Let A , B , C and D are any 4 points in space. If A B → × C D → + B C → × A D → + C A → × B D → = λ Δ A B C , where Δ A B C is the area of triangle A B C , then the value of λ is equal to

Question

Let A , B , C and D are any 4 points in space. If A B → × C D → + B C → × A D → + C A → × B D → = λ Δ A B C , where Δ A B C is the area of triangle A B C , then the value of λ is equal to, 2, 1 2, 4, 1 4

MARKS App · Accepted Answer

Let, the position vectors are A a → , B b → , C c → , D d → So, Δ A B C = 1 2 A B → × A C → = 1 2 b → - a → × c → - a → … 1 Now, A B → × C D → = b → - a → × d → - c → = b → × d → - b → × c → - a → × d → + a → × c → B C → × A D → = c → - b → × d → - a → = c → × d → - c → × a → - b → × d → + b → × a → C A → × B D → = a → - c → × d → - b → = a → × d → - a → × b → - c → × d → + c → × b → A B → × C D → + B C → × A D → + C A → × B D → = 2 a → × b → + b → × c → + c → × a → = 2 b → - a → × c → - a → = 4 Δ A B C ⇒ λ = 4

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