Question

Let $$\overrightarrow a ,\,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-coplanar vectors, and let and $$\overrightarrow p ,\,\overrightarrow q $$ and $$\overrightarrow r $$ be the vectors defined by the relations $$\overrightarrow p = \frac{{\overrightarrow b \times \overrightarrow c }}{{\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\overrightarrow q = \frac{{\overrightarrow c \times \overrightarrow a }}{{\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}}$$ and $$\overrightarrow r = \frac{{\overrightarrow a \times \overrightarrow b }}{{\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}}.$$ Then the value of the expression $$\left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow p + \left( {\overrightarrow b + \overrightarrow c } \right).\overrightarrow q + \left( {\overrightarrow c + \overrightarrow a } \right).\overrightarrow r $$ is equal to :

A. 0

B. 1

C. 2

D. 3

Answer : 3

Solution :
$$\eqalign{ & \overrightarrow a .\overrightarrow p = \frac{{\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)}}{{\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}} = \frac{{\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}}{{\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}} = 1 = \overrightarrow b .\overrightarrow q = \overrightarrow c .\overrightarrow r \cr & \overrightarrow b .\overrightarrow p = \frac{{\overrightarrow b .\left( {\overrightarrow b \times \overrightarrow c } \right)}}{{\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}} = \frac{0}{{\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}} = 0 = \overrightarrow c .\overrightarrow p = \overrightarrow a .\overrightarrow r \cr} $$
Therefore, the given expression is equal to $$1 + 0 + 1 + 0 + 1 + 0 = 3.$$

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