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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    The distance of the point \[B\,(\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] from the line which is passing through \[A\,(4\mathbf{i}+2\mathbf{j}+2\mathbf{k})\] and which is parallel to the vector \[\overrightarrow{C}=2\mathbf{i}+3\mathbf{j}+6\mathbf{k}\] is [Roorkee 1993]

    A) 10

    B) \[\sqrt{10}\]

    C) 100

    D) None of these

    Correct Answer: B

    Solution :

    • \[B{{M}^{2}}=A{{B}^{2}}-A{{M}^{2}}\]       ...(i)                   
    • \[\overrightarrow{AB}=-3\mathbf{i}+0\mathbf{j}+\mathbf{k}\]                   
    • \[A{{B}^{2}}={{\overrightarrow{AB}}^{2}}=9+1=10\]                   
    • \[AM=\]Projection of \[\overrightarrow{AB}\] in direction of\[\overrightarrow{C}\]                          
    • \[=2\mathbf{i}+3\mathbf{j}+6\mathbf{k}\]                   
    • \ \[AM=\frac{\overrightarrow{AB}\,.\,\overrightarrow{C}}{|\overrightarrow{C}|}=\frac{(-3\mathbf{i}+0\mathbf{j}+\mathbf{k})\,.\,(2\mathbf{i}+3\mathbf{j}+6\mathbf{k})}{7}=0\]                   
    • \ \[B{{M}^{2}}=10-0=10\]                   
    • \[\Rightarrow BM=\sqrt{(10)}\], {by (i)}.


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