Q. If $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{c}}$ are the $pth,\ qth,\ nth$ terms of an HP and $\overrightarrow{u} = (q - r)\widehat{\mathbf{i}} + (r - p)\widehat{\mathbf{j}} + (p - q)\widehat{\mathbf{k}}$ and $\overrightarrow{v} = \frac{\widehat{\mathbf{i}}}{a} + \frac{\widehat{\mathbf{j}}}{b} + \frac{\widehat{\mathbf{k}}}{c}$, then
A) $\overrightarrow{\mathbf{u}},\mathbf{\ }\overrightarrow{\mathbf{v}}$ are parallel vectors
B) $\overrightarrow{\mathbf{u}},\ \overrightarrow{\mathbf{v}}$ are orthogonal vectors
C) $\overrightarrow{\mathbf{u}}\mathbf{\bullet}\overrightarrow{\mathbf{v}} = 1$
D) $\overrightarrow{\mathbf{u}}\mathbf{\times}\overrightarrow{\mathbf{v}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}}$
Explanation
(b)
$\because$ $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}},and\ \overrightarrow{\mathbf{c}}$ are the $pth,\ qth,\ nth$ terms of an HP respectively.
$\frac{1}{a} = A + (p - 1)D,\frac{1}{b} = A + (q - 1)D\ and\frac{1}{c} = A + (r - 1)D$
$\therefore q - r = \frac{c - b}{bcD},\ r - p = \frac{a - c}{acD}$
And $q - r = \frac{b - a}{abD}$
$\Rightarrow \ \frac{(q - r)}{a} + \frac{(r - p)}{b} + \frac{(p - q)}{c} = 0$
$\Rightarrow \ \overrightarrow{\mathbf{u}}\mathbf{\bullet}\overrightarrow{\mathbf{v}} = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.