Question
Let $$\vec a,\,\vec b,\,\vec c$$ be unit vectors such that $$\vec a + \vec b + \vec c = \vec 0.$$ Which one of the following is correct ?
A.
$$\vec a \times \vec b = \vec b \times \vec c = \vec c \times \vec a = \vec 0$$
B.
$$\vec a \times \vec b = \vec b \times \vec c = \vec c \times \vec a \ne \vec 0$$
C.
$$\vec a \times \vec b = \vec b \times \vec c = \vec a \times \vec c \ne \vec 0$$
D.
$$\vec a \times \vec b,\,\vec b \times \vec c,\,\vec c \times \vec a$$ are mutually perpendicular
Answer :
$$\vec a \times \vec b = \vec b \times \vec c = \vec c \times \vec a \ne \vec 0$$
Solution :
Since, $$\vec a + \vec b + \vec c = \vec 0$$ and $$\vec a,\,\vec b,\,\vec c$$ are unit vectors, therefore $$\vec a,\,\vec b,\,\vec c$$ form an equilateral triangle.
$$\eqalign{
& \Rightarrow \vec a \times \left( {\vec a + \vec b + \vec c} \right) = \vec 0 \cr
& \Rightarrow \vec a \times \vec a + \vec a \times \vec b + \vec a \times \vec c = \vec 0 \cr
& \Rightarrow \vec a \times \vec b = \vec c \times \vec a \cr
& {\text{Similarly, }}\vec b \times \vec c = \vec c \times \vec a \cr
& \therefore \vec a \times \vec b = \vec b \times \vec c = \vec c \times \vec a \cr} $$
Also since $$\vec a,\,\vec b,\,\vec c$$ are non parallel (these form an equilateral $$\Delta $$ ).
$$\therefore \vec a \times \vec b = \vec b \times \vec c = \vec c \times \vec a \ne \vec 0$$