Analytical Geometry with Oblique Bases

The coordinate systems chosen in crystallography are generally defined by three nonorthogonal base vectors a,b,c of different lengths (a,6,c). These non-unitary systems introduce some complexity into the expressions used in analytical geometry, 

CONVENTION. A right-handed coordinate system is chosen Le. a, b and c are taken in the order of the thumb, index and middle finger of the right hand a is the angle between b and c, p is the angle between a and c, y is the angle between 

The normal to the plane hu- -kv- -lw = lis oriented from the origin towards the plane and may be calculated from the vector product (Fig, 1.2)  

Consider the pyramid whose vertices are the origin 0 and the intersections A, B, C of the plane with the axes. Its volume V is equal to one third of the area of the triangle formed by three vertices multiplied by the distance of the triangle from the fourth vertex. Let d be the distance of the plane from the origin. We thus obtain  

By labeling the base vectors a, a2 and a3, and the corresponding reciprocal vectors ajjaf and a, these relationships may be summarized by the following expression which defines the reciprocal vectors in terms of the base vectors and vice versa  

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