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JEE Main & Advanced JEE Main Solved Paper-2015

  • question_answer
    Let\[\vec{a},\vec{b}\]and\[\vec{c}\]be three non - zero vectors such that no two of them are collinear and\[\left( \vec{a}\times \vec{b} \right)\times \vec{c}=\frac{1}{3}\left| {\vec{b}} \right|\left| {\vec{c}} \right|\vec{a}.\]If \[\theta \]is the angle between vectors \[\vec{b}\]and \[\vec{c},\]then a value of \[\sin \theta \]is : [JEE Main Solved Paper-2015 ]

    A) \[\frac{2}{3}\]                                   

    B) \[\frac{-2\sqrt{3}}{3}\]

    C) \[\frac{2\sqrt{3}}{3}\]   

    D) \[\frac{-\sqrt{2}}{3}\]

    Correct Answer: C

    Solution :

                    \[\left( \vec{a}\times \vec{b} \right)\times \vec{c}=\frac{1}{3}\left| {\vec{b}} \right|\left| {\vec{c}} \right|\vec{a}\] \[\Rightarrow \]\[\left( \vec{a}.\vec{c} \right)\vec{b}-\left( \vec{b}.\vec{c} \right)\vec{a}=-\frac{1}{3}\left| {\vec{b}} \right|\left| {\vec{c}} \right|\] \[\Rightarrow \]\[\vec{b}.\vec{c}=-\frac{1}{3}\left| {\vec{b}} \right|\left| {\vec{c}} \right|\]\[\Rightarrow \]\[\frac{\vec{b}.\vec{c}}{\left| {\vec{b}} \right|\left| {\vec{c}} \right|}=-\frac{1}{3}\] \[\Rightarrow \]\[\cos =-\frac{1}{3}\]Hence \[\sin \theta =\frac{2\sqrt{2}}{3}\]                


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