2), I get 3. I want it to be the case when I evaluate it at (1, 2), I get 3, so that tells me that this constant k here should just equal 3.

So notice the way that I've written the function here is actually quite powerful. We have a lot of control; this term 2 was telling us the slope with respect to x. So when you move purely in the x direction—and that was kind of illustrated here—purely in the x direction, that's telling us the slope with respect to x.

This term one here was telling us the slope with respect to y. So when we moved purely in the y direction, that's telling us the slope there. We could just turn those knobs. If we change the two and we change the one, that's what's going to allow us to basically change what the slopes of the plane are.

I'm going to say slopes plural because it's with respect to the x and the y direction, and that'll give us control over various different planes that pass through—if I'm looking at the one, oh geez, I don't let's say the one right here.

Then the movement in the y direction is very shallow, so that would be turning this knob lower, and instead of one, it might be point zero one. And if I were looking at movement in the x direction, you know, this looks actually negative, so this would tell you that it's going to be some kind of negative number.

So you can kind of dial these knobs and that changes the different planes that pass through that same point. Then plugging in this (1, 2, 3) tells us what point we're specifying, but we're basically saying when you input x equals 1 and you input y equals 2, the whole thing should equal 3.

So this form right here is powerful enough that I want you to remember it for the next video. I want you to remember the idea of writing things down in this way, where you specify the point it's passing through with its x coordinate, y coordinate, and z coordinate placed where they are, and then you tweak the slopes using these coefficients out front.

So with that, I will see you next video.