Hi everyone. Hi and welcome to our last section. We're going to talk about linear equations. It's interesting that we started at the very, very beginning with studying in lines. But you got to remember that was on the xy-plane. Let's do a little then and now. Before, we were on the xy-plane, the good old days. We had a beautiful little line, how cute, it did its thing. We're experts in this now. Good old equation, y equals mx plus b. Now our goal is to take our knowledge here, expand it, and try to figure out an equation for this line in three dimensions. What I hope though you remember from our very earlier sections is that, the key things that make up a line are the slope m and its intercept b. That's one way to do it. If I give you a slope and I tell you a point that it goes through, then we're all thinking about the same line. However, one of the things that I hope you also remember from lecture is that this is not the equation of every single line. In particular, if you have a slope, then your slope is not undefined. Think about that for a second. If m is an actual number, well then it can't be that m is undefined. What lines have undefined slopes? These are your vertical lines. These are lines that go high to low, maybe x equals four or something like that. That's a perfectly good line, but it's not captured in the form y equals mx plus b. The reason for that is that the slope m is undefined. If you wanted a way to capture all types of lines, the general umbrella equation of a line, we have the form Ax plus By plus C is 0. This is what we will call a linear equation. This is a way to capture all lines on the plane. That is the right way to think about lines. There's no reason why we should restrict ourselves to certain kinds of lines, although they are the ones that come up the most often. Now, as I expand my knowledge and grow my thinking from two-dimensional to three-dimensional, if I have a line in space. Stare at the room you're at and look at the intersection of the ceiling and the wall, or some pipe that's going through, whatever you can imagine. There's some line that's passing through space. I want to describe this line. If I generalize this equation that I had in a linear equation for the XY plane, my linear equation is going to be Ax plus By plus Cz plus some D. I introduced the new Z term, and again, I set it equal to 0. It's important to realize that this is an equation. The problem with this equation is though it's not the equation of a line. This is the equation of what happens if I take a line and give it some dimension, some extra dimension, and some extra variable, z. Well, it's moving it, shaking a line back and forth where you actually get a plane. We're going to come back to this one in a second. This gives us the equation of a plane. This is not the way to get the equation of a line. We have to be clever and really try to generalize what does it mean to have a slope? What does it mean to have a point of intersection? We will come back to this equation. You might want to write it down now anyway, this is the equation of a plane in space. But how do I truly capture the same dimensional object, the same line, the same string in space that's floating around in a three-dimensional case. Well, we're going to need some equivalent of slope. We need some m, whatever that's going to be for 3D, and we also need a b, an intercept. Let's think about there. B, the intercept, is really just a point in space. It's just a point where something happens. I can easily replace that in three dimensions with some point P. P will have its coordinates, whatever it is, abc, perfectly fine. Then a slope. A slope is really saying like rise over run. It's the part of the curve, change of Y over change of X. That's now going to be replaced by a vector, and we're going to call this the direction vector. It tells us in what direction the line moves. Think about this for a second. Pick your favorite particle of dust in the air. If you fix your eyes on that spot, and then you shoot a little laser beam in a certain direction and say, I want you to point it out at the sun, I want you to point at the light bulb. You can imagine that we're all thinking about the same line. I've uniquely defined the equation of a line, and that is how we're going to do this. The first equation of a line, and we are going to study this one a bunch, is going to be using the form l of t. It's going to be instead of mx plus b, we're going to call it P for point, plus tv. I'm going to write it like this. P plus tv, you have to realize though recognize this stare at both of them back and forth, mx plus b, P plus tv, they're trying to do the same thing. T in this case is what's called the parameter. You want to think about it like time. Think about a little bug, a little particle of light, something moving along the line. It walks along the line. When t is 0, well, you just get that point P. P is always going to be your starting point. If I have a little line in space, I have some point P on it. If t is positive, the artist is drawing the line in the positive direction. If t is negative, you draw in the negative direction. Some people will say, "Well, wait t is time, can I have negative time?" Sure, yesterday. It's okay don't get hung up on if t could be positive or negative. These are two equations that we are going to see over and over and over again. If you're keeping track of a formula sheet, this is a great one to do. This is the equation of a line in space with P is some point that it goes through, and v is your direction vector. We will come back to this equation of the plane in the next video, we will study this more. But I want you to see that if you just generalize the equation of a line, you actually don't get a line back, you get the higher dimensional version, the plane. Let's play around with this equation of our line so far. Remember, we have our point that the line goes through, some given point P. We have our direction vector v, and we can do whatever we want with these things. Let's give them actual names. Let's get our hands dirty. Let's play around with these things. The point that it goes through will be fixed. We'll do x-naught, y-naught, and z-naught. A little subscript 0 is sometimes read as naught. You can say x-zero, y-zero, and z-zero, but you can also say x-naught and y-naught and z-naught. Our direction vector, some generic a, b, and c. Of course, we'll put all numbers behind this in a minute. But remember this is the direction of, sometimes you say that it's parallel to. Parallel to is another way to give this in word problems when you see this thing, so we have this. Let's just write our formula down, l of t. Now remember, like any function l for line, but I could totally call it any letter I want. It doesn't matter. These are dummy variables and t is the parameter because you think about this often in terms of physics with time, and a particle moving through space, whatever. But of course t is a dummy variable as well. That could be whatever you want. Let's put our formula down. Let's write it a bunch of times, practice through repetition. P is x naught, y naught, and z naught. We throw those things in, remember it's a good old point. Plus t and then we'll do the vector here, so a, b, c. I'm not great. I'm lazy about writing angle brackets versus parentheses. Hopefully, it's clear through context. If I write a little arrow over something, I really mean the vector. The point P, this is also technically a little vector, the position vector. I'm being lazy. If you're better than me, be consistent in your brackets, angle brackets or not. Different books stress different notation, but it's usually pretty clear through context. Anyway, let's play around with this using our vector skills. t is a parameter, t is a real number. Let's distribute that in. I'll be good and I'll write my angle brackets, x naught, y naught, z naught plus ta, tb, and tc. Don't be scared to do [inaudible] on vectors. They're just pairs of numbers, triples of numbers. Now, I take my two vectors that are being added and I add them to form a single vector. It's x naught plus ta, y naught plus tb, and where you add component-wise, z naught plus tc. It's a single vector. The thing that I want you to realize is how do you know when you're looking at the equation of a line and maybe not some other spiral or some weird shape in space. Each equation in the equation of a line, it may not be obvious from the P plus tv, but if you work it all out and put numbers in there, each one is a linear equation. Each one is like mx plus p, mx plus p, mx plus p. When you're working with a line in space, you basically have three linear equations, and it's all captured inside of one vector. This form because it has a parameter, this is called the parametric equation of the line. To get different points on the line, you have some line in space. If I want different points, every point corresponds to the number t. It's like a copy of the real number line just moved in whatever direction you want. When t equals 0 in particular, so for example, if you plug in 0, get p plus 0 times v, well, 0 times any vector, of course, is just p. You get the particular point p that you're given you're starting with. This is at t equals 0. This is where the artist is putting the pen on the paper and about to draw the line. As t moves forward, maybe if t equals 1, you get another point on here to use 2, to use negative 1. They all correspond to different values of t. T is the parameter and the parameterization of this line. Another way to describe lines is through something that's called the symmetric equation of a line. Let's start with our general line again. I'm going to mix up the notation just because so you don't get comfortable with it. But Rt. I'm using R now instead of L for line. Who cares? I'm going to give you some three linear equations, so x plus ta, y plus tb, and z naught plus tc. Remember, you can read off the point that it goes through by just plugging in t0, so it will be x naught, y naught, z naught, and then your vector a, your b, and your c. That's the direction vector. Oftentimes, what you can do is you can say, "Well, my x coordinate of any point in the line is going to be the first component, the y coordinate as well and the z coordinate as well." We can make the coordinates general for some specific moment in time. Let's set up these equations. You have x equals x naught plus ta, and you have y equals y naught plus tb, and then z is some fixed z naught plus tc. The next step you can do, may not be obvious why you want to do that, but you can certainly solve for t in these equations. If we rearrange some things, so subtract over the x naught, you get x minus x naught over a. You get y minus y naught over b, if is equal to t again, and then on the last side, t is equal to z minus z naught over c. This is called eliminating the parameter if you solve for t. One thing you realize is that, because there's a t in each one of these, it's a common term. You can express everything in terms of the parameter, and then because they're all equal to t, you can set them all equal to each other. This is another way to describe the properties of a line. All the information you need is inside of these equations. It's not as obvious to get them, or perhaps not as obvious that they're linear equations, but they're perfectly fine. If you need the first point of intersection, there's x naught, y naught, and z naught. If you need the direction vector, look at the denominators, a, b, and c. This is nice. Of course this only works when a, b, and c don't have any zero components inside of them, but this is another way to present the equation of a line. It's not as common, but it's certainly a perfectly fine way. These are called symmetric equation of a line. This is exact. Given this, you can go back, reverse the process, so it all equal to t, solve for x and put it back. It's one inside the same. When you have these lines, you can describe them in any way you want. I guess anything like radians and degrees, when there's more than one way to describe something, you want to be fluent and comfortable in both. There's one more formula that I want you to know as well. I probably should have a separate slide for this, but it's for line segments instead of lines. If you're given, somewhere in space, two points, there's a perfectly fine point P and there is another fine point Q, so you're given points P and Q, and you want just the segment between them, maybe you're studying something from part a to point b, then that is where we call that a line segment of course, instead of a line, you can still do everything the same. Because that segment is absolutely part of a much larger line. How do you restrict to just that piece? Well, you can follow the same process. I have a point P, I want to head over to Q, so let's get the direction vector v. Do you remember how to find the direction vector given two points? Remember head minus tail? This is like Q minus P. That's your direction vector. Then you can plug in. Again, if you have numbers, you can absolutely work out what this is, is called as ABC, and you still have your line, r of t is P plus tv. This one starts at P and heads over to Q. If for whatever reason you started at Q and went back to P, you just switch the variables. But this is another little equation, just another way to do it. You're restricting things. While this is perfectly fine, I want you to think of another parameterization. This is the one that we're going to use more often, if we do more in this multi-variable setting. For a line segment, I want you to think of the formula, 1 minus t P, plus tQ, and we're going to set 0 to t to 1. Stare at this for a second. Let's talk about this formula. Put this one down. This is the line segment that moves from P to Q. Why is this perfectly good as well? What happens if t is zero? Plug in zero, you get 1 minus 0, that's 1 times P, that's just P, plus 0 times Q, so you just get P. You start off at t is zero. When t is one, you get 1 minus 1, which is 0, the P goes away and your 1Q, 1 times Q is Q. This is another one as well. Just as all these little formulas that pop up here, keep this in mind for line segments. Because we have more room and space, some new things happen. Let me start this section off with a true or false question. Lines that do not touch must be parallel. True or false thing about this, lines that do not touch must be parallel. It's a little tricky question because it depends. If you're in R_2, lines that don't touch, the only way to do this, to have two lines that don't touch, is they must be parallel. In R_2, the answer is true. But if we're in R_3, remember R_2 is the x, y plane, R_3 is space, three copies of the real lines, x, y, and z, you can actually have non-parallel lines that don't touch. I can have some line maybe doing something up here and another one down here. They're certainly not parallel, but if you extend the line, they just never touch. Like two asteroids in space that are just flying past each other, they're not moving in the same direction, but they'll never crash. If you want to see this in the room you're looking at, look at maybe the line that connects the ceiling and the wall, and perhaps a line that connects an adjacent wall and the floor. They're just never going to touch. It is false inside of R_3. They don't have to be parallel and they also don't have to touch. When this happens, when you have non-parallel lines that don't touch, we define them to be skew lines. This is only for not inside the xy-plane. Skew lines are defined to be lines that do not touch, do not meet, and are not parallel. Not parallel just means not in the same plane. You can not talk about R_3 without talking about the difference between the xy-plane and R_3. Skew lines, just keep in mind that they're out there. Let's do some examples just to practice our equations. Have your equations ready as we go through this. Find the parametric equation, this is the one that's like, lt is P plus tv, of the line through a point 1, 0, negative 3, they're not labeling it, so I will, P, and parallel to 2i minus 4j plus 5k. Remember i, j and k, they are not variables, they're actually given vectors. I, j, and k, it's pretty common to see a question in side of i, j, and k. If you study physics, you might be more comfortable with it than I am. My physics background is not very strong. I like to get it out of i, j, k notation, i, j, k kind of guy, and I usually write it with angle brackets immediately, just my preferred way to think about it, and I see the vector this way. Remember i just means the first component, j is second component and k is third component. This is my direction vector. Part of this is just reading off the pieces. Once I have my P and my v, well then I'm basically done, it's just my job to put it all together in the formula. It's a P plus tv, put as one like tv, then we have 1, 0, minus 3, plus t times 2, minus 4, 5. Put it all together, we'll do in one shot. You get 1 plus 2t. I bring the t in and I add it to the first component. I'll do the same thing here. It's 0 plus t times negative 4, that's just minus 4t, then minus 3 plus 5t. Notice I have three linear equations that aren't even squared or anything like that, three nice equations of the line. That's my final answer. It's a little weird to realize that this is what the answer is. Just like before, if I asked you for equation of a line and you gave me mx plus b, with whatever the numbers are, same in here, only of course you're adding three of them. That should make sense because we're in 3D. Let's do another one. Find the parametric equation of the line through the origin and parallel to line x equals 2t, y equals 1 minus t, and z equals 4 plus 3t. Now we have a little more work to do. First of all, let's read this carefully. Find the parametric equation of a line. They're asking for the same formula again, so what do I need? I need to find some point, and then I need some direction vector through the origin. Through the origin, that's a fancy way to say that p is going to be 0, 0, 0, so look for that. So somehow some way this is going to be my direction vectors in here. Remember what this is. They're handing me some other line and I want parallel. I want to read off the direction vector from this. The direction vector in any equation of a line comes from the coefficients on the variable t. So I have a two for my x-coordinate, a minus one for my y and a three for the other one. You can just read these off 2, minus 1, and 3 and I get my v that way. Now I have my p, now I have my v put it all together. This one's actually not too bad. The hardest part of course is passing the information. Basically I have 0, 0, 0, I guess I really didn't have to write that, but I'll be a good person and do it anyway, 2, minus 1, 3, close it up, finish it all up. Basically there's not much more to say here then 2t, minus t, 3t, perfectly good equation of a line. How do I know the equation of a line? Three linear equations, each individual one is a line, put them all together, I get a line and space. Let's do another one. Find the parametric equation for a line through 2, 1, 0 and perpendicular to both i plus j and j plus k. Now this is getting interesting. J and K are my standard vectors. I'm not an i, j, k, kind of guy. I like to write them out using my angle brackets. Remember what i plus j is, the coefficients here, just one and one. There's no k so it's zero. This is basically 1, 1, 0 and j and k is, well, there's no i, so it's 0i, plus, 1j plus, 1k. So this question could easily have been replaced with 1, 1, 0 and 0, 1, 1, but they wrote it as an i, j, k notation. Now, they gave me the point P. The point P is 2, 1, 0. How do I find something that's perpendicular to two vectors. This part of the information is giving me my vector v. Do remember this? If I give you two vectors, how do we find a vector that is perpendicular to it? Let me draw a picture here to help steer your memory. If I have some vector and another vector, how do I find one that's perpendicular or perhaps better said, orthogonal to both? If I have like A and B, well, remember this, take the cross product, so we do need a cross product here, no big deal. Let's call the first one A and the second one B for no good reason. Now we'll call A cross B. Here it goes. Let's set up our little diagram, i, j, k, 1, 1, 0, 0, 1, 1. It doesn't matter the order because they don't specify anything other than perpendicular. If you happen to choose like, reverse the order, it's perfectly fine. Just get the vector go the other way and that satisfies the condition. So there's more than one answer here. Let's compute the cross product. Remember how this works. We take i and then we imagine that the row and column of i is not there. We look at the four numbers that remain, and we do our little x pattern and then multiply and subtract. We do 1, minus 0, and then we repeat as we go. Of course, the only thing to watch out for is that minus j, so we do a minus j, same thing, imagine that j is not there. We leave four numbers that remain, do our x pattern. We do 1 minus 0 again, and then plus k. You got to imagine that the k column is not there and we do our x pattern and we get 1, minus 0. So not too bad. This turns out to be just, well, i times 1, minus J times 1, plus k times 1. Take it out of i, j, k notation and you get 1, negative 1. It's not a bad idea right here to check that something is in fact parallel to both these vectors. Remember perpendicular. The way to check that is to do a very quick dot product. You can check by taking the dot product. What is 1, 1, 0? The vector that we started with take a dot product with the answer. Just a bunch of arithmetic, some negative signs float around. Is it in fact zero? How do you do a dot product? You multiply the components, add as you go. So multiplied together, you get 1 plus 1 times minus 1. So absolutely this is just 1 minus 1. Zero term just adds a zero, the other vector is 0, 1, 1. For all the same reasons, you're also going to get a yes as well. I like checking my answer. You get 1 plus minus 1 plus 1, that's 0. So yes to both, just make sure you didn't mess anything up and we have our vector. So this will be our vector v. Obviously, all the hard work comes in and getting the vector v. Then once we have that, we just put it all together for our Formula. Our line with respect to t, which is p plus tv, then becomes 2, 1, 0 plus t 1, negative 1, 1. You could leave your answer like that or you can clean it up. I like to clean it up so it's 2 plus t, 1 minus t, and then 0 plus 1t is just t and there we go, that's our final answer. Get comfortable handing that back in terms of a line. Cross product is coming to play here to find equations of lines. Great job putting everything together here and remind yourself keep these formulas handy for dot products, cross products, and all the things that go with it because we're just going to keep using these as we develop this theory. Great job on this video. I'll see you next time.
