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JEE Main & Advanced Mathematics Vector Algebra Question Bank Scalar triple product and their applications

  • question_answer
    If the vectors \[2\mathbf{i}-3\mathbf{j},\,\,\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[3\mathbf{i}-\mathbf{k}\] form three concurrent edges of a parallelopiped, then the volume of the parallelopiped is        [IIT 1983; RPET 1995; DCE 2001; Kurukshetra CEE 1998; MP PET 2001]

    A)             8

    B)             10

    C)             4

    D)             14

    Correct Answer: C

    Solution :

                    Here,     \[\overrightarrow{OA}=2\mathbf{i}-3\mathbf{j}=\mathbf{a}\]                 (say)                                                 \[\overrightarrow{OB}=\mathbf{i}+\mathbf{j}-\mathbf{k}=\mathbf{b}\,\](say)                                  and       \[\overrightarrow{OC}=3\mathbf{i}-\mathbf{k}=\mathbf{c}\]   (say)                                 Hence volume is \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]=\mathbf{a}.\,(\mathbf{b}\times \mathbf{c})=\left| \begin{matrix}    2 & -3 & 0  \\    1 & 1 & -1  \\    3 & 0 & -1  \\ \end{matrix} \right|=4\].


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