By Walter John Moore

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**Extra resources for Seven Solid States;: An Introduction to the Chemistry and Physics of Solids**

**Example text**

29) with Eqs. b2 a1 b1 a2 / should give the desired expression for the c2 . 0; 1; 0/ is a unit vector along the y axis. According to Fig. e. c3 must be positive. 1 1 0 0/ D ˙1 the plus sign must be chosen. 30) 30 1 Basic Knowledge The vector product of two vectors can most easily be remembered if written via a determinant which we introduced in Sect. 6; specifically, compare Eqs. 30). 21. e. show explicitly that in this case the vector a b is perpendicular to both vectors a and b, and that its length corresponds to the definition of Eq.

The vector b has the same direction as a if c > 0, opposite to that of a if c < 0, and is the null vector if c D 0. Dividing a vector a by its length jaj results in a unit vector: e D a= jaj which has the same direction as a. The sum of two vectors a and b is defined as follows: translate the vector a into ! ! the vector AB, vector b into BC, then the sum c D a C b is defined as the vector AC, see Fig. 11(a). It is easy to see using the geometrical representation of the vectors in the 3D Cartesian system, that Cartesian coordinates of the vector c are obtained by summing up the corresponding coordinates of a and b.

1; 4; 6; 4; 1/ for n D 4; and so on. One may notice that a set of coefficients for the power n C 1 can actually be obtained from the coefficients corresponding to the previous power n. This elegant rule known as Pascal’s triangle9 is illustrated in Fig. 15. e. for n D 5. 1; 5; 10; 10; 5; 1/, and they are constructed following two simple rules: (i) the first and the last coefficients (for the terms a5 b0 D a5 and a0 b5 D b5 ) are 1; (ii) any other coefficient is obtained as a sum of the two coefficients directly above it from the previous line (corresponding to n D 4 in our case).