Finding the Volume of a Unit Cell

[surajalok]

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.

 

[lanedance]

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)

 

[surajalok]

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 something in the book about det(A^T A)

 

[lanedance]

fair bump, but you could do the same thing with
$$u = (1,0,0)$$
$$v = (1,1,0)$$

 

[lanedance]

in fact i think it follow pretty quickly form there...

 

[surajalok]

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.

 

[surajalok]

anyone?