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

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

    A) \[7|\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,{{|}^{2}}\]

    B) \[42|\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,{{|}^{2}}\]

    C) 0

    D) None of these

    Correct Answer: B

    Solution :

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


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