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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    The vector c directed along the internal bisector of the angle between the vectors \[\mathbf{a}=7\mathbf{i}-4\mathbf{j}-4\mathbf{k}\] and \[\mathbf{b}=-2\mathbf{i}-\mathbf{j}+2\mathbf{k}\] with \[|\mathbf{c}|\,=5\sqrt{6},\] is   

    A) \[\frac{5}{3}\,(\mathbf{i}-7\mathbf{j}+2\mathbf{k})\]                    

    B) \[\frac{5}{3}\,(5\mathbf{i}+5\mathbf{j}+2\mathbf{k})\]

    C) \[\frac{5}{3}\,(\mathbf{i}+7\mathbf{j}+2\mathbf{k})\]      

    D) \[\frac{5}{3}\,(-5\mathbf{i}+5\mathbf{j}+2\mathbf{k})\]

    Correct Answer: A

    Solution :

    • The required vector c is given by \[\lambda \left( \frac{\mathbf{a}}{|\mathbf{a}|}+\frac{\mathbf{b}}{|\mathbf{b}|} \right)\]                   
    • Now, \[\frac{\mathbf{a}}{|\mathbf{a}|}=\frac{1}{9}(7\mathbf{i}-4\mathbf{j}-4\mathbf{k})\]                   
    • and  \[\frac{\mathbf{b}}{|\mathbf{b}|}=\frac{1}{3}(-2\mathbf{i}-\mathbf{j}+2\mathbf{k})\]                   
    • \[\Rightarrow \mathbf{c}=\lambda \left( \frac{1}{9}\mathbf{i}-\frac{7}{9}\mathbf{j}+\frac{2}{9}\mathbf{k} \right)\]                   
    • \[\Rightarrow |\mathbf{c}{{|}^{2}}={{\lambda }^{2}}.\frac{54}{81}\]                   
    • \[\Rightarrow {{\lambda }^{2}}=225\]or\[\lambda =\pm 15\] .                   
    • Therefore, \[\mathbf{c}=\pm \frac{5}{3}(\mathbf{i}-7\mathbf{j}+2\mathbf{k}).\]


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