KVPY Sample Paper KVPY Stream-SX Model Paper-21

  • question_answer
    The position vectors of three points A, B, C are \[\hat{i}+2\hat{j}+3\hat{k}\cdot 2\hat{i}+3\hat{j}+\hat{k}\And 3\hat{i}+\hat{j}+2\hat{k}.\] A unit vector perpendicular to the plane of the triangle ABC is:

    A) \[\left( -\frac{1}{\sqrt{3}} \right)\,\,(\hat{i}+\hat{j}+\hat{k})\]

    B) \[\left( \frac{1}{\sqrt{3}} \right)\,\,(\hat{i}-\hat{j}+\hat{k})\]

    C) \[\left( \frac{1}{\sqrt{3}} \right)\,\,(\hat{i}+\hat{j}-\hat{k})\]

    D) none of these

    Correct Answer: A

    Solution :

    unit vector \[\bot \] to plane ABC
    \[=\frac{\overrightarrow{AB}\times \overrightarrow{AC}}{|\overrightarrow{AB}\times \overrightarrow{AC}|}\]\[=\frac{1}{|\overrightarrow{AB}\times \overrightarrow{AC}|}\,\,\left| \begin{matrix}    {\hat{i}} & {\hat{j}} & {\hat{k}}  \\    1 & 1 & -\,2  \\    2 & -1 & -1  \\ \end{matrix} \right|\]\[=\frac{\hat{i}\,(-1-2)-\hat{j}\,(-1+4)+\hat{k}\,(-1-2)}{|\overrightarrow{AB}\times \overrightarrow{AC}|}\]\[=\frac{3\hat{i}-3\hat{j}-3\hat{k}}{\sqrt{27}}=\frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}\]

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