Question
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 :
A.
$$\frac{2}{3}$$
B.
$$\frac{{ - 2\sqrt 3 }}{3}$$
C.
$$\frac{{ 2\sqrt 2 }}{3}$$
D.
$$\frac{{ - \sqrt 2 }}{3}$$
Answer :
$$\frac{{ 2\sqrt 2 }}{3}$$
Solution :
$$\eqalign{
& \left( {\vec a \times \vec b} \right) \times \vec c = \frac{1}{3}\left| {\vec b} \right|\left| {\vec c} \right|\vec a \cr
& \Rightarrow - \vec c\left( {\vec a \times \vec b} \right) = \frac{1}{3}\left| {\vec b} \right|\left| {\vec c} \right|\vec a \cr
& \Rightarrow - \left( {\vec c.\vec b} \right)\vec a + \left( {\vec c.\vec a} \right)\vec b = \frac{1}{3}\left| {\vec b} \right|\left| {\vec c} \right|\vec a \cr
& \Rightarrow - \left| {\vec b} \right|\left| {\vec c} \right|\,\cos \,\theta \,\vec a + \left( {\vec c.\vec a} \right)\vec b = \frac{1}{3}\left| {\vec b} \right|\left| {\vec c} \right|\vec a \cr} $$
$$\because \,\vec a,\,\vec b,\,\vec c$$ are non collinear, the above equation is possible only when
$$\eqalign{
& - \cos \,\theta = \frac{1}{3}\,\,{\text{and }}\,\vec c.\vec a = 0 \cr
& \Rightarrow \cos \,\theta = - \frac{1}{3}\,\,\, \Rightarrow \sin \,\theta = \frac{{2\sqrt 2 }}{3}\,;\theta \, \in \,{\text{II}}\,{\text{quad}} \cr} $$