Question
If the vectors $$\vec a,\,\vec b$$ and $${\vec c}$$ form the sides $$BC,\,CA$$ and $$AB$$ respectively of a triangle $$ABC,$$ then -
A.
$$\vec a.\vec b + \vec b.\vec c + \vec c.\vec a = 0$$
B.
$$\vec a \times \vec b = \vec b \times \vec c = \vec c \times \vec a$$
C.
$$\vec a.\vec b = \vec b.\vec c = \vec c.\vec a$$
D.
$$\vec a \times \vec b + \vec b \times \vec c + \vec c \times \vec a = 0$$
Answer :
$$\vec a \times \vec b = \vec b \times \vec c = \vec c \times \vec a$$
Solution :
$$\eqalign{
& {\text{Given }}\vec a + \vec b + \vec c = 0\,\,\,\,\,\,\,\,\,\,\left( {{\text{by triangle law}}} \right) \cr
& \therefore \vec a \times \left( {\vec a + \vec b + \vec c} \right) = \vec a \times \vec 0 = \vec 0 \cr
& \Rightarrow \vec a \times \vec a + \vec a \times \vec b + \vec a \times \vec c = \vec 0 \cr
& \Rightarrow \vec a \times \vec b = \vec c \times \vec a\,\,\,\,\,\,\,\,\,\,\,\left[ {\because \,\vec a \times \vec a = 0} \right] \cr
& {\text{Similarly, }}\vec a \times \vec b = \vec b \times \vec c\,; \cr
& {\text{Therefore }}\vec a \times \vec b = \vec b \times \vec c = \vec c \times \vec a \cr} $$