This exercise considers a crystal whose unit cell has base vectors that are not necessarily mutually orthogonal (a) The basis vectors of the unit cell of a crystal, with the origin O at one corner, are denoted by ei, e2 , ез. The matrix G has elements Gijs where Gij ei-ej and Hj are the elements of the matrix HG1. Show that the vectors f , Hije, are the reciprocal vectors and that Hj f f (b) If the vectors u and v are given by 0 1-1 obtain expressions for lul, vl, and u v. T/3, write down G and hence obtain H. points p ei, qe2, es, and (ii) the angle between this normal and e,. (c) If the basis vectors are each of length a and the angle between each pair is (d) Calculate (i) the length of the normal from O onto the plane containing the

Find a and that solve the vector equation: Suppose the above equation is expressed as Ax = b with x = coeffieints of the equations in one matrix form as (Ab)= Rows: 2 Columns: 3 b) c) d) 21 a (²) + ^ ( ;) = (²1) B α= B = Complete the missing column of the following matrix derived by doing row operation of the augmented matrix in a): 1 -2 (²). 0 1 Express the

2. Consider the function y(x) = 5x over the interval [0, 10]. (a) If the points A(0, y(0)), B(5, y(5)), and C(10, y(10)) are on the curve represented by the above function, find the slope of the lines (secants) AB, and AC, respectively. (b) If x represents time and y represents the displacement of a particle moving along a line, what is the physical meaning of your results to part (a)? (c) Now consider a point D(10 – h, y(10 – h)) on the curve represented by the given function, find the slope of the line DC. (d) As h → 0, what is the limiting value of your result to part (c)? (e) What is its physical meaning of your result to part (d) if again x represents time and y represents the displacement?

Consider the vectors a = 6i + 7j + 6k and b = 4i − j + 8k, where i, j, and k are mutuallyperpendicular unit vectors forming a right-handed system. (b) (i) If c = c1i + c2j + 12k is parallel to the vector a above, determine the values of c1 and c2. (ii) If d = d1i + 12j + d3k is perpendicular to both a and b above, determine the values of d1 and d3.