Linear algebra

  • Thread starter surajalok
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  • #1
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Homework Statement


Unit cell of a crystal lattice is a parallelepiped spanned by the vectors u, v, w vectors have lengths of 1, 2 resp3 (le). angles between the vectors is
[u, v] = pi / 4
[u, w] = pi / 3
[v, w] = pi / 6
Determine the volume of the unit cell.



Homework Equations


How do you solve this problem?



The Attempt at a Solution


volume = V (w, u, v) = w scalar with (u x v)
I can determine | uxv | = sqrt (2)
if I can find out the angle between
w and uxv then the problem is solved but it is not possible to get this angle.
 

Answers and Replies

  • #2
lanedance
Homework Helper
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may not be elegant, but how about noticing u is perpindicular to v, then if
|u| = 1
|v| = 2
|w| = 3

then choose an orthonormal basis such that
u = (1,0,0)
v = (0,2,0)

not let w = (a,b,c) with a^2 + b^2 + c^2 = 3^2, now you know
u.w = |u||w|cos(pi/3)
v.w = |v||w|cos(pi/6)
 
  • #3
19
0
may not be elegant, but how about noticing u is perpindicular to v, then if
|u| = 1
|v| = 2
|w| = 3

then choose an orthonormal basis such that
u = (1,0,0)
v = (0,2,0)

not let w = (a,b,c) with a^2 + b^2 + c^2 = 3^2, now you know
u.w = |u||w|cos(pi/3)
v.w = |v||w|cos(pi/6)

[u, v] = pi / 4
[u, v] is not pi/2

i said someting in the book about det(A^T A)
 
  • #4
lanedance
Homework Helper
3,304
2
fair bump, but you could do the same thing with
[tex] u = (1,0,0) [/tex]
[tex] v = (1,1,0) [/tex]
 
  • #5
lanedance
Homework Helper
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2
in fact i think it follow pretty quickly form there...
 
  • #6
19
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i don't know what i should do now?
we don't know anything about w.
I the book it said that examine det(A^T A)
colonnvektors in A are u,v,w.
 
  • #7
19
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anyone???
 

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