If p th, q th, r th terms of a geometric progression are the positive numbers a, b and c respectively, then the angle between the vectors ( a^2 ) i + ( b^2 ) j + ( c^2 ) k and (q-r) i +(r-p) j +(p-q) k is

Step-by-step solution

Let first term of a GP be u and common ratio z . T_p=u z^ p-1 =a aligned & array l u+(p-1) z= a T_q=u z^ q-1 =b array & u+(q-1) z= b & and T_r=u z^ r-1 =c aligned u+(r-1) z= c Let be the angle between ( a^2 ) i + ( b^2 ) j + ( c^2 ) k and (q-r) i +(r-p) j +(p-q) k is = [ array r ( a^2 )(q-r)+ ( b^2 )(r-p) + ( c^2 )(p-q) array ] [ array r ( a^2 )^2+ ( b^2 )^2+ ( c^2 )^2 (q-r)^2+(r-p)^2+(p-q)^2 array ] From Eqs. (i), (ii) and (iii) gathered q-r= b- c, r-p= c- a p-q= a- b gathered From Eq. (iv), taking numerator term