Question
If $$a,\,b,\,c$$ are the $${p^{th}},\,{q^{th}},\,{r^{th}}$$ terms of an HP and $$\overrightarrow u = \left( {q - r} \right)\hat i + \left( {r - p} \right)\hat j + \left( {p - q} \right)\hat k,\,\overrightarrow v = \frac{{\overrightarrow i }}{a} + \frac{{\overrightarrow j }}{b} + \frac{{\overrightarrow k }}{c}{\text{ then :}}$$
A.
$$\overrightarrow u ,\,\overrightarrow v $$ are parallel vectors
B.
$$\overrightarrow u ,\,\overrightarrow v $$ are orthogonal vectors
C.
$$\vec u.\vec v = 1$$
D.
$$\vec u \times \vec v = \overrightarrow i + \overrightarrow j + \overrightarrow k $$
Answer :
$$\overrightarrow u ,\,\overrightarrow v $$ are orthogonal vectors
Solution :
$$\eqalign{
& \frac{1}{a} = A + \left( {p - 1} \right)D\,; \cr
& \frac{1}{b} = A + \left( {q - 1} \right)D\,; \cr
& \frac{1}{c} = A + \left( {r - 1} \right)D \cr
& \therefore \,q - r = \frac{{c - b}}{{bcD}},\,r - p = \frac{{a - c}}{{acD}},\,p - q = \frac{{b - a}}{{abD}} \cr
& \Rightarrow \frac{{q - r}}{a} + \frac{{r - p}}{b} + \frac{{p - q}}{c} = 0 \cr
& \Rightarrow \vec u.\vec v = 0 \cr} $$