If ABC is a right-angled triangle in which BC is the longest side and the position vector of B and C are respectively 3 i - 2 j + k and 5 i + j - 3 k , then the value of AB AC + BA BC + CA CB is

Step-by-step solution

Since ABC is a right-angled triangle and BC is the longest side, the right angle must be at A . Therefore, AB AC = 0 . The given expression is AB AC + BA BC + CA CB . Substituting AB AC = 0 , we get: 0 + BA ( BA + AC ) + CA ( CA + AB ) = | BA |^2 + BA AC + | CA |^2 + CA AB Since BA AC = 0 and CA AB = 0 , the expression simplifies to: | BA |^2 + | CA |^2 By Pythagoras theorem, | BA |^2 + | CA |^2 = | BC |^2 . Given the position vectors of B and C : B = 3 i - 2 j + k C = 5 i + j - 3 k BC = C - B = (5 - 3) i + (1 - (-