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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    The position vectors of the vertices of a quadrilateral ABCD are \[a,\,b,\,c\] and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is [Roorkee 2000]

    A) \[\frac{1}{4}\,|a\times b+b\times d+d\times a|\]

    B) \[\frac{1}{4}\,\left| b\times  c+c\times d+a\times d+b\times a \right|\]

    C) \[\frac{1}{4}\,\left| a\times  b+b\times c+c\times d+d\times a \right|\]

    D) \[\frac{\text{1}}{\text{4}}\text{ }\!\!|\!\!\text{ b }\!\!\times\!\!\text{ c+c }\!\!\times\!\!\text{ d+d }\!\!\times\!\!\text{ b }\!\!|\!\!\text{ }\].

    Correct Answer: C

    Solution :

    • It is given that \[\mathbf{a},\mathbf{b},\mathbf{c}\] and \[\mathbf{d}\] are the position vectors of vertices of a quadrilateral ABCD respectively.                  
    • Let E, F, G and H are the middle points of sides AB, BC, CD and DA respectively.                   
    • The position vectors of these points will be \[\overrightarrow{OE}=\frac{1}{2}(\mathbf{a}+\mathbf{b}),\,\,\,\,\,\overrightarrow{OF}=\frac{1}{2}(\mathbf{b}+\mathbf{c})\],           
    • \[\overrightarrow{OG}=\frac{1}{2}(\mathbf{c}+\mathbf{d}),\,\,\,\,\,\,\,\overrightarrow{OH}=\frac{1}{2}(\mathbf{a}+\mathbf{d})\]                   
    • Then    \[\overrightarrow{EF}=\overrightarrow{OF}-\overrightarrow{OE}=\left( \frac{\mathbf{c}-\mathbf{a}}{2} \right)\]                   
    • and     \[\overrightarrow{FG}=\frac{1}{2}(\mathbf{d}-\mathbf{b}),\overrightarrow{GH}=\frac{1}{2}(\mathbf{a}-\mathbf{c}),\,\overrightarrow{GH}=\frac{1}{2}(\mathbf{b}-\mathbf{d})\]                   
    • It is clear that \[\overrightarrow{EF}\]is parallel to \[\overrightarrow{GH}\] and \[\overrightarrow{FG}\] is parallel to \[\overrightarrow{HE}\]. Thus EFGH is a parallelogram.                   
    • \[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\overrightarrow{EF}\times \overrightarrow{FG}=\frac{1}{4}\{(\mathbf{c}-\mathbf{a})\times (\mathbf{d}-\mathbf{b})\}\]                                                
    • \[=\frac{1}{4}(\mathbf{c}\times \mathbf{d}-\mathbf{c}\times \mathbf{b}-\mathbf{a}\times \mathbf{d}+\mathbf{a}\times \mathbf{b})\]                                                
    • \[=\frac{1}{4}(\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{d}+\mathbf{d}\times \mathbf{a})\]                          \[\therefore \]   
    • Area of parallelogram EFGH is,                   
    • \[A=|\overline{EF}\times \overline{FG}|\]\[=\frac{1}{4}|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{d}+\mathbf{d}\times \mathbf{a}|\].


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