Mathematical Methods for Physics and Engineering - Matematica.NET

7.9 RECIPROCAL VECTORSthe line to the plane is zero unlessin which case the distance, d, will bewhere r is any point in the plane.b · ˆn =0,d = |(a − r) · ˆn|,◮A line is given by r = a + λb, wherea = i +2j +3k and b =4i +5j +6k. Findthecoordinates of the point P at which the line intersects the planex +2y +3z =6.A vector normal to the plane isn = i +2j +3k,from which we find that b · n ≠ 0. Thus the line does indeed intersect the plane. To findthe point of intersection we merely substitute the x-, y- andz- values of a general pointon the line into the equation of the plane, obtaining1+4λ +2(2+5λ)+3(3+6λ) =6 ⇒ 14 + 32λ =6.This gives λ = − 1 , which we may substitute into the equation for the line to obtain4x =1− 1 (4) = 0, y =2− 1 (5) = 3 and z =3− 1 (6) = 3 . Thus the point of intersection is4 4 4 4 2(0, 3 4 , 3 2 ). ◭ 7.9 Reciprocal vectorsThe final section of this chapter introduces the concept of reciprocal vectors,which have particular uses in crystallography.The two sets of vectors a, b, c and a ′ , b ′ , c ′ are called reciprocal sets ifanda · a ′ = b · b ′ = c · c ′ = 1 (7.47)a ′ · b = a ′ · c = b ′ · a = b ′ · c = c ′ · a = c ′ · b =0. (7.48)It can be verified (see exercise 7.19) that the reciprocal vectors of a, b and c aregiven bya ′ =b × ca · (b × c) , (7.49)b ′ =c × aa · (b × c) , (7.50)c ′ =a × ba · (b × c) , (7.51)where a · (b × c) ≠ 0. In other words, reciprocal vectors only exist if a, b and c are233