We are interested then, in making a distinction between those two.

So we will be interested in describing those directions and

coming up with a consistent way of doing that.

And what we're going to do then is to look at a unit cube.

And what I have described up here is a simple unit cube in which I've labeled O,

that represents the origin, and the other corners, A, B, C, D, F and G.

And what we'll be able to do then is to describe

The co-ordinates of each one of those points because in order for

us to be able to get directions we need to describe the co-ordinates

that tell us something about the distances between the two points of interest.

So first let's look at the position that we're referring to as the origin.

The way we describe this in crystallographic terms is

that the origin is given as 0,0,0.

So we're looking at the origin, and

we're positioning at that origin at that position given as 0.

Now when we come over here and we look at position A.

Position A is along the X axis and we will call that position since it is

at the boundary of the unit cell, it will be the position one comma zero comma zero.

Now we turn our attention to B and what we now have is the position

which it will be described as the 0, 1, 0.

Again, these integers are delineated by commas.

And up here we look at C.

That´s along the Z axis and that position will be 0,0,1.

And we look at D and that position is 1,1,1.

And we look at E, which is on the XY plane.

So that will be 1,1,0.

And we look at F.

That's on that back corner, and what we see here is it is in the plane of Z and Y.

And what we will see is that that is going to be 0,1,1.

And up here we look at G and

we'll see that that's going to be at the position 1,0,1.

The last position, which is indicated as a face position,

position H, what we see is we are coming out in the direction of X and Y.

But in the case of X, we're coming out at a distance of one

half along the x, one alone the Y, and one half along the Z.

And as a result,

what we will see is the positions that are idicated over here on the right.

So the last position of H is 1/2, 1, 1/2.

And remember these are integers, or they're fractions of an integer, and

that puts them inside of the unit cell of interest.

Now that we've come up with a way of identifying the positions of interest.

If, for example, we're interested in talking about the vector that is

described by the position AO or the vector AO.

We can do that by recognizing.

First, we get the two points of interest.

We have the 0,0,0, and the 1,0,0.

And what we're going to do is to note those two positions, and

then what we do is we subtract the tail of

the arrow from the tip, and we clear any fractions.