Finding Components of Vectors in a Crystal Lattice

[Eduard1]

Homework Statement

Be the vectors a, b, c such as:

| a | = | b | = | c | = 10.5 Angstron

The angles between these vectors are:

alpha = beta = gamma = 109.5 degree

These vectors represent the lattice vectors of a crystal.

Find out their components (a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3).

Homework Equations

The Attempt at a Solution

So, we know the length of the vectors and the angle between them. We want to find out the components of each vector. I know one has to use the trigonometric functions, but I am not sure how. Should I project the vector a on b (to find a_2) and on c (to find a_3) ? Or how do I find the components of a ?

Thank you very much for any fast answer/hint/suggestion,
Eduard

 

[obafgkmrns]

Looks like one of the vectors can be arbitrary, so let a = {10.5, 0.0, 0.0}. Vector b can be assumed to lie in the 1-2 plane without loss of generality, so let b = {10.5 cos(109.5), 10.5 sin(109.5), 0.0}.

You can find vector c using spherical trigonometry. The most expedient way appears to be to solve the spherical right triangle consisting of the head of vector a, the head of vector c, and a point half-way between the heads of vectors a and b. Hope that helps!