Question

The three following coordinate vectors are given in unitary coordinates (in [m]): a = (5; 0;...

The three following coordinate vectors are given in unitary coordinates (in [m]):
a = (5; 0; 0)
b = (-1; 4; 1)
c = (0; 1; 3)

a) Determine the new coordinate system, giving |a|, |b|, |c|, alpha, beta and gamma. Use vector operations to obtain the values.

b) Determine the metric matrix for this coordinate system, and the volume of the parallelepiped spanned by a, b, c. For the volume calculation, use the determinant of the metric matrix.

c) Calculate the distance between two end points of the vectors given in relative coordinates (u; v; w) (1/2, 1/2, 0) and (1/2, 1/4, 1/4), and determine the angle between them! Use the metric matrix for this calculation, and compare the results to the method using the unitary coordinate system.

d) Determine the associated reciprocal vectors a*, b*, c* and the reciprocal angles alpha*, beta*, gamma*. Use the vector relationships to calculate the vectors in the reciprocal unitary system. Determine the reciprocal coordinate system by inversion of the metric matrix of the direct lattice.

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