Question
If $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$ are three noncoplanar vectors then $$\left[ {\overrightarrow a + \overrightarrow b + \overrightarrow c \,\,\overrightarrow a - \overrightarrow c \,\,\overrightarrow a - \overrightarrow b } \right]$$ is equal to :
A.
$$0$$
B.
$$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$
C.
$$ - 3\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$
D.
$$2\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$
Answer :
$$ - 3\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$
Solution :
$$\eqalign{
& \,\,\,\,\,\,\,\left\{ {\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) \times \left( {\overrightarrow a - \overrightarrow c } \right)} \right\}.\left( {\overrightarrow a - \overrightarrow b } \right) \cr
& = \left\{ { - \overrightarrow a \times \overrightarrow c + \overrightarrow b \times \overrightarrow a - \overrightarrow b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a } \right\}.\left( {\overrightarrow a - \overrightarrow b } \right) \cr
& = \left[ {\overrightarrow a \,\,\overrightarrow c \,\,\overrightarrow b } \right] - \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right] - \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right] \cr
& = - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] \cr
& = - 3\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] \cr} $$