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Determinant and area of a parallelogram

Video transcript
I've got a 2 by 2 matrix here, and let's just say its entries are a, b, c, and d. And it's composed of two column vectors. We've done this before, let's call this first column v1 and let's call the second column v2. So we can rewrite here. We can say v1 one is equal to ac, and we could write that v2 is equal to bd. And these are both members of r2, and just to have a nice visualization in our head, let's graph these two. Let me draw my axes. That's my vertical axis. That's my horizontal axis. And maybe v1 looks something like this. v1 might look something like that. So that is v1. It's horizontal component will be a, its vertical coordinant -- give you this as maybe a position vector, or just how we're drawing it, is c. And then v2, let's just say it looks something like this. Let's say that they're not the same vector. So v2 looks like that. v2, its horizontal coordinate is going to b, and its vertical coordinate is going to be d. Now, what we're going to concern ourselves with in this video is the parallelogram generated by these two guys. What I mean by that is, imagine that these two guys are position vectors that are specifying points on a parallelogram, and then of course the -- or not of course but, the origin is also another point in the parallelogram, so what will be the last point on the parallelogram? Well, you can imagine. A parallelogram, we already have two sides of it, so the other two sides have to be parallel. So one side look like that, parallel to v1 the way I've drawn it, and the other side will look like this. That's our parallelogram. The parallelogram generated by v2 and v1. What we're going to concern ourselves with specifically is the area of the parallelogram generated by v1 and v2. So how do we figure that out? In general, if I have just any parallelogram-- this is kind of a tilted one, but if I just have any parallelogram, let me just draw any parallelogram right there-- the area is just equal to the base-- so that could be the base-- times the height. So it's equal to base -- I'll write capital B since we have a lowercase b there-- base times height. That's what the area of a parallelogram would be. Now what are the base and the height in this situation? Let me write this down. The area of our parallelogram is equal to the base times the height. And actually-- well, let me just write it here. So what is the base here? The base here is going to be the length of our vector v. So this is our base. So this right here is going to be the length of vector v1, the length of this orange vector right here. And what's the height of this parallelogram going to be? We could drop a perpendicular here, and that, the length of this line right here, is going to be our height. So how can we figure out that, you know, we know what v1 is, so we can figure out the base pretty easily. But how can we figure out the height? Well, one thing we can do is, if we can figure out this guy right here, we could use the Pythagorean theorem. Because the length of this vector squared, plus H squared, is going to be equal to the length of v2 squared. So let's see if we can do that. What is this guy? What is this green guy right here? Well if you imagine a line-- let's imagine some line l. So let's say l is a line spanned by v1. Which means you take all of the multiples of v1, and all of the positions that they specify will create a set of points, and that is my line l. So you take all the multiples of v1, you're going to get every point along this line. Just like that. Now if we have l defined that way-- that line right there is l, I don't know if you can see it. Let me do it a little bit better -- and it goes through v1 and it just keeps going over there. What is this green line right there? This green line that we're concerned with, that's the projection onto l of what? Well, we have a perpendicular here, you can imagine the light source coming down-- I don't know if that analogy helps you-- but it's kind of the shadow of v2 onto that line. So it's a projection of v2, of your vector v2 onto l is this green line right there. So, if we want to figure out H, we can just use the Pythagorean theorem. So we can say that the length of H squared-- well I'm just writing H as the length, I'm not even specifying it as a vector. So we could say that H squared, where that is the length of this line, plus the length of this vector squared-- and the length of that vector squared is the length of the projection onto l of v2. I'll do that in a different color. So the length of the projection onto l of v2 squared-- all right? We're just doing the Pythagorean theorem. This squared plus this squared is going to equal that squared. It's going to be equal to the length of v2 squared. That's just the Pythagorean theorem. Nothing fancy there. So how can we simplify? We want to solve for H. And actually, let's just solve for H squared for now because it'll keep things a little bit simpler. So we can say that H squared is equal to this guy, is equal to the length of my vector v2 squared minus the length of the projection squared. So minus -- I'll do that in purple -- minus the length of the projection onto l of v2 squared. Remember, this thing is just this thing right here, we're just doing the Pythagorean theorem. So let's see if we can simplify this, or write it in terms that we understand. So the length of a vector squared, this is just equal to-- let me write it this way-- this is just equal to v2 dot v2. You take a vector, you dot it with itself, and you get the length of that vector squared, we saw that many, many videos ago. And then what is this guy going to be equal to? Well, the projection-- I'll do it over here. The projection onto l of v2 is going to be equal to v2 dot the spanning vector, which is v1. So it's v2 dot v1 over the spanning vector dotted with itself, v1 dot v1. We saw this several videos ago when we learned about projections. And this is just a number right there. And then it's going to be times the spanning vector itself. So times v1. That is what the projection is. So it's going to be this minus the length of the projection squared. What is the length of the projection squared? Well that's this guy dotted with himself. Let me write it this way, let me take it step by step. So this is going to be minus-- I'm want to make sure I can still see that up there so I don't have to rewrite it. The projection is going to be, it looks a little complicated but hopefully things will simplify, v2 dot v1 over v1 dot v1 times-- switch colors-- times the vector-- this is all just going to end up being a number, remember you take dot products, you get numbers-- times the vector v1. And we're going to take the length of that whole thing squared. And all of this is going to be equal to H squared. Once again, just the Pythagorean Theorem. This or this squared, which is the height squared, is equal to your hypotenuse squared, this is your hypotenuse squared, minus the other side squared. This is the other side squared. Looks a little complicated, but it was just a projection of this guy on to that right there. So let's see if we can simplify this a little bit. We're just going to have to break out some algebra or let s can do here. Well actually, not algebra, some linear algebra. So what is this guy? Well this guy is just the dot product of this with itself. So this is just equal to-- we know, I mean any vector, if you take the square of its length, it's just that vector dotted with itself. So this thing, if we are taking the square of this guy's length, it's just equal to this guy dotted with himself. Let me write everything over again. So we get H squared is equal to v2 dot v2, and then minus this guy dotted with himself. So minus v2 dot v1 over v1 dot v1, times the vector v1, dotted with itself. Dotted with v2 dot v1-- remember, this green part is just a number-- over v1 dot v1 times v1. And what is this equal to? Let's just simplify this. These are just scalar quantities, and we saw that the dot product is associative with respect to scalar quantities, so we can just change the order here. So this is going to be equal to the scalar quantity times itself. So we could say this is equal to v2 dot v1. Let me write it this way. V2 dot v1, that's going to be-- and we're going to multiply the numerator times itself, v2 dot v1. And then all of that over v1 dot v1 times v1 dot v1. Remember, I'm just taking these two terms and multiplying them by each other. I'm just switching the order, and then we know that the scalars can be taken out, times these two guys dot each other. Times v1 dot v1. That's what this simplifies to. Now this is now a number. We had vectors here, but when you take a dot product, you just get a number. And this number is the same as this number. So we can simplify a little bit. So we can cross those two guys out, and then we are left with that our height squared is equal to this expression times itself. We have it times itself twice, so it's equal to-- let me start over here. It's equal to v2 dot v2 minus this guy times itself. So v2 dot v1 squared, all of that over just one of these guys. v1 dot v1. That is what the height squared is. Now we have the height squared, we could take the square root if we just want to solve for the height. But to keep our math simple, we know that area is equal to base times height. Let's just say what the area squared is equal to. Because then both of these terms will get squared. So if the area is equal to base times height-- we saw that at the beginning of the video-- then the area squared is going to be equal to these two guys squared. It's going to be equal to base squared times height squared. Now what is the base squared? Let me do it like this. So the base squared-- we already saw, the base of our parallelogram is the length of vector v1. Now what is the base squared? The base squared is going to be the length of vector v1 squared. Or another way of writing that is v1 dot v1. And we already know what the height squared is, it's this expression right there. Let me write that down. The height squared is the height squared right there. So what is our area squared going to be? Our area squared-- let me go down here where I'll have more space-- our area squared is going to be equal to our base squared, which is v1 dot v1 times our height squared. Let with me write it like this. No, I was using the wrong color. Times this guy over here. v2 dot v2 minus v2 dot v1 squared over v1 dot v1. Now what does this simplified to? Well, this is just a number, these are all just numbers. So if I multiply, if I distribute this out, this is equal to what? This times this is equal to v1-- let me color code it-- v1 dot v1 times this guy times v2 dot v2. And then when I multiplied this guy times that guy, what happens? Well I have this guy in the numerator and that guy in the denominator, so they cancel out. So I'm just left with minus v2 dot v1 squared. Now let's remind ourselves what these two vectors were. v1 was the vector ac and v2 is the vector bd. Let me rewrite it down here so we have it to work with. So v1 was equal to the vector ac, and v2 is equal to the vector bd. So what is v1 dot v1? That is equal to a dot a, a times a, a squared plus c squared. That's this, right there. And that's what? v2 dot v2. So we're going to have that times v2 dot v2. v2 dot v2 is v squared plus d squared. And then we're going to have minus v2 dot v1 squared. So what's v2 dot v1? It's b times a, plus d times c, or a times b plus -- we're just dotting these two guys. So it's ab plus cd, and then we're squaring it. And let's see what this simplifies to. Hopefully it simplifies to something. Let me switch colors. So if we just multiply this out, let me write it here. Our area squared is equal to a squared times b squared. a squared times d squared, plus c squared times b squared, plus c squared times d squared. And then minus this guy squared. What is that going to be equal to? ab squared is a squared, b squared. And then I'm going to multiply these guys times each other twice, so that's going to be plus 2abcd. And then you're going to have a plus c squared, d squared. I just foiled this out, that's the best way you could think about it. And then, if I distribute this negative sign, what do I have? Let me rewrite everything. It's equal to a squared b squared, plus a squared d squared, plus c squared b squared, plus c squared d squared, minus a squared b squared, minus 2abcd, minus c squared, d squared. All I did is, I distributed the minus sign. Now it looks like some things will simplify nicely. And now remember, all this is equal to our area squared. We have a ab squared, we have a minus ab squared. They cancel out. We have a minus cd squared and a cd squared, so they cancel out. So all we're left with is that the area of our parallelogram squared is equal to a squared d squared minus 2abcd plus c squared b squared. Now this might look a little bit bizarre to you, but if you made a substitution right here, if you said that x is equal to ad, and if you said y is equal to cb, then what does this become? This is equal to x squared minus 2 times xy plus y squared. Right? Hopefully you recognize this. And this is just the same thing as x minus y squared. So, if this is our substitutions we made-- I did this just so you can visualize this a little bit better. So we have our area squared is equal to x minus y squared or ad minus cb, or let me write it, bc squared. That's what the area of our parallelogram squared is. And if you don't quite understand what I did here, I just made these substitutions so you can recognize it better. But just understand that this is the same thing as this. If you want, you can just multiply this guy out and you'll get that right there. But what is this? What is this thing right here? It's the determinant. This is the determinant of our original matrix. Well, I called that matrix A and then I used A again for area, so let me write it this way. Area squared -- let me write it like this. Area squared is equal to ad minus bc squared. So this is area, these A's are all area. But what is this? This is the determinant of my matrix. That is the determinant of my matrix A, my original matrix that I started the problem with, which is equal to the determinant of abcd. Right? The determinant of this is ad minus bc, by definition. So your area-- this is exciting! So the area of your parallelogram squared is equal to the determinant of the matrix whose column vectors construct that parallelogram. Is equal to the determinant of your matrix squared. Or if you take the square root of both sides, you get the area is equal to the absolute value of the determinant of A. Which is a pretty neat outcome, especially considering how much hairy algebra we had to go through. Let's go back all the way over here, go back to the drawing. So if we want to figure out the area of this parallelogram right here, that is defined, or that is created, by the two column vectors of a matrix, we literally just have to find the determinant of the matrix. The area of this is equal to the absolute value of the determinant of A. And you have to do that because this might be negative. You can imagine if you swapped these guys around, if you swapped some of the rows, this guy would be negative, but you can 't have a negative area. And it wouldn't really change the definition, it really wouldn't change what spanned. If you switched v1 and v2, you're still spanning the same parallelogram, you just might get the negative of the determinant. But that is a really neat outcome. And you know, when you first learned determinants in school-- I mean, we learned the first motivation for a determinant was this idea of when we take the inverse of a 2 by 2, this thing shows up in the denominator and we call that the determinant. But now there's this other interpretation here. There's actually the area of the parallelogram created by the column vectors of this matrix.