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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

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

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


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