7.10 EXERCISES
7.2 A unit cell of diamond is a cube of sideA, with carbon atoms at each corner, at
the centre of each face and, in addition, at positions displaced by^14 A(i+j+k)
from each of those already mentioned;i,j,kare unit vectors along the cube axes.
One corner of the cube is taken as the origin of coordinates. What are the vectors
joining the atom at^14 A(i+j+k) to its four nearest neighbours? Determine the
angle between the carbon bonds in diamond.
7.3 Identify the following surfaces:
(a) |r|=k;(b)r·u=l;(c)r·u=m|r|for− 1 ≤m≤+1;
(d)|r−(r·u)u|=n.Herek,l,mandnare fixed scalars anduis a fixed unit vector.
7.4 Find the angle between the position vectors to the points (3,− 4 ,0) and (− 2 , 1 ,0)
and find the direction cosines of a vector perpendicular to both.
7.5 A, B, CandDare the four corners, in order, of one face of a cube of side 2
units. The opposite face has cornersE, F, GandH,withAE, BF, CGandDH
as parallel edges of the cube. The centreOofthecubeistakenastheorigin
and thex-,y-andz-axes are parallel toAD,AEandAB, respectively. Find the
following:
(a) the angle between the face diagonalAFand the body diagonalAG;
(b) the equation of the plane throughBthat is parallel to the planeCGE;
(c) the perpendicular distance from the centreJof the faceBCGFto the plane
OCG;
(d) the volume of the tetrahedronJOCG.7.6 Use vector methods to prove that the lines joining the mid-points of the opposite
edges of a tetrahedronOABCmeet at a point and that this point bisects each of
the lines.
7.7 The edgesOP,OQandORof a tetrahedronOP QRare vectorsp,qandr,
respectively, wherep=2i+4j,q=2i−j+3kandr=4i− 2 j+5k. Show that
OPis perpendicular to the plane containingOQR. Express the volume of the
tetrahedron in terms ofp,qandrand hence calculate the volume.
7.8 Prove, by writing it out in component form, that
(a×b)×c=(a·c)b−(b·c)a,and deduce the result, stated in equation (7.25), that the operation of forming
the vector product is non-associative.
7.9 Prove Lagrange’s identity, i.e.
(a×b)·(c×d)=(a·c)(b·d)−(a·d)(b·c).7.10 For four arbitrary vectorsa,b,candd, evaluate
(a×b)×(c×d)in two different ways and so prove thata[b,c,d]−b[c,d,a]+c[d,a,b]−d[a,b,c]= 0.Show that this reduces to the normal Cartesian representation of the vectord,
i.e.dxi+dyj+dzk,ifa,bandcare taken asi,jandk, the Cartesian base vectors.
7.11 Show that the points (1, 0 ,1), (1, 1 ,0) and (1,− 3 ,4) lie on a straight line. Give the
equation of the line in the form
r=a+λb.