Let A , B , C and D are 4 points in space satisfying the equation A B → ⋅ C D → = k A D → 2 + B C → 2 - A C → 2 - B D → 2 , then the value of k is equal to

Step-by-step solution

Let position vectors of A , B , C , D are a → , b → , c → , d → respectively A B → ⋅ C D → = b → - a → ⋅ d → - c → = b → ⋅ d → - b → ⋅ c → - a → ⋅ d → + a → ⋅ c → … 1 A B → 2 + B C → 2 - A C → 2 - B D → 2 = d → 2 + a → 2 - 2 a → ⋅ d → + c → 2 + b → 2 - 2 b → ⋅ c → - a → 2 - c → 2 + 2 a → ⋅ c → - b → 2 - d → 2 + 2 b → ⋅ d → = 2 b → ⋅ d → - b → ⋅ c → - a → ⋅ d → + a → ⋅ c → … 2 From 1 & 2 , k = 0.5