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JEE Main & Advanced Mathematics Vector Algebra Question Bank Mock Test - Vector Algebra

  • question_answer
    Let the position vectors of the points P and Q be \[4\hat{i}+\hat{j}+\lambda \hat{k}\] and \[2\hat{i}-\hat{j}+\lambda \hat{k}\], respectively. Vector \[\hat{i}-\hat{j}+6\hat{k}\]is perpendicular to the plane containing the origin and the point?s P and Q. then\[\lambda \]equals

    A) \[-\,1/2\]

    B) 1/2

    C) 1                     

    D) none of these

    Correct Answer: A

    Solution :

    [a] A vector perpendicular to the plane of O, P and Q is \[\overrightarrow{OP}\times \overrightarrow{OQ}.\] Now, \[\overrightarrow{OP}\times \overrightarrow{OQ}=\left| \begin{matrix}    {\hat{i}} & {\hat{j}} & {\hat{k}}  \\    4 & 1 & \lambda   \\    2 & -1 & \lambda   \\ \end{matrix} \right|=2\lambda \hat{i}-2\lambda \hat{j}-6\hat{k}\] Therefore, \[\hat{i}-\hat{j}+6\hat{k}\] is parallel to \[2\lambda \hat{i}-2\lambda \hat{j}-6\hat{k}\] Hence, \[\frac{1}{2\lambda }=\frac{-1}{-2\lambda }=\frac{6}{-6}\] \[\therefore \lambda =-\frac{1}{2}\]


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