• # 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)        $\frac{2\sqrt{2}}{3}$

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$a.c=0$ $\Rightarrow$               $\frac{1}{3}bc=-b\cos \theta$ $\Rightarrow$               $\cos \theta =-\frac{1}{3}$ $\Rightarrow$               ${{\cos }^{2}}\theta =\frac{1}{9}$ $\Rightarrow$               $1-{{\sin }^{2}}\theta =\frac{1}{9}$ $\Rightarrow$               ${{\sin }^{2}}\theta =1-\frac{1}{9}=\frac{8}{9}$ $\Rightarrow$               $\sin \theta =\frac{2\sqrt{2}}{3}$                           $(\because 0\le \theta \le \pi )$