If (a + b), (b + c) and (c + a) are coplanar then a × (b × c) is?

1) 0

2) 1

3) a.b

4) b.c

Answer: (1) 0

Solution: Given, vectors (a + b), (b + c) and (c + a) are coplanar

Therefore,

(a+b) [(b+c) x (c + a)] = 0

⇒ (a+b) (b x c + b x a + c x c + c x a) = 0

⇒ (a + b) (b x c + b x a + c x a) = 0

⇒ a.(b x c) + a.(b x a)+ a.(c x a) + b.(b x c) + b.(b x a) + b.(c x a) = 0

⇒ [a b c] + 0 + 0+ 0+0 = 0

⇒ [a b c] = 0