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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    If \[\mathbf{a}=2\mathbf{i}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=4\mathbf{i}-3\mathbf{j}+7\mathbf{k}.\] If \[\mathbf{d}\times \mathbf{b}=\mathbf{c}\times \mathbf{b}\] and \[\mathbf{d}\,.\,\mathbf{a}=0,\] then d will be [IIT 1990]

    A) \[\mathbf{i}+8\mathbf{j}+2\mathbf{k}\]

    B) \[\mathbf{i}-8\mathbf{j}+2\mathbf{k}\]

    C) \[-\mathbf{i}+8\mathbf{j}-\mathbf{k}\]

    D) \[-\mathbf{i}-8\mathbf{j}+2\mathbf{k}\]

    Correct Answer: D

    Solution :

    • \[\mathbf{d}\times \mathbf{b}=\mathbf{c}\times \mathbf{b}\] gives \[\left| \,\begin{matrix}    \mathbf{i} & \mathbf{j} & \mathbf{k}  \\    x & y & z  \\    1 & 1 & 1  \\ \end{matrix}\, \right|=\left| \,\begin{matrix}    \mathbf{i} & \mathbf{j} & \mathbf{k}  \\    4 & -3 & 7  \\    1 & 1 & 1  \\ \end{matrix}\, \right|\,,\] where \[\mathbf{d}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\]  (say)                   
    • On solving, \[x=-1,\] \[y=-8,\] \[z=2\]                   
    • Hence \[\mathbf{d}=-\mathbf{i}-8\mathbf{j}+2\mathbf{k}.\]


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