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JEE Main & Advanced Sample Paper JEE Main Sample Paper-12

  • question_answer
    Given \[\vec{A}=2\hat{i}+3\hat{j}+6\hat{k},\]\[\vec{B}=\hat{i}+\hat{j}-2\vec{k}\]and \[\vec{C}=\hat{i}+2\hat{j}+\hat{k}\]Compute the value of \[|\vec{A}\times [\vec{A}\times (\vec{A}\times \vec{B})].\vec{C}|.\]

    A)  343                       

    B)  512

    C)  221                       

    D)  243

    Correct Answer: A

    Solution :

     \[\vec{V}=\vec{A}\times \left[ (\vec{A}.\vec{B})\vec{A}-(\vec{A}.\vec{A})\vec{B} \right].\vec{C}\] \[=\left( \underbrace{\vec{A}\times (\vec{A}.\vec{B})\vec{A}}_{zero}-(\vec{A}.\vec{A})\vec{A}\times \vec{B} \right).\vec{C}=-|\vec{A}{{|}^{2}}[\vec{A}\vec{B}\vec{C}]\]Now, \[|\vec{A}{{|}^{2}}=4+9+36=49\] \[[\vec{A}\vec{B}\vec{C}]=\left| \begin{matrix}    2 & 3 & 6  \\    1 & 1 & -2  \\    1 & 2 & 1  \\ \end{matrix} \right|\] \[=2(1+4)-2(3-12)+1(-6-6)\] \[=10+9-12=7\] \[\therefore \]\[\left| -|\vec{A}{{|}^{2}}[\vec{A}\vec{B}\vec{C}] \right|=49\times 7=343\]


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