Question
The scalar $$\vec A.\left( {\vec B + \vec C} \right) \times \left( {\vec A + \vec B + \vec C} \right)$$ equals :
A.
$$0$$
B.
$$\left[ {\vec A\,\vec B\,\vec C} \right] + \left[ {\vec B\,\vec C\,\vec A} \right]$$
C.
$$\left[ {\vec A\,\vec B\,\vec C} \right]$$
D.
None of these
Answer :
$$0$$
Solution :
$$\eqalign{
& \vec A.\left( {\vec B + \vec C} \right) \times \left( {\vec A + \vec B + \vec C} \right) \cr
& = \vec A.\left[ {\vec B \times \vec A + \vec B \times \vec B + \vec B \times \vec C + \vec C \times \vec A + \vec C \times \vec B + \vec C \times \vec C} \right] \cr
& = \vec A.\vec B \times \vec A + \vec A.\vec B \times \vec C + \vec A.\vec C \times \vec A + \vec A.\vec C \times \vec B\,\,\,\,\left[ {{\text{Using }}\vec a \times \vec a = 0} \right] \cr
& = 0 + \left[ {\vec A\,\vec B\,\vec C} \right] + 0 + \left[ {\vec A\,\vec C\,\vec B} \right] \cr} $$
(as $$\left[ {\vec a\vec b\vec c} \right] = 0$$ if any two vector are equal out of $$\vec a,\,\vec b,\,\vec c$$ )
$$\eqalign{
& = \left[ {\vec A\,\vec B\,\vec C} \right] - \left[ {\vec A\,\vec B\,\vec C} \right]\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {{\text{using }}\left[ {\vec a\vec b\vec c} \right] = - \left[ {\vec a\vec c\vec b} \right]} \right] \cr
& = 0 \cr} $$