ABSTRACT
If the positions of atoms in 3-D space are given by (x, y, z) coordinates, we can use the standard Pythagorean calculation of distance. If atom A has coordinates a = ( , , )a a ax y z T and atom B has coordinates b = ( , , )b b bx y z T, the distance between A and B is given by
d a b a b a bx x y y z z( , ) ( ) ( ) ( ) .A B = − + − + −2 2 2 (7.1)
is is the same as the norm calculation:
|| || ( ) ( ).a b a b a b− = − − T
(7.2)
7.3.2 Bond Angle
Recall from linear algebra that an inner product of normalized vectors u and v can be viewed as the cosine of the angle between these vectors. is is expressed using the following formula:
cos , (|| |||| ||).θ = 〈 〉u v u v (7.3)
Consider two atoms A and B (with coordinate vectors a and b), both bonded to a third atom C with coordinates given by vector c as in Figure 7.1.
