Vector Algebra Question 198
Question: Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies equation $ \mathbf{p}\times {(\mathbf{x}-\mathbf{q})\times \mathbf{p}}+\mathbf{q}\times {(\mathbf{x}-\mathbf{r})\times \mathbf{q}}+\mathbf{r}\times {(\mathbf{x}-\mathbf{p})\times \mathbf{r}}=0, $ then x is given by
[IIT 1997 Cancelled]
Options:
A) $ \frac{1}{2},(\mathbf{p}+\mathbf{q}-2\mathbf{r}) $
B) $ \frac{1}{2}(\mathbf{p}+\mathbf{q}+\mathbf{r}) $
C) $ \frac{1}{3}(\mathbf{p}+\mathbf{q}+\mathbf{r}) $
D) $ \frac{1}{3}(2\mathbf{p}+\mathbf{q}-\mathbf{r}) $
Show Answer
Answer:
Correct Answer: B
Solution:
- $ |\mathbf{p}|,=,|\mathbf{q}|,=,|\mathbf{r}|,=c $ , (say)
and $ \mathbf{p}.\mathbf{q}=0=\mathbf{p}.\mathbf{r}=\mathbf{q}.\mathbf{r} $
$ \mathbf{p}\times |(\mathbf{x}-\mathbf{q})\times \mathbf{p}|+\mathbf{q}\times |(\mathbf{x}-\mathbf{r})\times \mathbf{q}|+\mathbf{r}\times |(\mathbf{x}-\mathbf{p})\times \mathbf{r}|=0 $
$ \Rightarrow (\mathbf{p}.\mathbf{p})(\mathbf{x}-\mathbf{q})-{\mathbf{p}.(\mathbf{x}-\mathbf{q})}\mathbf{p}+………=0 $
$ \Rightarrow c^{2}(\mathbf{x}-\mathbf{q}+\mathbf{x}-\mathbf{r}+\mathbf{x}-\mathbf{p})-(\mathbf{p}.\mathbf{x})\mathbf{p}-(\mathbf{q}.\mathbf{x})\mathbf{q}-(\mathbf{r}.\mathbf{x})\mathbf{r}=0 $
$ \Rightarrow c^{2}{3\mathbf{x}-(\mathbf{p}+\mathbf{q}+\mathbf{r})}-[(\mathbf{p}.\mathbf{x})\mathbf{p}+(\mathbf{q}.\mathbf{x})\mathbf{q}+(\mathbf{r}.\mathbf{x})\mathbf{r}]=0 $
Which is satisfied by $ \mathbf{x}=\frac{1}{2}(\mathbf{p}+\mathbf{q}+\mathbf{r}) $ .
BETA
