Question
$$\left[ {\overrightarrow a \,\,\overrightarrow b + \overrightarrow c \,\,\overrightarrow a + \overrightarrow b + \overrightarrow c } \right]$$ is equal to :
A.
$$0$$
B.
$$2\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$
C.
$$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]$$
D.
none of these
Answer :
$$0$$
Solution :
$$\eqalign{
& \,\,\,\,\,\,\,\left\{ {\overrightarrow a \times \left( {\overrightarrow b + \overrightarrow c } \right)} \right\}.\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) \cr
& = \left( {\overrightarrow a \times \overrightarrow b + \overrightarrow a \times \overrightarrow c } \right).\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) \cr
& = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] + \left[ {\overrightarrow a \,\,\overrightarrow c \,\,\overrightarrow b } \right] \cr
& = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] \cr
& = 0 \cr} $$