This question is similar to Chapter 10 Class 12 Vector Algebra - Ex 10.3

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https://www.teachoo.com/3412/746/Ex-10.3--10---If-a--b--c-are-such--a---b-is-perpendicular-to-c/category/Ex-10.3/

 

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Question 24 (A) If vectors 𝑎 ⃗=2𝑖 ˆ+2𝚥 ˆ+3𝑘 ˆ,𝑏 ⃗=−𝚤 ˆ+2𝚥 ˆ+𝑘 ˆ and 𝑐 ⃗=3𝑖 ˆ+𝚥 ˆ are such that 𝑏 ⃗+𝜆𝑐 ⃗ is perpendicular to 𝑎 ⃗, then find the value of 𝜆.Given 𝑎 ⃗ = 2𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂ 𝑏 ⃗ = −𝑖 ̂ + 2𝑗 ̂ + 𝑘 ̂ 𝑐 ⃗ = 3𝑖 ̂ + 𝑗 ̂ Now, (𝒃 ⃗ + 𝜆𝒄 ⃗) = (−𝑖 ̂ + 2𝑗 ̂ + 𝑘 ̂) + 𝜆 (3𝑖 ̂ + 𝑗 ̂) = −𝑖 ̂ + 2𝑗 ̂ + 𝑘 ̂ + 3𝜆𝑖 ̂ + 𝜆𝑗 ̂ = (3𝜆 − 1) 𝒊 ̂ + (2 + 𝜆) 𝒋 ̂ + 𝒌 ̂ Since (𝒃 ⃗ + 𝜆𝒄 ⃗) is perpendicular to 𝒂 ⃗ And, since Dot product of perpendicular vectors is 0 Therefore, (𝒃 ⃗ + 𝜆𝒄 ⃗). 𝒂 ⃗ = 0 ["(3𝜆 − 1) " 𝑖 ̂" + (2 + 𝜆) " 𝑗 ̂" + " 𝑘 ̂ ] . (2𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂) = 0 (3𝜆 − 1) × 2 + (2 + 𝜆) × 2 + 1 × 3 = 0 6𝜆 − 2 + 4 + 2𝜆 + 3 = 0 8𝜆 + 5 = 0 8𝜆 = −5 𝜆 = (−𝟓)/𝟖

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.