Intersecting Planes: (110) & (101) - Direction?

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In summary, the materials science student needed to find the direction of the intersection between two planes by taking the cross product of the normal vectors to the planes.
  • #1
JSBeckton
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Have a question from materials science class:

Using appropriate sketches determine the direction of the intersection of the following planes: (110) and (101)

do I need to actually draw out these planes, then find 2 vectors for each. take the cross product, then use the lines perpindicular to the planes to find th edirection? Or can i simply use the planes to get the cross product?


[tex]
\begin{array}{*{20}c}
i & j & k \\
1 & 1 & 0 \\
1 & 0 & 1 \\
\end{array}
[/tex]
 
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  • #2
You're going to have to let us know what your notation means. There are many different ways to specify a plane. What exactly do those numbers represent?
 
  • #3
this is the standard materials science notation for a lattice plane. You take the inverse of the individual coordinates and that gives you the x, y and z intercepts.
 
  • #4
Well, if I understand your notation correctly then the planes are very simple and just drawing a picture will do it without calculation. You need to find 2 points that are on both planes, and one of them is a given: (1, 0, 0).
 
  • #5
I'm pretty sure that they want us to use the cross-product.
 
  • #6
Then why would they ask you to draw a picture? The second point is obvious from the picture.
 
  • #7
The notation (ABC) refers to the plane with equation Ax+ By+ Cz= 1 so that (1/A, 0, 0), (0, 1/B, 0), (0, 0, 1/C) are the x,y,z intercepts.

Yes, Ai+ Bj+ Ck is perpendicular to that plane. Also Xi+ Yj+ Zk is perpendicular to the plane Xx+ Yy+ Zz= 1 and so their cross product it perpendicular to both and points in the direction of their line of intersection. You still need to find a point on that line.
 
  • #8
we are using miller indicies, I'm pretty sure that taking the cross product is right
 
  • #9
Yes, that's what I just said! Have you solved the problem yet?
 
  • #10
I took the crossproduct of the two planes and believe that the vector resulting from the cross product is the direction of the intersection, right?
 
  • #11
JSBeckton said:
I took the crossproduct of the two planes and believe that the vector resulting from the cross product is the direction of the intersection, right?
Yes! You've been told that three times now. Have you solved the problem yet?

By the way- please don't say "the crossproduct of the two planes". You don't take the crossproduct of planes- crossproduct is only defined for vectors. I know you meant "the crossproduct of the normal vectors to the two planes" but I would hope that materials science values precise statement as much as mathematics.
 
  • #12
For the pairs of planes: (110) and (101)

I crossed the normal vectors of the two planes and got [tex]
\left[ {1\overline 1 \overline 1 } \right]
[/tex]

I am sorry for the confusion but I did not understand why you said that I still needed to find a point? Thats why I asked if my method was correct again, because I did not find any point.
 
  • #13
Oh those material scientists! Why can't they use the same notation as normal people?

You have determined (I think!) that a vector perpendicular to both planes is i- j- k (those overlines are negatives? WHY??)

You now know a vector pointing in the direction of the line of intersection. But there are an infinite number of (parallel) lines pointing in that direction. You must know a point on the line to choose between them.

Your two planes are x+ y= 1 and x+ z= 1. Any point on the intersection line must satisfy both of those. But there are an infinite number of points on the line and you only need ONE. Let x= 0 (just because it is easy). Then y= 1 and z= 1. The point (0, 1, 1) satisfies both equations so it lies on both planes. It must, then, be on the line of intersection.

Now what are parametric equations for the line through (0, 1, 1) in the direction of the vector i- j- k?
 
  • #14
Well, he only was asking for the direction of the line of intersection so I don't think he actually needs the line.
 
  • #15
Yes, just the direction. By the way I agree with you about the notation, I am Mech Eng major, so hopefully I won't see this a whole lot after this class.
 

1. How do you determine the angle between two intersecting planes?

The angle between two intersecting planes can be determined by finding the angle between their corresponding normal vectors. This can be done using the dot product formula: cosθ = (a1a2 + b1b2 + c1c2) / (|a1b1c1| * |a2b2c2|), where a1, b1, c1 and a2, b2, c2 are the direction vectors of the two planes.

2. What is the direction of the intersection line between two planes?

The direction of the intersection line between two planes is perpendicular to both planes and can be determined by taking the cross product of their normal vectors. The result is a vector that is parallel to the intersection line.

3. Can two planes with different orientations intersect?

Yes, two planes with different orientations can intersect as long as they are not parallel. This means that their normal vectors cannot be parallel to each other.

4. How many intersection lines can there be between two planes?

There can be either one or an infinite number of intersection lines between two planes. If the planes are parallel, there will be no intersection line. If the planes are not parallel, there will be one intersection line. If the planes are coincident, meaning they are the same plane, there will be an infinite number of intersection lines.

5. How does the direction of the intersection line change if one of the planes is rotated?

The direction of the intersection line between two planes will change if one of the planes is rotated. This is because the normal vector of the rotated plane will also change, resulting in a different perpendicular direction. The angle between the two planes will also change, which will affect the angle of the intersection line.

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