Let a = 9 i ^ − 13 j ^ + 25 k ^ , b = 3 i ^ + 7 j ^ − 13 k ^ and c = 17 i ^ − 2 j ^ + k ^ be three given vectors. If r is a vector such that r × a = ( b + c ) × a and r ⋅ ( b − c ) = 0 , then (

Question

Let a = 9 i ^ − 13 j ^ ​ + 25 k ^ , b = 3 i ^ + 7 j ^ ​ − 13 k ^ and c = 17 i ^ − 2 j ^ ​ + k ^ be three given vectors. If r is a vector such that r × a = ( b + c ) × a and r ⋅ ( b − c ) = 0 , then ( 593 ) 2 ∣593 r + 67 a ∣ 2 ​ is equal to___________

TestHub · Accepted Answer

​ a = 9 i ^ − 13 j ^ ​ + 25 k ^ b = 3 i ^ + 7 j ^ ​ − 13 k ^ c = 17 i ^ − 2 j ^ ​ + k ^ b + c = 20 i ^ + 5 j ^ ​ − 12 k ^ b − c = − 14 i ^ + 9 j ^ ​ − 14 k ^ ( r − ( b + c ) ) × a ​ = 0 r − ( b + c ) = λ a r = λ a + b + c But r ⋅ ( b − c ) = 0 ⇒ ( λ a + b + c ) ⋅ ( b − c ) = 0 ⇒ λ a ⋅ b + b ⋅ b + c ⋅ b − λ a ⋅ c − b ⋅ c − c ⋅ c = 0 λ = a ⋅ b − a ⋅ c c ⋅ c − b ⋅ b ​ = − 389 = 204 294 − 227 ​ = 593 − 67 ​ ∴ r = b + c − 593 67 ​ a ⇒ 593 r + 67 a = 593 ( b + c ) ⇒ ∣ b + c ∣ 2 = 569 ​

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