Question
If $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$ are any three vectors in space then $$\left( {\overrightarrow c + \overrightarrow b } \right) \times \left( {\overrightarrow c + \overrightarrow a } \right).\left( {\overrightarrow c + \overrightarrow b + \overrightarrow a } \right)$$ is equal to :
A.
$$3\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$
B.
$$0$$
C.
$$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$
D.
none of these
Answer :
$$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$
Solution :
$$\eqalign{
& \,\,\,\,\,\,\left( {\overrightarrow c + \overrightarrow b } \right) \times \left( {\overrightarrow c + \overrightarrow a } \right).\left( {\overrightarrow c + \overrightarrow b + \overrightarrow a } \right) \cr
& = \left( {\overrightarrow c \times \overrightarrow a + \overrightarrow b \times \overrightarrow c + \overrightarrow b \times \overrightarrow a } \right).\left( {\overrightarrow c + \overrightarrow b + \overrightarrow a } \right) \cr
& = \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right] + \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right] + \left[ {\overrightarrow b \,\,\overrightarrow a \,\,\overrightarrow c } \right] \cr
& = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] + \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] \cr
& = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] \cr} $$