If a, b, c are p^ t h and q^ th and r^ t h terms of a GP, then the vectors a i + b j + c k and (q-r) i +(r-p) j +(p-q) k are

Step-by-step solution

Let the first term and common ratio of a GP be and , then aligned a= & ^ p-1 , b= ^ q-1 and c= ^ r-1 & a= +(p-1) , & b= +(q-1) and & c= +(r-1) aligned The dot product of the given two vectors is aligned & (q-r) a+(r-p) b+(p-q) c & (q-r)[ +(p-1) ]+(r-p) & [ +(q-1) ]+(p-q)[ +(r-1) ] & [q-r+r-p+p-q]+ & [(p-1)(q-r)+(r-p)(q-1) & +(r-1)(p-q)] & =0+0=0 aligned The two vectors are perpedicular.