If four distinct points with position vectors a → , b → , c → and d → are coplanar, then [ a → b → c → ] is equal to

Question

If four distinct points with position vectors a → , b → , c → and d → are coplanar, then [ a → b → c → ] is equal to, d → b → a → + a → c → d → + d → b → c →, a → d → b → + d → c → a → + d → b → c →, d → c → a → + b → d → a → + c → d → b →, b → c → d → + d → a → c → + d → b → a →

MARKS App · Accepted Answer

Given that a → , b → , c → , d → are coplanar. ⇒ b → - a → , c → - a → , d → - a → are also coplanar. ⇒ b → - a → c → - a → d → - a → = 0 ⇒ b → - a → . c → - a → × d → - a → = 0 ⇒ b → - a → . c → × d → - c → × a → - a → × d → = 0 ( Since a → × a → = 0 ) ⇒ b → . c → × d → - b → . c → × a → - b → . a → × d → - a → . c → × d → + a → . c → × a → + a → . a → × d → = 0 ⇒ b → c → d → - b → c → a → - b → a → d → - a → c → d → = 0 We know that x → y → z → = y → z → x → = z → x → y → and x → y → z → = - x → z → y → ⇒ b → c → d → + a → b → d → + a → d → c → = a → b → c → ⇒ c → d → b → + b → d → a → + d → c → a → = a → b → c → Hence, the required answer is d → c → a → + b → d → a → + c → d → b →

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