# Perpendicular distance between planes

1. Sep 16, 2012

### unscientific

1. The problem statement, all variables and given/known data

The question is as attached in the picture.

3. The attempt at a solution

I understand and got the answer to part (a) but from (b) onwards I am completely lost. Here are my assumptions:

1)Edges of 'cube' are still of length a

2)Successive planes are parallel

But, there is some things remain unknown:

1)Where is the origin at? The centre of the cube? or at the bottom leftmost point?

2)Why does the plane pass through (a/2, 0, a/2)?

3) Do we assume that the plane contains the origin? Then won't the equation be

r . n = 0?

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Last edited: Sep 16, 2012
2. Sep 18, 2012

### unscientific

would appreciate any help..

3. Sep 18, 2012

### voko

The description states, the center of the cube.

There is an atom at (a/2, 0, 0) through which a plane has to pass; so it passes through (a/2, 0, z) for any z because it is parallel to the z axis. I am not sure why they did not specify the obvious point, though.

It is another plane that satisfies the equation, but not for a positive p. They actually use this plane implicitly when they claim that the p found is the distance between the planes. Can you see how?

4. Sep 21, 2012

### unscientific

Based on the face-centred structure, there is no atom at (a/2, 0, a/2) as that 'centre-face' atom only exists on alternate layers...

So what I would do is:

a. Taking the plane 1 to be the plane containing origin,

b. Find equation of plane 3 (that contains the atom at (a/2, 0, a) ).

c. Find distance between plane 3 and plane 1 then divide by 2.

This gives me a distance of a/sqrt(32), which is half of what the answer has..

(this is assuming the vertical height between layers is always a/2)

This whole question can be resolved if it said something like "All layers of the crystal are arranged in the same manner". Then it will no longer be alternate layers..

Last edited: Sep 21, 2012
5. Sep 21, 2012

### voko

I don't see how you get this. $$\frac 1 {\sqrt{2}} (1, 1, 0) (\frac a 2, 0, a) = \frac 1 {\sqrt{2}} (1 \cdot \frac a 2 + 1 \cdot 0 + 0 \cdot a) = \frac 1 {\sqrt{2}} \frac a 2 = \frac a {\sqrt{8}}$$

6. Sep 21, 2012

### unscientific

This is the perpendicular distance between the 1st and 3rd layer....So We must divide by 2

7. Sep 21, 2012

### voko

Why is it 3rd? Where is the second?

8. Sep 21, 2012

### unscientific

Because looking at the cubic structure, the atom that is in the centre of a face occurs only at every alternate layer..

UNLESS

every layer is the same..

9. Sep 21, 2012

### voko

I do not think we are getting anywhere. The first plane with the given normal vector passes through the atom at (0, 0, a/2), and that plane passes through the origin. There is a plane that passes through (a/2, 0, 0), which also passes through (a/2, 0, a). You say this the third plane. Where is the second plane? What atom does it pass through?

10. Sep 21, 2012

### unscientific

No, no what I meant was that:

First plane: passes through (0,0,0) and (a/2, 0, 0)
Second plane: the 'usual' base/top of a cube plane
Third plane: the plane that passes through the atom on every alternate level that is centred on a face (a/2, 0, a)

Assumptions: vertical distance between planes is a/2.

11. Sep 21, 2012

### voko

These planes do not have (1, 1, 0) as their normal vector.

12. Sep 21, 2012

### unscientific

Hmm? I thought they are parallel with one another? Then the normal vector should be the same? (Or does the normal vector depend on perpendicular distance to origin? )

Last edited: Sep 21, 2012
13. Sep 21, 2012

### voko

I fail to see how. The plane passing through the origin with the normal (1, 1, 0) cannot possibly pass through (a/2, 0, 0). This is seen from its equation: (1, 1, 0)(0, 0, 0) = 0, and (1, 1, 0)(a/2, 0, 0) = a/2 ≠ 0.

14. Sep 21, 2012

### unscientific

That is true...But doesnt the first plane 'cut' the cube into half and by right should contain the atom that is the centre of a face?

15. Sep 21, 2012

### voko

The first plane cuts the cube diagonally. Its projection onto the XY plane is a straight line passing through the upper left corner to the bottom right corner. It does pass though an atom at the middle of the face, but that is (0, 0, a/2), (0, 0, -a/2), etc.

16. Sep 21, 2012

### unscientific

Is this what you mean?

17. Sep 21, 2012

### voko

Exactly.

18. Sep 21, 2012

### voko

Now if you add an atom in the center of the "frontal" or the "right" face, and draw a plane passing through it parallel to the plane in blue, you will get the plane they considered in the solution.

19. Sep 21, 2012

### unscientific

I see!

So, it must pass through the point (a/2,a/2,0) that is directly below the red point as in the answer!

I think the question assumes that the vertical separation is a/2..

20. Sep 21, 2012

### unscientific

But im not sure how they got the answer to the last part with normal (1,1,1)..