Let a = 2 ^i - ^j + 3 k^ , b = 3 ^i - 5 ^j + k^ and c be a vector such that a \times c = c \times b and ( a + c ) ⋅ ( b + c ) = 168. Then the maximum value of ∣ c ∣ 2 is :
Step-by-Step Solution
Key Concept: Apply the core result for dot product, cross product and projections and simplify using the given constraints.
\to ^ ^ ^ a = 2 i - j + 3k (4) \to ^ ^ ^ b = 3 i - 5 j + 3k \to \to \to \to a \times c = c \times b \to \to \to \to a \times c + b \times c = 0 \to \to \to ( a + b ) \times c = 0 \to \to \to \Rightarrow c = \lambda( a + b ) \to ^ ^ ^ c = \lambda(5 i - 6 j + 4k) \ldots . . (1) \to 2 2 | c | = \lambda (25 + 36 + 16) \to 2 2 | c | = 77\lambda \to \to \to \to ( a + c ) ⋅ ( b + c ) = 168 \to \to \to \to \to \to \to 2 a ⋅ b + a ⋅ c + c ⋅ b + | c | = 168 \to \to 14 + c ⋅ (a + b) + 77\lambda \to 2 = 168 using equation (1) ^ ^ ^ 2 2 \lambda|5 i - 6 j + 4k| + 77\lambda = 154 2 77\lambda + 77\lambda - 154 = 0 2 \lambda + \lambda - 2 = 0 \lambda = -2, 1 \to 2 \therefore Maximum value of | c | occurs when \lambda = -2 \to 2 2 | c | = 77\lambda = 77 \times 4 = 308
Correct Answer: 4