Rigid transformations of parallel lines

0:00- [Instructor] The goal of this video is to convince you that if I were to take parallel lines like these two lines here, line l and line s, and I were to conduct a rigid transformation the same rigid transformation, on both of them, that the resulting lines are still going to be parallel.

0:19So let's just remind ourselves what parallel lines are. So these two parallel lines, they're in the plane, and they're on the screen. If you're watching, well, I'm assuming it's on some kind of a screen right now. And the fact that they're parallel, they are never going to intersect.

0:33Another way to think about it is they have the exact same slope. There's no reason why these things will ever going to actually touch each other, or another way to think about it, they go in the exact same direction. Now, let's try a very basic rigid transformation.

0:49Let's do a translation of them. So I'll just copy and paste these parallel lines. So let's just actually start where we were. And if we were to translate it, you could just show up right over there. But notice, no matter where I move this thing, they are still going to be parallel.

1:04That's because relative to each other, and actually, when we're talking about a translation, they're still actually parallel to the original lines as well, but I'll call this l prime and s prime right over here, not only are they still parallel to each other, but as I said, they're still pointing in the same direction.

1:22So they're parallel to the original l and s as well. That's not gonna be true of all the rigid transformations, but what we're gonna focus on is, are they going to be parallel relative to each other? And that is true for all rigid transformations. So when we did a translation, when we did a translation, relative to each other, they are still parallel.

1:43They are still not going to intersect. Now, let's say we wanted to do a rotation. So if I just wanted to rotate it in place, like this, I could rotate it, and notice they are still parallel. But let me actually do a rotation and a translation here. So they are still.

2:04I'll put it right over here, and maybe I will call this, and I already used the prime, so maybe I'll use prime prime or something like that, just to give you a sense that these are different. But these are still parallel to each other. They're no longer parallel to the original ones.

2:19So l prime prime isn't parallel to l, but they are still parallel to each other. They still do not intersect. So this right over here, this is a rotation. It's arguably a rotation plus a translation. But even if you do a rotation and a translation, you are still going to be parallel.

2:38Now the last one I will think about is a reflection, and this one, I'm just going to try to do by hand. But let's say we wanted to reflect l and s across, I don't know, let's say we wanted to reflect it across this right over here. So lemme just do a dotted line.

2:56So if I did that, let's see, I would be about that far. s would look something like this. s, let's call that s prime, and l would look something like this. That would be l prime. And they still would be parallel. They would not intersect. So hopefully, this gives you at least an intuitive sense that no matter what you do, a translation, a rotation, or a reflection, all of these are rigid transformations.

3:26They preserve how these lines relate to each other, or the structure between them is one way to think about it, or the distance between them is not going to change as you do these rigid transformations.