Question
If $$\left( {\vec a \times \vec b} \right) \times \vec c = \vec a \times \left( {\vec b \times \vec c} \right)$$ where $$\vec a,\,\vec b$$ and $${\vec c}$$ are any three vectors such that $$\vec a.\vec b \ne 0,\,\vec b.\vec c \ne 0$$ then $${\vec a}$$ and $${\vec c}$$ are :
A.
inclined at an angle of $$\frac{\pi }{3}$$ between them
B.
inclined at an angle of $$\frac{\pi }{6}$$ between them
C.
perpendicular
D.
parallel
Answer :
parallel
Solution :
$$\eqalign{
& \left( {\vec a \times \vec b} \right) \times \vec c = \vec a \times \left( {\vec b \times \vec c} \right),\,\,\,\,\,\vec a.\vec b \ne 0,\,\,\,\vec b.\vec c \ne 0 \cr
& \Rightarrow \left( {\vec a.\vec c} \right).\vec b - \left( {\vec b.\vec c} \right).\vec a = \left( {\vec a.\vec c} \right).\vec b - \left( {\vec a.\vec b} \right).\vec c \cr
& \Rightarrow \left( {\vec a.\vec b} \right).\vec c = \left( {\vec b.\vec c} \right).\vec a \cr
& \Rightarrow \vec a\,||\,\vec c \cr} $$