Now we've talked about position inside is going to be fractions or integers along the x, y, and z direction. We're doing in the form a, b, c, that corresponds to x, y, z. That'd be relative to the origin. We always want to define the origin. There is our origin. Then we'd go find now point. So I'm going to go 1/2 along the x, 1/2 along the y, and 0 displacement along the z. That marks the spot. Then that's going to tell me that is the point 1/2, 1/2, 0. Now I can do that to each lattice point. We'll just grab one. Let's take this guy. I'm going to go out one along the x, one along the y, and one along the z. That gives me the 1, 1, 1. Pretty straightforward. For negative, we are going to put it, indices don't necessarily have to, but if you want to keep it inside the unit cell, you're going to reduce it down. You have to be mindful. The question can ask, just give you the vector. That's fine. Give it to him. But if it says in the direction in the unit cell, you have to make sure you give fractions or maximum integer of one. Negative indices are represented by bar. The bar resides on top of the scalar value. Now let's take this particular case. I'm going to move my origin here. To get to that back point, I'm going to go minus 1 in the x plus y plus z. That's going to be bar 1, 1, 1, 1. Now I have to put square brackets to say that's a direction. That gives me the bar 1, 1, 1 direction. The origin, I can move. In this particular case, the origin was here. Remember, I go one in the negative x, positive y, positive z. We'll do one. This is one of the activities that you'll be doing in recitation. We'll do an example. Let's take this guy. Here's my origin. I'm going to go 1/2. No displacement in the x, 1/2 in the y, and 1 in the z. This displacement, 0 displacement in the x, positive 1/2, and positive 1 in the z. Again, define the origin if you have to draw the vector or if the vector is given, or the directions given. If it's given to you, you have to define the indices. If you're given the indices, you have to draw the vector. We'll do this one. You had to draw the 1, 0, 0. I start with the origin. That's going to give me a displacement of 1 along the z. One displacement in the x of one, no displacement in the y, no displacement in the z, and that's going to give me the direction 1, 0, 0. Let's take a moment for inquiry. Here we want to give the indices for the various directions. Let's start off with the black here. But we're going to define our origin here. We're going to go 1 in the x, 0 displacement in the y, and minus 1 in the z. Hopefully that is going to be this particular case. We're going to discard these guys. Black is going to be 1, 0, bar 1. That guy is correct. Now, the purple one. We're going to do red next. Red, I'm going to put my origin here. I'm going to go one in the x, one in the y, and 0 in the z. That works out. Green, I put my origin here. I'm going to go negative 1 in the x in part, no displacement in the y, and a positive 1 in the z. That gives me negative 1, 0, 1. Checked out. Then a little purple royal goal. I got to do it from down here. There's my origin. I'm going to go minus 1 in the x, minus 1 in the y, and I'm going to go minus 1 in the z. I'm going to be left with bar 1, bar 1, bar 1. It does check out. Remember, define your origin. That best helps you to draw the vector inside the unit cell. In this nose here, I move my origin up here because I knew I had to go a negative x, I had to go a negative y, I got to go a negative z. I'm going to bring it all, put my origin here so I can make those translations inside the unit cell. Again, let's do that one again. Here's my direction. I move my origin here. That allows me to make that translation of minus 1x, minus y, and minus z inside the unit cell. I cannot stress enough to practice. Practice drawing in the unit cell. Practice drawing outside of the unit cell, if need be. Also just think about moving that origin. Well, again, now I hope we have some idea of how to define directions inside the unit cell. How to identify points inside the unit cell. When we talk about a direction, we use the square bracket. Thank you.
