Question
If $$a,\,b,\,c$$ are the $${p^{th}},\,{q^{th}},\,{r^{th}}$$ terms of an HP and $$\overrightarrow u = \left( {q - r} \right)\overrightarrow i + \left( {r - p} \right)\overrightarrow j + \left( {p - q} \right)\overrightarrow k ,\,\overrightarrow v = \frac{{\overrightarrow i }}{a} + \frac{{\overrightarrow j }}{b} + \frac{{\overrightarrow k }}{c},$$ then :
A.
$$\overrightarrow u ,\,\overrightarrow v $$ are parallel vectors
B.
$$\overrightarrow u ,\,\overrightarrow v $$ are orthogonal vectors
C.
$$\overrightarrow u .\overrightarrow v = 1$$
D.
$$\overrightarrow u \times \overrightarrow v = \overrightarrow i + \overrightarrow j + \overrightarrow k $$
Answer :
$$\overrightarrow u ,\,\overrightarrow v $$ are orthogonal vectors
Solution :
$$\overrightarrow u .\overrightarrow v = \frac{{q - r}}{a} + \frac{{r - p}}{b} + \frac{{p - q}}{c}.....(1)$$
From the question, $$\frac{1}{a} = x + \left( {p - 1} \right)y,\,\frac{1}{b} = x + \left( {q - 1} \right)y,\,\frac{1}{c} = x + \left( {r - 1} \right)y$$
Putting these in $$(1)$$ and simplifying, $$\overrightarrow u .\overrightarrow v = 0$$
Clearly, $$\left( {\overrightarrow i + \overrightarrow j + \overrightarrow k } \right).\overrightarrow v = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \ne 0.$$ So, $$\overrightarrow u \times \overrightarrow v \ne \overrightarrow i + \overrightarrow j + \overrightarrow k $$