Question
If $$\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b } \right) = \overrightarrow b \times \left( {\overrightarrow b \times \overrightarrow c } \right)$$ and $$\overrightarrow a .\overrightarrow b \ne 0$$ then $$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$ is equal to :
A.
0
B.
1
C.
2
D.
none of these
Answer :
0
Solution :
$$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow a .\overrightarrow a } \right)\overrightarrow b = \left( {\overrightarrow b .\overrightarrow c } \right)\overrightarrow b - \left( {\overrightarrow b .\overrightarrow b } \right)\overrightarrow c .$$ Taking dot product with $$\overrightarrow b \times \overrightarrow c ,\,\left( {\overrightarrow a .\overrightarrow b } \right)\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = 0\,\,\,\,\,\,\,\, \Rightarrow \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = 0.$$