Vector Algebra Question 105
Question: If a, b, c are the $ p^{th},q^{th}.{r^{th}} $ terms of an HP and $ \vec{u}=(q-r)\vec{i}+(r-p)\vec{j}+(p-q)\vec{k},\vec{v}=\frac{{\vec{i}}}{a}+\frac{{\vec{j}}}{b}+\frac{{\vec{k}}}{c} $ then
Options:
A) $ \vec{u},\vec{v} $ are parallel vectors
B) $ \vec{u},\vec{v} $ are orthogonal vectors
C) $ \vec{u}.\vec{v}=1 $
D) $ \vec{u}\times \vec{v}=\vec{i}+\vec{j}+\vec{k} $
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] $ \frac{1}{a}=A+(p-1)D;\frac{1}{b}=A+(q-1)D; $ $ \frac{1}{c}=A+(r-1)D $
$ \therefore q-r=\frac{c-b}{bcD},r-p=\frac{a-c}{acD} $ $ p-q=\frac{b-a}{abD}\Rightarrow \frac{q-r}{a}+\frac{r-p}{b}+\frac{p-q}{c}=0 $
$ \Rightarrow \overset{\to }{\mathop{u}},\cdot \overset{\to }{\mathop{v}},=0 $
BETA
