• # question_answer 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] 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) $\frac{1}{3}(2\mathbf{p}+\mathbf{q}-\mathbf{r})$

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$\mathbf{x}=\frac{1}{2}(\mathbf{p}+\mathbf{q}+\mathbf{r})$.

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