In a ∆ A B C , the sides B C ,   C A and A B are consecutive positive integers in increasing order. Let a → , b → and c → are position vectors of the vertices A , B and C respectively. If c → - a → · b → - c → = 0 , then the value of a → × b → + b → × c → + c → × a → is equal to

Step-by-step solution

c → − a → ⋅ b → − c → = 0 ⇒ A C → ⋅ C B → = 0 ⇒ Δ A B C is right angle at C . From diagram k 2 + k + 1 2 = k + 2 2 k 2 + k 2 + 2 k + 1 = k 2 + 4 k + 4 k 2 - 2 k - 3 = 0 ⇒ k = 3 , - 1 ∴ k ≠ - 1 So, sides are 3 , 4 , 5 Area of ∆ A B C = 1 2 a → × b → + b → × c → + c → × a → = 1 2 × 3 × 4 ⇒ a → × b → + b → × c → + c → × a → = 12