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BCECE Engineering BCECE Engineering Solved Paper-2014

  • question_answer
    Let a, b and c be non-zero vectors such that no two are collinear and \[(a\times b)\times c=\frac{1}{3}\]\[b||c|\]\[|a.\] If  \[\theta \] is the acute angle between the vectors b and c, then sin \[\theta \] is equal to

    A)  \[\frac{2\sqrt{2}}{3}\]  

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

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

    D) \[\frac{1}{3}\]

    Correct Answer: A

    Solution :

    We have , \[(a\times b)\times c=\frac{1}{3}|b|c|a\] \[\Rightarrow \]               \[(a.c)b-(b.c).a=\frac{1}{3}|b|c|a\] \[\Rightarrow \]               \[(a.c)b-\left\{ (b.c)+\frac{1}{3}|b|c| \right\}a=0\] As a and b are not parallel, \[(a.c)=0\]and \[b.c+\frac{1}{3}|b|c|=0\] \[\Rightarrow \]               \[|b|c|\cos \theta +\frac{1}{3}|b|c|=0\] \[\Rightarrow \]               \[\cos \theta =-\frac{1}{3}\] \[\therefore \]  \[\sin \theta =\sqrt{1-{{\cos }^{2}}\theta }=\sqrt{1-\frac{1}{9}}\] \[=\sqrt{\frac{8}{9}}=\frac{2\sqrt{2}}{3}\]                


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