Question
$$\overrightarrow a ,\overrightarrow b ,\,\overrightarrow c $$ are noncoplanar vectors and $$\overrightarrow p ,\overrightarrow q ,\,\overrightarrow r $$ are defined as $$\overrightarrow p = \frac{{\overrightarrow b \times \overrightarrow c }}{{\left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right]}},\,\overrightarrow q = \frac{{\overrightarrow c \times \overrightarrow a }}{{\left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right]}},\,\overrightarrow r = \frac{{\overrightarrow a \times \overrightarrow b }}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}}.\left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow p + \left( {\overrightarrow b + \overrightarrow c } \right).\overrightarrow q + \left( {\overrightarrow c + \overrightarrow a } \right).\overrightarrow r $$ is equal to :
A.
0
B.
1
C.
2
D.
3
Answer :
3
Solution :
$$\eqalign{
& {\text{Expression}} = \frac{{\left( {\overrightarrow a + \overrightarrow b } \right).\left( {\overrightarrow b \times \overrightarrow c } \right)}}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}} + \frac{{\left( {\overrightarrow b + \overrightarrow c } \right).\left( {\overrightarrow c \times \overrightarrow a } \right)}}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}} + \frac{{\left( {\overrightarrow c + \overrightarrow a } \right).\left( {\overrightarrow a \times \overrightarrow b } \right)}}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}} + \frac{{\left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right]}}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}} + \frac{{\left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right]}}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}} \cr
& = 1 + 1 + 1 \cr
& = 3 \cr} $$