What is the scalar product of the vector î + ĵ + k̂ with the unit vector along the sum of vectors 2î + 4ĵ − 5k̂ and λî + 2ĵ + 3k̂ if it is equal to 1? A. 1 B. −1 C. 2 D. 0 Answer: A) 1 Explanation: Sum vector S = (2+λ)î + 6ĵ − 2k̂. Unit vector u = S / |S|. Dot product: (î+ĵ+k̂).u = (2+λ + 6 − 2) / √((2+λ)² + 36 + 4) = 1. So, (λ+6) = √((λ+2)² + 40). Squaring: λ² + 12λ + 36 = λ² + 4λ + 4 + 40. 8λ = 8 → λ = 1.

imo class 12 vector algebra

What is the scalar product of the vector î + ĵ + k̂ with the unit vector along the sum of vectors 2î + 4ĵ − 5k̂ and λî + 2ĵ + 3k̂ if it is equal to 1?

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What is the scalar product of the vector î + ĵ + k̂ with the unit vector along the sum of vectors 2î + 4ĵ − 5k̂ and λî + 2ĵ + 3k̂ if it is equal to 1?

A. 1
B. −1
C. 2
D. 0

Answer: A) 1

Explanation: Sum vector S = (2+λ)î + 6ĵ − 2k̂. Unit vector u = S / |S|. Dot product: (î+ĵ+k̂).u = (2+λ + 6 − 2) / √((2+λ)² + 36 + 4) = 1. So, (λ+6) = √((λ+2)² + 40). Squaring: λ² + 12λ + 36 = λ² + 4λ + 4 + 40. 8λ = 8 → λ = 1.

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