 In this video, we're going to talk about how we can find vector equations and parametric equations to represent certain types of flats in a vector space like lines and planes and the such. So for example, consider we have a line in R3 that contains the point 1, 2, 3, but it's parallel to the vector 5, negative 3, 1. Now in this context, when we talk about parallel, this is describing the direction of the line. If the line runs parallel to a vector, and we think of this vector as space, this is telling in space here, this is telling us that the vector here is our spanner. So when you see the word parallel, that's just telling you that the flat, when it's parallel to these vectors, the spanner vectors, because they're the spanners. And so therefore, constructing the vector form of the equation of this line is fairly straightforward. X is going to equal X naught plus T times V right here, where X is some generic vector X1, X2, X3. X naught is a specific vector 1, 2, 3, that's on the line. T is some unspecified parameter here, and then the directional vector, the slope vector, our spanner will be 5, negative 3, 1. So this is our vector equation. But if we go one more step here, we essentially have our parametric equations here. We have that the first equation X1 is equal to 1 plus 5T. You'll notice if you look at just this equation right now, this is a linear equation with respect to the variables X1 and T. The other coordinate X2 is given as 2 minus 3T. And then lastly, X3 is given as 3 plus T. And so this gives us the parametric equations of our line right here. And so that's all there really is to it. If I tell you the point on the line and I tell you the spanner, that's all there's going to be. We can come up with the vector equation, we can come up with the parametric equations like so. Let's look at another example of such a thing. This time, let's look for a line in 4 space, R4. How does it change when the picture itself is four-dimensional? Well, it really doesn't change that much at all, really. It's going to pass through the origin. This tells us that our X naught, we can actually take as the zero vector, right? The origin is just a zero vector, which would be 0, 0, 0, 0 in 4 space. And then again, this is parallel to the vector 4, negative 3, 2, negative 1. Oftentimes, you're going to see when we talk about this coordinate geometry here that the points are often written not as vectors, but as points. But really, in the vector space, R4, we don't distinguish between vectors and points. So the exact same thing. So therefore, our vector equation is going to look like any vector equation of a line. X equals X naught plus Tv. It always looks like that. Specifically, the vector X will look like, well, you could call it X1, X2, X3, X4. If you wanted to, you could call it X, Y, Z, W. But it's just a generic point in 4 space. Because the point is the origin itself, you can actually remove it from consideration adding to it. It's not going to do much. And then you get T times our spanner, which is 4, negative 3, 2, and negative 1. If we write this as parametric equations, you're going to get that X1 equals 4T. You get that X2 equals negative 3T. You're going to see that X3 equals 2T. And then finally, X4 equals negative T. You can see that lines that pass to the origin are a lot easier to describe than an arbitrary line. But as a third example here, this time, let's look for a plane that's passing through 4-dimensional space here. It gives us, well, if we're looking for a plane, the equation will look something like a general vector on the plane will be a specific particular vector on the plane. Plus a linear combination of two independent spanners. Since it's a plane, a two-flat, we're going to need two vectors that will then span the plane right here. And so they give us the vector X0. So X would look like X1, X2, X3, X4 because we're in 4-space. X0 is given by the vector 26, 3, negative 13, negative 18. And then we take the span of our two spanners because our plane is parallel to them. S of U, which is 1, negative 3, negative 2, negative 1, and then plus T of V there, which is 0010. And then from here, our parametric equations are essentially immediate. We get that X1 equals 26 plus S plus 0T, which is just, it just disappears. X2, which equals 3 minus 3S plus, again, 0T. We get X3, which equals negative 13 minus 2S plus 1T this time, just a T. And then we get X4, which equals negative 18 minus 7 plus 0T again. And so this is where we can describe these flats.