Let O be the origin and O A → = 2 i ^ + 2 j ^ + k ^ ,    O B → = i ^ - 2 j ^ + 2 k ^ and O C → = 1 2 O B → - λ O A → for some λ > 0 . If O B → × O C → = 9 2 , then which of the following statements is (are) TRUE?
C. Area of the triangle A B C is 9 2
O A → = 2 i ^ + 2 j ^ + k ^ O B → = i ^ − 2 j ^ + 2 k ^ O C → = 1 2 O B → − λ O A → ; λ > 0 O B → × O C → = 9 2 ⇒ O B → × 1 2 O B → − λ O A → = 9 2 ⇒ 1 2 O B → × O B → − λ O A → = 9 2 ⇒ 1 2 λ O A → × O B → = 9 2 ⇒ λ 2 O A → × O B → = 9 2             . . . 1 now, O A → × O B → = i ^ j ^ k ^ 2 2 1 1 - 2 2 = 6 i ^ − 3 j ^ − 6 k ^ ⇒ O A → × O B → = 36 + 9 + 36 = 9                 . . . 2 Put in (1) λ 2 × 9 = 9 2 ⇒ λ = 1 ⇒ λ = ± 1                   ∵   λ > 0 ⇒ λ = 1 ⇒ O C → = 1 2 O B → − O A → O C → = 1 2 − i ^ − 4 j ^ + k ^ (A) Projection of O C → on O A → = O C → ⋅ O A → O A → = 1 2 − i ^ − 4 j ^ + k ^ ⋅ 2 i ^ + 2 j ^ + k ^ 4 + 4 + 1 = 1 2 − 2 − 8 + 1 3 = − 9 6 = − 3 2 (B) Area of Δ O A B = 1 2 O A → × O B → = 9 2 sq unit (C) Area of A B E = 1 2 | A B → × A C → | A B → = - i ^ - 4 j ^ + k ^ A C → = - 5 2 i ^ - 4 j ^ - k ^ 2 A B → × A C → = i ^ j ^ k ^ - 1 - 4 1 - 5 2 - 4 - 1 2 = 6 i ^ - 3 j ^ - 6 k ^ | A B → × A C → | = 36 + 9 + 36 = 9 Area of ∆ A B C = 1 2 A B → × A C → = 9 2   sq . unit (D) Apply angle between diagonals O D → & C A → cos θ = a → + c → ⋅ a → − c → a → + c → a → − c → a → = 2 i ^ + 2 j ^ + k ^ c → = 1 2 − i ^ + 4 j ^ + k ^ a → + c → = 3 2 i ^ + 3 2 k ^ a → + c → = 9 4 + 9 4 = 3 2 2 a → − c → = 5 2 i ^ + 4 j ^ + 1 2 k ^ a → − c → = 25 4 + 16 + 1 4 = 90 4 = 3 10 2 cos θ = 15 4 + 3 4 3 2 2 3 10 2 = 9 2 9 4 × 20 = 1 5 ≠ π 3
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