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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    If \[\mathbf{a}\times \mathbf{r}=\mathbf{b}+\lambda \mathbf{a}\] and \[\mathbf{a}\,\,.\,\,\mathbf{r}=3,\] where \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[\mathbf{b}=-\mathbf{i}-2\mathbf{j}+\mathbf{k},\] then r and l are equal to 

    A) \[\mathbf{r}=\frac{7}{6}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{6}{5}\]

    B) \[\mathbf{r}=\frac{7}{6}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{5}{6}\]

    C) \[\mathbf{r}=\frac{6}{7}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{6}{5}\]

    D) None of these

    Correct Answer: B

    Solution :

    • Given, \[\mathbf{a}\times \mathbf{r}=\mathbf{b}+\lambda \,\mathbf{a}\Rightarrow (\mathbf{a}\times \mathbf{r})\,.\,\mathbf{a}=\mathbf{b}\,.\,\mathbf{a}+\lambda \,\mathbf{a}\,.\,\mathbf{a}\]                   
    • \[\Rightarrow 0=\mathbf{b}\,.\,\mathbf{a}+\lambda \,|\mathbf{a}{{|}^{2}}\Rightarrow \lambda =-\frac{\mathbf{b}\,.\,\,\mathbf{a}}{|\mathbf{a}{{|}^{2}}}=\frac{5}{6}\]                   
    • Also, \[(\mathbf{a}\times \mathbf{r})\times \mathbf{a}=\mathbf{b}\times \mathbf{a}+\lambda \,\mathbf{a}\times \mathbf{a}\Rightarrow \mathbf{r}=\frac{7}{6}\mathbf{i}+\frac{2}{3}\mathbf{j}\].


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