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JEE Main & Advanced Mathematics Vector Algebra Question Bank Self Evaluation Test - Vector Algebra

  • question_answer
    If \[\overrightarrow{OA}=\vec{a};\overrightarrow{OB}=\vec{b};\overrightarrow{OC}=2\vec{a}+3\vec{b}\,;\] \[\overrightarrow{OD}=\vec{a}-2\vec{b}\], the length of \[\overrightarrow{OA}\] is three times the length of \[\overrightarrow{OB}\] and \[\overrightarrow{OA}\] is perpendicular to \[\overrightarrow{DB}\] then \[\left( \overrightarrow{BD}\times \overrightarrow{AC} \right).\left( \overrightarrow{OD}\times \overrightarrow{OC} \right)\] is

    A) \[7{{\left| \vec{a}\times \vec{b} \right|}^{2}}\]

    B) \[42|\vec{a}\times \vec{b}{{|}^{2}}\]

    C) 0

    D) None of these

    Correct Answer: B

    Solution :

    [b] \[\overrightarrow{BD}=\overset{\to }{\mathop{a}}\,-3\overset{\to }{\mathop{b}}\,,\,\,\overrightarrow{AC}=\overset{\to }{\mathop{a}}\,+3\overset{\to }{\mathop{b}}\,\] \[\overrightarrow{BD}\times \overrightarrow{AC}=(\overset{\to }{\mathop{a}}\,-3\overset{\to }{\mathop{b}}\,)\times (\overset{\to }{\mathop{a}}\,+3\overset{\to }{\mathop{b}}\,)=6\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,\] \[\overrightarrow{OD}\times \overrightarrow{OC}=(\overset{\to }{\mathop{a}}\,-2\overset{\to }{\mathop{b}}\,)\times (2\overset{\to }{\mathop{a}}\,+3\overset{\to }{\mathop{b}}\,)=7\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,\] \[(\overrightarrow{BD}\times \overrightarrow{AC}).(\overrightarrow{OD}\times \overrightarrow{OC})=42{{(\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,)}^{2}}\]


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