If the projection of the vector i ^ + 2 j ^ + k ^ on the sum of the two vectors 2 i ^ + 4 j ^ - 5 k ^ and - λ i ^ + 2 j ^ + 3 k ^ is 1 , then λ is equal to _______.

Question

If the projection of the vector i ^ + 2 j ^ + k ^ on the sum of the two vectors 2 i ^ + 4 j ^ - 5 k ^ and - λ i ^ + 2 j ^ + 3 k ^ is 1 , then λ is equal to _______.,

MARKS App · Accepted Answer

Sum of two vectors = 2 i ^ + 4 j ^ - 5 k ^ + ( - λ i ^ + 2 j ^ + 3 k ^ ) = ( 2 - λ ) i ^ + 6 j ^ - 2 k ^ Projection = 1 = 1 ( 2 - λ ) + 2 ( 6 ) + 1 × ( - 2 ) ( 2 - λ ) 2 + 6 2 + 2 2 ⇒ 12 - λ λ 2 - 4 λ + 44 = 1 ⇒ ( 12 - λ ) 2 = λ 2 - 4 λ + 44 ⇒ 144 + λ 2 - 24 λ = λ 2 - 4 λ + 44 ⇒ 100 = 20 λ ⇒ λ = 5

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