Question
Let $$\vec a,\,\vec b,\,\vec c$$ be three non-coplanar vectors and $$\vec p,\,\vec q,\,\vec r$$ are vectors defined by the relations $$\vec p = \frac{{\vec b \times \vec c}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}},\,\vec q = \frac{{\vec c \times \vec a}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}},\,\,\vec r = \frac{{\vec a \times \vec b}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}}$$ then the value of the expression $$\left( {\vec a + \vec b} \right).\vec p + \left( {\vec b + \vec c} \right).\vec q + \left( {\vec c + \vec a} \right).\vec r$$ is equal to :
A.
$$0$$
B.
$$1$$
C.
$$2$$
D.
$$3$$
Answer :
$$3$$
Solution :
Given that $$\vec a,\,\vec b,\,\vec c$$ are non coplanar
$$\therefore \left[ {\vec a\,\vec b\,\vec c} \right] \ne 0$$
Also $$\vec p = \frac{{\vec b \times \vec c}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}},\,\vec q = \frac{{\vec c \times \vec a}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}},\,\,\vec r = \frac{{\vec a \times \vec b}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}}.....(1)$$
$$\eqalign{
& {\text{Now, }}\left( {\vec a + \vec b} \right).\vec p + \left( {\vec b + \vec c} \right).\vec q + \left( {\vec c + \vec a} \right).\vec r \cr
& = \left( {\vec a + \vec b} \right).\frac{{\vec b \times \vec c}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}} + \left( {\vec b + \vec c} \right).\frac{{\vec c \times \vec a}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}} + \left( {\vec c + \vec a} \right).\frac{{\vec a \times \vec b}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}} \cr
& = \frac{{\vec a.\vec b \times \vec c}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}} + \frac{{\vec b.\vec c \times \vec a}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}} + \frac{{\vec c.\vec a \times \vec b}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}} \cr
& \left[ {{\text{Using }}\,\vec b.\vec b \times \vec c = \vec c.\vec c \times \vec a = \vec a.\vec a \times \vec b = 0} \right] \cr
& = \frac{{\left[ {\vec a\,\vec b\,\vec c} \right]}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}} + \frac{{\left[ {\vec a\,\vec b\,\vec c} \right]}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}} + \frac{{\left[ {\vec a\,\vec b\,\vec c} \right]}}{{\left[ {\vec a\,\vec b\,\vec c} \right]}} \cr
& = 1 + 1 + 1 \cr
& = 3 \cr} $$