JEE Main & Advanced AIEEE Solved Paper-2004

  • question_answer
    Let a,b and c be non-zero vectors such that \[(a\times b)\times c=\frac{1}{3}|b||c|a.\]If\[\theta \]is the acute angle between the vectors b and c, then\[\sin \theta \]equals

    A) \[\frac{1}{3}\]      

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

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

    D)        \[\frac{2\sqrt{2}}{3}\]

    Correct Answer: D

    Solution :

    Since, \[\frac{1}{3}|b||c|a=(a\times b)\times c\] We know that \[(a\times b)\times c=(a.c)b-(b.c)a\] \[\therefore \] \[\frac{1}{3}|b||c|a=(a.c)b-(b.c)a\] On comparing the coefficients of a and b, we get \[\frac{1}{3}|b|c|=-b.c\]and\[a.c=0\] \[\Rightarrow \]               \[\frac{1}{3}bc=-b\cos \theta \] \[\Rightarrow \]               \[\cos \theta =-\frac{1}{3}\] \[\Rightarrow \]               \[{{\cos }^{2}}\theta =\frac{1}{9}\] \[\Rightarrow \]               \[1-{{\sin }^{2}}\theta =\frac{1}{9}\] \[\Rightarrow \]               \[{{\sin }^{2}}\theta =1-\frac{1}{9}=\frac{8}{9}\] \[\Rightarrow \]               \[\sin \theta =\frac{2\sqrt{2}}{3}\]                           \[(\because 0\le \theta \le \pi )\]

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