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

  • question_answer
    The value of l for which the four points \[2\mathbf{i}+3\mathbf{j}-\mathbf{k},\] \[\mathbf{i}+2\mathbf{j}+3\mathbf{k}\], \[3\mathbf{i}+4\mathbf{j}-2\mathbf{k},\,\,\mathbf{i}-\lambda \mathbf{j}+6\mathbf{k}\] are coplanar                                     [MP PET 2004]

    A)             8

    B)             0

    C)             ? 2

    D)             6

    Correct Answer: C

    Solution :

                    Let four points \[A,B,C,D\]represent the given points                                 So, \[\overrightarrow{AB}=-\mathbf{i}-\mathbf{j}+4\mathbf{k},\,\,\overrightarrow{BC}=2\mathbf{i}+2\mathbf{j}-5\mathbf{k},\]                                       \[\overrightarrow{CD}=-2\mathbf{i}-(\lambda +4)\mathbf{j}+3\mathbf{k}\]                                 From the condition, \[[\overrightarrow{AB}\,\,\overrightarrow{BC}\,\,\overrightarrow{CD}]=0\]                                 Þ  \[\left| \,\begin{matrix}    -1 & -1 & 4  \\    2 & 2 & -5  \\    -2 & -(\lambda +4) & 3  \\ \end{matrix}\, \right|=0\]                                     Þ  \[-1[2.3-5(\lambda +4)]+1[6-10]+4[-2(\lambda +4)+4]=0\]                                 \[\Rightarrow \lambda =-2\].


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