Question
If $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$ are any three vectors such that $$\left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow c = \left( {\overrightarrow a - \overrightarrow b } \right).\overrightarrow c = 0$$ then $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c $$ is :
A.
$$\overrightarrow 0 $$
B.
$$\overrightarrow a $$
C.
$$\overrightarrow b $$
D.
none of these
Answer :
$$\overrightarrow 0 $$
Solution :
$$\eqalign{
& \left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow c = \left( {\overrightarrow a - \overrightarrow b } \right).\overrightarrow c = 0\,\,\,\,\, \Rightarrow \overrightarrow a .\overrightarrow c = \overrightarrow b .\overrightarrow c = 0 \cr
& \therefore \,\overrightarrow c \bot \overrightarrow a {\text{ as well as }}\overrightarrow b \cr
& \therefore \,\overrightarrow c ||\overrightarrow a \times \overrightarrow b .{\text{ Hence, }}\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow a = \overrightarrow 0 . \cr} $$