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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    The point of intersection of \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}\] and \[\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b}\], where \[\mathbf{a}=\mathbf{i}+\mathbf{j}\] and \[\mathbf{b}=2\mathbf{i}-\mathbf{k}\] is [Orissa JEE 2004]

    A) \[3\mathbf{i}+\mathbf{j}-\mathbf{k}\]

    B) \[3\mathbf{i}-\mathbf{k}\]

    C) \[3\mathbf{i}+2\mathbf{j}+\mathbf{k}\]

    D) None of these

    Correct Answer: A

    Solution :

    • We have \[\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b}\]and \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}\]                   
    • Adding \[\mathbf{r}\times (\mathbf{a}+\mathbf{b})=0\]i.e., r is parallel to \[\mathbf{a}+\mathbf{b}\]                   
    • or   \[\mathbf{r}=\lambda (\mathbf{i}+\mathbf{j}+2\mathbf{i}-\mathbf{k})\]                         
    • \[\mathbf{r}=\lambda (3\mathbf{i}+\mathbf{j}-\mathbf{k})\] for \[\lambda =1\Rightarrow \mathbf{r}=(3\mathbf{i}+\mathbf{j}-\mathbf{k})\].


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