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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    The volume of the tetrahedron, whose vertices are given by the vectors \[-\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[\mathbf{i}+\mathbf{j}-\mathbf{k}\]with reference to the fourth vertex as origin, is

    A) \[\frac{5}{3}\]cubic unit

    B) \[\frac{2}{3}\] cubic unit

    C) \[\frac{3}{5}\]cubic unit

    D) None of these

    Correct Answer: B

    Solution :

    • Volume of tetrahedron \[ABCD\] is, \[\frac{1}{6}|\overrightarrow{AB}\times \overrightarrow{AC}\,.\,\overrightarrow{AD}|\], where \[A(-1,\,1,\,1),\] \[B(1,\,-1,\,1),\] \[C(1,\,1,\,-1)\] and \[D(0,\,0,\,0).\]                   
    • \[=\frac{1}{6}|(2\mathbf{i}-2\mathbf{j})\times (2\mathbf{i}-2\mathbf{k})\,.\,(\mathbf{i}-\mathbf{j}-\mathbf{k})|\]                   
    • \[=\frac{1}{6}\left| \,\begin{matrix}    2 & -2 & 0  \\    2 & 0 & -2  \\    1 & -1 & -1  \\ \end{matrix}\, \right|=\frac{1}{6}(-4)=-\frac{2}{3}=\frac{2}{3}\] cubic unit.


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