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JEE Main & Advanced Mathematics Vector Algebra Question Bank Vector or Cross product of two vectors and its application

  • question_answer
    If the vectors  \[\mathbf{i}-3\mathbf{j}+2\mathbf{k}\], \[-\mathbf{i}+2\mathbf{j}\] represents the diagonals of a parallelogram, then its area will be              [Roorkee 1976]

    A)                 \[\sqrt{21}\]

    B)                 \[\frac{\sqrt{21}}{2}\]

    C)                 \[2\sqrt{21}\]

    D)                 \[\frac{\sqrt{21}}{4}\]

    Correct Answer: B

    Solution :

               Let \[\mathbf{a}=\mathbf{i}-3\mathbf{j}+2\mathbf{k},\] \[\mathbf{b}=-\mathbf{i}+2\mathbf{j}\]                    \[\Rightarrow \mathbf{a}\times \mathbf{b}=\left| \begin{matrix}    \mathbf{i} & \mathbf{j} & \mathbf{k}  \\    1 & -3 & 2  \\    -1 & 2 & 0  \\ \end{matrix} \right|=-4\mathbf{i}-2\mathbf{j}-\mathbf{k}\]                                 Hence area is equal to \[\frac{1}{2}|\mathbf{a}\times \mathbf{b}|\,=\frac{\sqrt{21}}{2}.\]


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